%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% ribetofficial.tex --- Scribe notes for Ken Ribets Spring 1996 course. %% %% %% %% This file should compile with standard latex, the amssymb package, %% %% and the diagrams package. The diagrams package is contained in the %% %% file "diagrams.tex". Obtain it and put it in the same directory %% %% as this file. %% %% %% %% This file is being maintained by William Stein (was@math.berkeley.edu).%% %% This file was assembled by Lawren Smithline (lawren@math.berkeley.edu).%% %% %% %% Last modification date: 9/25/96. %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{report} %% Uncomment the following two lines if your system doesn't %% define the \mathfrak font. %\font\german=eufm10 scaled \magstep 1 %\def\mathfrak{\german} \input diagrams \usepackage{amssymb} \marginparwidth 0pt \oddsidemargin 0pt \evensidemargin 0pt \marginparsep 0pt \topmargin 0pt \textwidth 6.5in \textheight 8.5 in \def\nn{{\rm N}} % Roman N in math mode \def\aa{{\cal A}} \def\rr{{\cal R}} \def\flt{F{\sc l}T} % Fermat's little theorem \def\pf{{\sc Proof.\ }} \def\st{\ \cdot\backepsilon\cdot\ } % ``such that'' symbol \def\vs{\vspace{4pt}} \def\A{{\bf A}} % Adele ring \def\C{{\bf C}} % complex nos. \def\Z{{\bf Z}} % integers \def\F{{\bf F}} % field \def\P{{\bf P}} % projective land \def\N{{\mathfrak N}} % index of an ideal in a number ring (norm) \def\H{{\mathfrak H}} % the complex upper halfplane \def\R{{\bf R}} % reals \def\Q{{\bf Q}} % rationals \def\O{{\cal O}} % ring of integers \def\T{{\bf T}} % Hecke algebra \def\G{{\bf G}} % group scheme \def\X{{X(N)}} % Modular curve \def\E{{E_j}} \def\K{{\overline{K}}} \def\ff{{\cal F}} % Modular function field \def\Qp{\Q_p} % p-adic numbers \def\Zp{\Z_p} % p-adic integers \def\0{{\bf 0}} % zero structure \def\a{\alpha} % your basic Greeks \def\b{\beta} \def\c{\gamma} \def\d{\delta} \def\e{\epsilon} \def\f{\zeta} \def\G{\Gamma} \def\ka{\kappa} \def\la{\lambda} \def\t{\tau} \def\om{\omega} \def\Om{\Omega} \def\rh{{\overline % rho bar {\rho_E}}} \def\rhla{{\rho_\lambda}} \def\th{{\theta}} \def\g{\gamma} \def\p{{\mathfrak p}} % Gothic p, a prime ideal \def\q{{\mathfrak q}} % Gothic q \def\M{{\mathfrak m}} % maximal ideal \def\Tm{{\T_\M}} % T localized at m \def\ia{{\mathfrak a}} \def\ib{{\mathfrak b}} \def\si{\sigma} \def\cent{{\bf C}} % centralizer \def\qed{\hfill $\blacksquare$\smallskip} \def\tate{{\rm Ta}} % Tate module \def\tatel{{{\rm Ta}_\l}} % l-adic Tate module \def\tatem{{{\rm Ta}_\M}} % m-adic Tate module \def\tatels{{{\rm Ta}_\l^*}} % contrav. l-adic T. m. \def\tatems{{{\rm Ta}_\M^*}} % contrav. m-adic T. m. \def\ve{\varepsilon} % epsilon the character \def\lan{\langle} % angle brackets \def\ran{\rangle} \def\<{{\langle}} % < bracket \def\>{{\rangle}} % > bracket \def\l{\ell} % script l as in l-adic. \def\Ql{{\Q_\l}} % l-adic numbers \def\Zl{{\Z_\l}} % fewer l-adic numbers \def\sigmaonf{ {}^\sigma\!f} % sigma acting on f, from upper left hand corner \def\GL{{\rm GL}} % general linear group \def\SL{{\rm SL}} % special linear group \def\gal{{\cal G}al} % Galois group \def\GQ{\gal(\overline\Q/\Q)} % abs Galois gp of Q \def\GQp{\gal(\overline\Qp/\Qp)} % abs Galois gp of Qp \def\modgp{\SL_2(\Z)} % modular group \def\abcd{\left( % 2 x 2 matrix a b // c d \begin{array}{cc} a&b\\c&d\end{array}\right)} \def\isom{\cong} \def\tensor{\otimes} \def\tr{{\rm tr}} \def\frob{{\rm frob}} \def\mod{\ {\rm mod}\,} \def\im{{\rm im}} \def\ord{{\rm ord}} \def\rank{{\rm rank}} \def\aut{{\rm Aut}} \def\Hom{{\rm Hom}} \def\plim{{\displaystyle\lim_{\longleftarrow}\,}} % Projective limit \def\dlim{{\displaystyle\lim_{\longrightarrow}\,}} % Direct limit \def\plimr{{\displaystyle\lim_{ \buildrel\longleftarrow\over r}\,}} % Projective limit over r \def\Div{{\rm Div}} \def\det{{\rm det }} % Determinant \def\endo{{\rm End}} \def\Cot{{\rm Cot}} % contangent space \def\ver{{\rm ver}} % Vershibung endmorphism \def\inj{\hookrightarrow} \def\into{\rightarrow} \def\onto{\twoheadrightarrow} \def\isomap{{\buildrel \sim\over\rightarrow}} \newarrow{To} ----> \newarrow{Line} ----- \newtheorem{thm}{Theorem} \newtheorem{dfn}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lem}{Lemma} \newtheorem{cor}{Corollary} \newtheorem{res}{Result} \newtheorem{eg}{Example} \newtheorem{claim}{Claim} \newtheorem{exercise}{Exercise} \newtheorem{remark}{Remark} \newtheorem{conj}{Conjecture} \newtheorem{note}{Note} \begin{document} \title{Scribe notes for Ken Ribet's Math 274} \author{} \date{\null} \maketitle \section*{January 17, 1996} \noindent{Scribe: Lawren Smithline, \tt } \bigskip \noindent Here are some topics to be discussed in this course: \begin{tabular}{l} Galois representations and modular forms, \\ Hecke algebras, \\ modular curves, and Jacobians (Abelian varieties). \end{tabular} \\ This lecture is a brief overview of some connection between these concepts, and also an exercise in name-dropping. We can describe an an elliptic curve, or the Jacobian of a higher genus curve, or abelian variety using a lattice. For $E$, the lattice is $L = H_1(E(\C),\Z) \hookrightarrow \C,$ by the map $\c \mapsto \int_\c \omega.$ Weil considered curves over a finite field, $k$ of characteristic $p$. There is an algebraic definition of $L/nL$ for $n \geq 1, \ \gcd(n, p) = 1.$ $$E[n] = \{ P \in E(\bar k) : nP = 0\} = \textstyle\frac1n L/L = L/nL.$$ For example, let $n = \l^\nu$ for $\nu \geq 1$, and $\l$ a prime different from $p.$ Weil further considered the limit $$E[\l^\infty] = \bigcup_{\nu = 1}^\infty E[\l^\nu].$$ Tate oberserved there is a map $E[\l^\nu] \stackrel{\l}\rightarrow E[\l^{\nu-1}],$ and so of course this inverse limit, $$ \lim_\leftarrow E[\l^\nu] = T_\l(E)$$ is called the Tate module. As $E[n]$ is free of rank 2 over $\Z / n\Z$, so is $T_\l(E)$ free of rank 2 over $\Z_\l$. Also, $V_l(e) = T_\l(E) \otimes \Q_\l$ is a 2 dimensional vector space over $\Q_\l$. This is the first example of $\l$-adic \'etale cohomology. For topological space $X$, $X \mapsto H_{\mathaccent 19 et}^i(X/\bar k, \Q_\l).$ Here are the names of some cool folks: Taniyama Shimura Mumford Tate. Now, elliptic curve $E/\Q$ gets an action of $G = \gal(\bar \Q/\Q)$, and so does $E[n]$. I.e. $\si(P+Q) = \si(P) + \si(Q).$ So we have a homomorphism $\rho: G \rightarrow {\rm Aut}(E[n]) = GL_2(\Z/n\Z).$ Since we have an exact sequence $$1 \rightarrow \ker \rho \rightarrow G \rightarrow {\rm im}\, \rho \rightarrow 0,$$ we get a tower of fields $$\bar \Q \rightarrow K \rightarrow \Q,$$ and ${\cal G}al(K/\Q) = {\rm im}\, \rho \subset GL_2( \Z/n\Z).$ \smallskip And now for something completely different. We can also get to these Galois representations via modular forms. Let $k$ be the weight, such as 2. Let $N$ be the level. The complex vector space $S_k(N)$ is the set of cusp forms on $\Gamma_1(N)$, a finite dimensional vector space, namely, the set of holomorphic functions $f$ on $\H$ such that $$ f((az+b)/(cz+d)) = (cz+d)^kf(z)$$ for $$\left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in \Gamma_1(N), \ \ a,d \equiv 1, c \equiv 0 (N).$$ Such an $f$ has a power series (or Fourier series) expansion in $q = \exp(2\pi i z)$: $$f(z) = \sum_1^\infty c_n q^n.$$ Here is a famous example observed by Ramanujan, and proved by Mordell using (his) Hecke operators: $$ q\prod_1^\infty (1 - q^n)^{24} = \sum_1^\infty \tau(n)q^n,$$ for Ramanujan's $\tau$ function. Now, $\tau(n)\tau(m) =\tau(nm)$ for $\gcd(n,m) = 1$. Also, there is a recurrence for prime powers. Amusingly, the normalized basis element of $S_{12}(1)$ is $$\Delta = \sum_1^\infty \tau(n)\exp(2\pi i nz).$$ Even more amusingly, $$\tau(n) \equiv \sum_{d \mid n} d^{11} \ \ (691).$$ \smallskip Experience and Shimura have shown that there exist $f \in S_k(N)$ such that $$T_n(f) =c_n \cdot f$$ for all $n \geq 1$ for some scalars $c_n$, and $$f = \sum c_n q^n$$ for the same $c_n$, and that these $c_n$ are algebraic integers in a finitely generated number field. That is, $[\Q(c_n: n\geq 1): \Q]$ is finite. How can we study and interpret this? We start with the Hecke ring, $$\Q[T_n] \subseteq End(S_k(N)).$$ Serre in 1968 said there should be Galois representations attached to forms of arbitrary weight. Deligne constructed them. In a broad stroke, one can say that we get between Galois representations and modular forms via Frobenius elements. Next time, we continue with the semihistorical overview. \section*{January 19, 1996} \noindent{Scribe: William Stein, \tt } \bigskip \noindent {\bf\large Modular Representations and Modular Curves} \smallskip \subsection*{Arithmetic of Modular Forms} Suppose $f=\sum_{n=1}^{\infty}a_n q^n$ is a cusp form in $S_k(N)$ which is an eigenform for the Hecke operators. The Mellin transform associates to $f$ the $L$-function $L(f,z)=\sum_{n=1}^{\infty} n^{-s}{a_n}$. Let $K=\Q(a_1,a_2,\ldots)$. One can show that the $a_n$ are algebraic integers and $K$ is a number field. When $k=2$, $f$ is associated to $f$ an abelian variety $A_f$ over $\Q$ of dimension $[K:\Q]$, and $A_f$ has a $K$ action. (See Shimura, {\em Introduction to the Arithmetic Theory of Automorphic Functions}, Theorem 7.14.) \begin{eg}[Modular Elliptic Curves] If $a_n\in\Q$ for all $n$, then $K=\Q$ and $[K:\Q]=1$. In this case, $A_f$ is a one dimensional abelian variety, which is an elliptic curve, since it has nonzero genus. An elliptic curve arising in this way is called modular. \end{eg} \begin{dfn} Elliptic curves $E_1$ and $E_2$ are {\em isogenous} if there is a morphism $E_1\into E_2$ of algebraic groups, which has a finite kernel. \end{dfn} The following conjecture motivates much of the theory. \begin{conj} Every elliptic curve over $\Q$ is modular, that is, isogenous to a curve constructed in the above way. \end{conj} For $k\geq 2$, Serre and Deligne found a way to associate to $f$ a family of $\l$-adic representations. Let $\l$ be a prime number and $K$ be as above. It is well known that $$K\otimes_{\Q} \Q_{\l}\isom \prod_{\la|\l}K_{\la}.$$ One can associate to $f$ a family of representations $$ \rho_{\l,f}:G=\gal(\overline{\Q}/\Q) \rightarrow\GL(K\otimes_{\Q}\Q_{\l}) $$ unramified at all primes $p\not|\l N$. By unramified we mean that for all primes $P$ lying over $p$, the inertia group of the decomposition group at $P$ is contained in the kernel of $\rho$. (The decomposition group $D_P$ at $P$ is the set of those $g\in G$ which fix $P$ and the inertia group is the kernel of the map $D_P\rightarrow \gal(\O/P)$, where $\O$ is the ring of all algebraic integers.) Now $I_P\subset D_P \subset \gal(\overline{\Q}/\Q)$ and $D_P / I_P$ is cyclic, since it is isomorphic to a subgroup of the galois group of a finite extension of finite fields. So $D_P / I_P$ is generated by a Frobenious automorphism $\frob_p$ lying over $p$. We have $$ \tr(\rho_{\l,f}(\frob_p)) = a_p\in K \subset K\otimes \Q_{\l} $$ and \begin{equation}\label{detrho} \det(\rho_{\l}) = \chi_{\l}^{k-1}\ve, \end{equation} where $\chi_{\l}$ is the $\l$th cyclotomic character and $\ve$ is a Dirichlet character. \subsection*{Characters} Let $f\in S_k(N)$. For all $\abcd \in \modgp$ with $c\cong 0 \mod{N}$ we have $$ f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k \ve(d) f(z), $$ where $\ve:(\Z/n\Z)^*\rightarrow \C^*$ is a Dirichlet character mod $N$. If $f$ is an eigenform for the diamond bracket operator $$, (so that $f| = \ve(d) f$) then $\ve$ actually takes values in $K$. Let $\phi_n$ be the mod $n$ cyclotomic character. The map $\phi_n: G \rightarrow (\Z/n\Z)^*$ takes $g\in G$ to the automorphism induced by $g$ on the $n$th cyclotomic extension $\Q(\mu_n)$ of $\Q$, where we identify $\gal(\Q(\mu_n)/\Q)$ with $(\Z/n\Z)^*$. The $\ve$ appearing in (\ref{detrho}) is really the composition $$ G\stackrel{\phi_n}\longrightarrow(\Z/n\Z)^* \stackrel{\ve}\longrightarrow \C^*. $$ For each positive integer $\nu$ we consider the $\l^{\nu}$th cyclotomic character on $G$, $$ \phi_{\l^{\nu}}:G\rightarrow (\Z/\l^{\nu}\Z)^*. $$ Putting these together give a map $$ \phi_{\l^{\infty}}=\lim_{\stackrel\longleftarrow\nu} \phi_{\l^{\nu}}:G\stackrel{\chi_{\l}}\longrightarrow\Z_{\l}^{*}. $$ \subsection*{Parity Conditions} Let $c\in\gal(\overline{\Q}/\Q)$ be complex conjugation. We have $\phi_n(c)=-1$, so $\ve(c) = \ve(-1)$ and $\chi_{\l}(c) = (-1)^{k-1}$. Let $$\abcd =\left(\begin{array}{cc} -1&0\\0&-1 \end{array}\right).$$ For $f\in S_k(N)$, $$f(z) = (-1)^k\ve(-1)f(z),$$ so $(-1)^k\ve(-1) = 1$. Thus, $$\det(\rho_{\l}(c)) = \epsilon(-1)(-1)^{k-1} = -1.$$ The $\det$ character is odd so the representation $\rho_{\l}$ is odd. \begin{remark} (Vague Question) How can one recognize representations like $\rho_{\l,f}$ ``in nature''? Mazur and Fontaine have made relevant conjectures. The Shimura-Taniyama conjecture can be reformulated by saying that for any representation $\rho_{\l,E}$ comming from an elliptic curve $E$ there is $f$ so that $\rho_{\l,E}\isom \rho_{\l,f}$. \end{remark} \subsection*{Conjectures of Serre (mod $\l$ version)} Suppose $f$ is a modular form, $\l$ a rational prime, $\la$ a prime lying over $\l$, and the representation $$\rho_{\la,f}:G\rightarrow \GL_2(K_{\la})$$ (constructed by Serre-Deligne) is irreducible. Then $\rho_{\la,f}$ is conjugate to a representation with image in $\GL_2(\O_{\la})$, where $\O_{\la}$ is the ring of integers of $K_{\la}$. Reducing mod $\l$ gives a representation $$\overline{\rho}_{\la,f}:G\rightarrow\GL_2(\F_{\la})$$ which has a well-defined trace and det, i.e., the det and trace don't depend on the choice of conjugate used to reduce mod $\la$. One knows from representation theory that if such a representation is semisimple then it is completely determined by its trace and det. Thus if $\overline{\rho}_{\la,f}$ is irreducible it is unique in the sense that it doesn't depend on the choice of conjugate. We have the following conjecture of Serre which remains open. \begin{conj}[Serre] All irreducible representation of $G$ over a finite field which are odd, i.e., $det(\sigma(c))=-1$, $c$ complex conjugation, are of the form $\overline{\rho}_{\la,f}$ for some representation $\rho_{\la,f}$ constructed as above. \end{conj} \begin{eg} Let $E/\Q$ be an elliptic curve and let $\sigma_{\l}:G\rightarrow\GL_2(\F_{\l})$ be the representation induced by the action of $G$ on the $\l$-torsion of $E$. Then $\det \sigma_{\l} = \phi_{\l}$ is odd and $\sigma_{\l}$ is usually irreducible, so Serre's conjecture would imply that $\sigma_{\l}$ is modular. From this one can, under Serre's conjecture, prove that $E$ is modular. \end{eg} \begin{dfn} Let $\sigma:G\rightarrow \GL_2(\F)$ ($\F$ is a finite field) be a represenation of the galois group $G$. The we say that the {\em representions $\sigma$ is modular} if there is a modular form $f$, a prime $\la$, and an embedding $\F\hookrightarrow \overline{\F}_{\la}$ such that $\sigma\isom\overline{\rho}_{\la,f}$ over $\overline{\F}_\la$. \end{dfn} \subsection*{Wile's Perspective} Suppose $E/\Q$ is an elliptic curve and $\rho_{\l,E}:G\rightarrow\GL_2(\Z_{\l})$ the associated $\l$-adic representation on the Tate module $T_{\l}$. Then by reducing we obtain a mod $\l$ representation $$\overline{\rho}_{\l,E}=\sigma_{\l,E}:G \rightarrow \GL_2(\F_{\l}).$$ If we can show this is modular for infinitely many $\l$ then we will know that $E$ is modular. \begin{thm}[Langlands and Tunnel] If $\sigma_{2,E}$ and $\sigma_{3,E}$ are irreducible, then they are modular. \end{thm} This is proved by using the fact that $\GL_2(\F_2)$ and $\GL_2(\F_3)$ are solvable so we may apply ``base-change''. \begin{thm}[Wiles] If $\rho$ is an $\l$-adic representation which is irreducible and modular mod $\l$ with $\l>2$ and certain other reasonable hypothesis are satisfied, then $\rho$ itself is modular. \end{thm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{January 22, 1996} \noindent{Scribe: Lawren Smithline, \tt } (Note. These were done about two months after the fact, since the assigned person didn't. So they are terse. It was a pretty easy lecture.) \bigskip Today, we have a limited goal: to explain modular forms as functions on lattices -- or elliptic curves. (See Serre's {\em Course in Arithmetic}, or Katz's paper in the Proceedings of the Antwerp Conference in the Springer LNM series.) Let the level $N = 1$. Consider a weight $k$ cusp form $f$. For $\t \in \H$, we have $$f( a\t+b / c\t+d) = (c\t+d)^k f(\t).$$ So $f(\t+1) = f(\t)$. By the map $\t \mapsto q= \exp(2\pi i \t)$, we map $\H$ to the punctured disc $ \{ z : 0 < |z| < 1\}$. We abuse notation and think of $f$ as a function on this disc. Since $f$ is a cusp form, $f$ extends to 0, and $f(0) = 0$. So we have a $q$-expansion $$ f = \sum _{n=1}^{\infty} a_n q^n. $$ \subsection*{Lattices inside $\C$} Let $L = \Z\om_1 \oplus Z\om_2$. We may assume that $\om_1 / \om_2 \in \H$. Let ${\mathfrak R}$ be the set of lattices in $\C$. $\SL_2 \Z$ acts on the left of $M = \{ (\om_1, \om_2) : \om_1, \om_2 \in \H\}$ by multiplication of the column vector. This action fixes the lattice. Here is the relation with elliptic curves. A lattice $L$ determines a complex torus $\C / L$. There is a Weierstrass $\wp$ function on this torus. Consider an elliptic curve $E$ over $\C$. There is a lattice given by the inclusion $H_1(E(\C),\Z) \hookrightarrow \C$. Choose a nonzero $\om \in H^0(E, \Om'_E).$ Then $\c \in H_1$ maps to $\int_\c \om \in \C$. So maybe we should think of ${\mathfrak R}$ as the set of pairs $\{ (E, \om) \}.$ We have a map $M / \C^\times \into \H$ sending $(\om_1, \om_2)$ to $\om_1 / \om_2.$ Now take the quotient on the left by $\SL_2 \Z$: $$ {\mathfrak R}/\C^{times} = \SL_2 \Z \backslash M / \C^\times \longrightarrow \SL_2 \Z \backslash \H.$$ But this just is the space of elliptic curves over $\C$. So $f:\H \rightarrow \C$ which is a modular form and $F:M \rightarrow \C$ satisfying $F(\la L) = \la^{-k}F(L)$ amount to the same thing by a simple calculation. \subsection*{Hecke Operators} Let $F$ be a function on lattices. Define the Hecke operator $T_n$ as $$T_n F (L) = \sum_{L' \subset L, (L:L') = n} F(L) n^{k-1}.$$ The essential case on elliptic curves is for $n = \l$ a prime. In this case, the $L'$ correspond to the $\l+1$ subgroups of order $\l$ of $(\Z / \l\Z)^ 2$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{January 24, 1996} \noindent{Scribe: William Stein, \tt } \bigskip \noindent {\bf \large More On Hecke Operators}\smallskip We consider modular forms $f$ on $\Gamma_1(1)=\modgp$, that is, holomorphic functions on $\H\cup\{\infty\}$ which satisfy $$f(\tau)=f(\frac{a\tau+b}{c\tau+d})(c\tau+d)^{-k}$$ for all $\abcd\in\modgp$. Using a Fourier expansion we write $$f(\tau)=\sum_{n=0}^{\infty} a_ne^{2\pi i\tau n},$$ and say $f$ is a cusp form if $a_0=0$. There is a correspondence between modular forms $f$ and lattice functions $F$ satisfying $F(\lambda L)=\lambda^{-k}F(L)$ given by $F(\Z\tau+\Z)=f(\tau)$. \subsection*{Explicit Description of Sublattices} The $n$th Hecke operator $T_n$ of weight $k$ is defined by $$T_n(L)=n^{k-1}\sum_{{L'\subset L,\ (L:L')=n}} L'.$$ What are the $L'$ explicitly? Note that $L/L'$ is a group of order $n$ and $$L'/nL\subset L/nL=(\Z/n\Z)^2.$$ Write $L=\Z\om_1+\Z\om_2$, let $Y_2$ be the cyclic subgroup of $L/L'$ generated by $\om_2$ and let $d=\#Y_2$. Let $Y_1=(L/L')/Y_2$. $Y_1$ is generated by the image of $\om_1$ so it is a cyclic group of order $a=n/d$. We want to exhibit a basis of $L'$. Let $\om_2'=d\om_2\in L'$ and use the fact that $Y_1$ is generated by $\om_1$ to write $a\om_1=\om_1'+b\om_2$ for $b\in\Z$ and $\om_1'\in L'$. Since $b$ is only well-defined modulo $d$ we may assume $0\leq b\leq d-1$. Thus $$ \left(\begin{array}{c}\om_1'\\ \om_2'\end{array}\right) = \left(\begin{array}{cc}a&b\\0&d\end{array}\right) \left(\begin{array}{cc}\om_1\\ \om_2\end{array}\right) $$ and the change of basis matrix has determinent $ad=n$. Since $$\Z\om_1'+\Z\om_2'\subset L' \subset L=\Z\om_1+\Z\om_2$$ and $(L:\Z\om_1'+\Z\om_2')=n$ (since the change of basis matrix has determinent $n$) and $(L:L')=n$ we see that $L'=\Z\om_1'+\Z\om_2'$. Thus there is a one-to-one correspondence between sublattices $L'\subset L$ of index $n$ and matrices $\bigl(\begin{array}{cc}a&b\\0&d\end{array}\bigr)$ with $ad=n$ and $0\leq b\leq d-1$. In particular, when $n=p$ is prime there $p+1$ of these. In general, the number of such sublattices equals the sum of the positive divisors of $n$. \subsection*{Action of Hecke Operators on Modular Forms} Now assume $f(\tau)=\sum_{m=0}^{\infty} c_m q^m$ is a modular form with corresponding lattice function $F$. How can we describe the action of the Hecke operator $T_n$ on $f(\tau)$? We have $$\begin{array}{ll} T_nF(\Z\tau+\Z) & = n^{k-1}\displaystyle\sum_{ \stackrel{\stackrel{\stackrel{a,b,d}{ab=n}}{0\leq b} \bigskip Following the notation of the last few lectures, let $M_k$ denote the space of modular forms of weight $k$ for $SL_2(\Z)$ and $S_k$ denote the subspace of cusp forms. Then we have: \begin{prop}. $M_k$ is a finite dimensional $\C$-vector space and is generated by modular forms having the coefficients of their Fourier expansion in $\Q$. \end{prop} {\sc Sketch of Proof}. (For details, refer Serre's ``A course in Arithmetic'' or Lang's ``Introduction to modular forms''.) \smallskip \noindent The key ingredient that goes into proving finite dimensionality is the following result, which can be obtained by contour integration: Let $f\in M_k$ and let $D=\{z\in \C :Im(z)>0, \mid z\mid \geq 1, \mid Re(z)\mid \leq 1/2\}$ be the fundamental domain for $SL_2(\Z)$. Then $$\sum_{p \in D\cup\infty} \frac{1}{e_p} ord_p(f) = \frac{k}{12} $$ where $$ e_p = \frac{1}{2} \# \{\gamma \in SL_2(\Z) : \gamma p =p \} = \frac{1}{2} \# Aut(E_p)$$ Here, the latter equality follows from the observation that the category of elliptic curves over $\C$ with isogenies is the same as the category of lattices in $\C$ upto homothety with maps being multiplication by elemets of $\C$. One can show that the invertible maps that preserve the lattice $\Z \oplus \Z p$ are in one-to-one correspondence with the set $\{\gamma \in SL_2(\Z) : \gamma p =p \}$ and hence the latter equality. In particular, $$ e_p = \left\{ \begin{array}{ll} 2 & \mbox{if $p=i$} \\ 3 & \mbox{if $p=\sqrt[3]{-1}$} \\ 1 & \mbox{otherwise} \end{array} \right. $$ Using this formula and relating the dimensions of $M_k$ and $S_k$, one can show that $M_k$ is finite dimensional and also explicitly calculate its dimension. To get a basis with Fourier coefficients in $\Q$, first observe that $M_k$ is generated by the set of Eisenstein series $G_k$ for all $k$. As a function on the complex upper half plane, the Eisenstein series $G_k$ for $k \in \Z$ and $k>1$ is given by $$G_k(\tau) = \sum_{(m,n) \in \Z^2 \backslash (0,0) } \frac{1}{(m\tau+n)^k}$$ One can then show that $$G_k(\tau) = \frac{1}{2}\zeta(1-k) + \sum_{k=1}^\infty \sigma_{k-1}(n)q^n$$ where $q=e^{2\pi i \tau}$, $\zeta$ is the Riemann zeta function and $$\sigma_k(n) = \sum_{d\mid n}d^k$$ There is a theorem due to Euler which states that $\zeta(1-k)= -\frac{b_k}{k}$ where $b_k$ are the Bernoulli numbers defined by the following power series expansion: $$\frac{x}{e^x-1} = \sum_{k=0}^\infty \frac{b_kx^k}{k!} $$ The constant term of $G_k$ is thus $-b_k/{2k}$, which is rational. Thus the Fourier expansion of $G_k$ has rational coefficients and thus we have found a basis with rational coefficients. \bigskip Next, let $V$ be a subspace of $M_k$ which is stable under the action of all the Hecke operators $T_n$. For example, observing that $T_n(G_k)=\sigma_{k-1}(n)G_k$, we see that $V=\C(G_k)$ is one such subspace. Let $\T=\T(V)$ = $\C$-algebra generated by the $T_n$'s inside $End(V)$ = $\C$-vector space generated by the $T_n$'s inside $End(V)$. The latter equality of sets follows because the product of two Hecke operators can be expressed as a linear combination of finitely many Hecke operators. For $k>0$, we define a bilinear map $\T \times V \rightarrow \C$ by $$(T,f) \mapsto a_1(f\mid T).$$ \begin{prop}. The induced maps $\T \rightarrow Hom(V,\C)$ and $V \rightarrow Hom(\T, \C)$ are isomorphisms. \end{prop} \pf We first show that the maps are injective.\\ Injectivity of the second map: $$\begin{array}{l} f \in V \mapsto 0 \\ \Rightarrow a_1(f \mid T) = 0 \ \forall T\in \T \\ \Rightarrow a_1(f \mid T_n) = 0 \ \forall n \\ \Rightarrow a_n(f) = 0 \ \forall n \geq 1 \\ \Rightarrow f\mbox{ is constant} \\ \Rightarrow f=0\mbox{ if }k>0 \end{array}$$ Injectivity of the first map: $$\begin{array}{l} T \in \T \mapsto 0 \\ \Rightarrow a_1(f \mid T)=0 \ \forall f\in V \\ \Rightarrow a_1((f\mid T_n)\mid T) = 0 \ \forall f\in V \\ \Rightarrow a_1((f\mid T)\mid T_n) = 0 \ \forall f \in V \\ \Rightarrow a_n(f\mid T) = 0 \ \forall n>0, \forall f\in V \\ \Rightarrow f\mid T = 0 \ \forall k>0, \forall f\in V \\ \Rightarrow T = 0 \end{array}$$ In the fourth line, $f$ is replaced by $f \mid T_n$. Next observe that $V$, being a subspace of $M_k$, is finite dimensional. Hence we have from the injectivities of both the above maps that each map is actually an isomorphism. \smallskip A map $\phi: M \rightarrow N$ of $\T$-modules is said to be $\T$-equivariant if $\phi (Tm) = T\phi(m) \ \forall m\in M$. \begin{prop}. The isomorphisms $\T \cong Hom(V,\C)$ and $V \cong Hom(\T, \C)$ as defined above are $\T$-equivariant. \end{prop} \pf Consider the first map. Here is the $\T$-module structure on $Hom(V,\C)$. Given $\psi \in Hom(V,\C)$, i.e. $\psi: V \rightarrow \C$, define $T\psi: V \rightarrow \C$ by $(T\psi) (f)=\psi (f\mid T)$. Let $\beta$ denote the map $\T \rightarrow Hom(V, \C)$. Then given $T'\in \T$ and $T \in \T$, we have to show that $\beta (T(T'))=T(\beta (T'))$. Let $f \in V$. Now $(\beta (TT'))(f) = a_1(f\mid TT')$, while $(T(\beta T'))(f) = \beta (T') (f\mid T) = a_1((f\mid T) \mid T') = a_1(f\mid TT')$. Thus $(\beta (TT'))(f)=(T(\beta (T')))(f) \ \forall f\in V$ and we are done. \noindent Next consider the second map. We define the $\T$-module structure on $Hom(\T,\C)$. Given $\phi \in Hom(\T,\C)$, i.e. $\phi: \T \rightarrow \C$, define $T\phi: \T \rightarrow \C$ by $(T\phi) (T')=\phi (TT')$. Let $\alpha$ denote the map $V \rightarrow Hom(\T, \C)$. Then given $f\in V$ and $T \in \T$, we have to show that $\alpha (T(f))=T(\alpha (f))$. Let $T' \in \T$. Now $$(\alpha (Tf))(T') = a_1((f\mid T) \mid T') = a_1(f\mid TT'),$$ while $$(T(\alpha f))(T') = \alpha (f) (TT') = a_1(f\mid TT').$$ Thus $$(\alpha (Tf))(T')=(T(\alpha f))(T') \ \forall T'\in \T$$ and we are done. \begin{dfn}. An element $f$ of $M_k$ is said to be an eigenform if it is an eigenfunction for all the Hecke operators.\\ i.e. $f\mid T_n = \lambda_n f$ for some $\lambda_n \in \C \ \forall n\geq 1$. \end{dfn} Let $f$ is an eigenform with eigenvalues $\lambda_n$. Then $a_n(f) = a_1(f\mid T_n) = a_1(\lambda_n f) = \lambda_n a_1(f) \ \forall n\geq 1$. \\ Thus if $a_1(f) = 0$, then $a_n(f) = 0 \ \forall n\geq 1$. \\ If $k>0$ then this implies $f=0$. Hence if $k>0$, then if $f\neq 0$, we can normalize $f$ to $\frac{1}{a_1(f)}f$. \begin{dfn}. An eigenform $f$ is said to be normalized if $a_1(f) = 1$. \end{dfn} If $f$ is a normalized eigenform with eigenvalues $\lambda_n$, then $a_n(f)=\lambda_n$ and $f\mid T_n = \lambda_n f = a_n(f)f$. % Given $f \in M_k$ define $\phi_f: \T \rightarrow \C$ by % $\phi_f(T) = a_1(f\mid T)$. Note that if $\alpha$ denotes the map % $V \rightarrow Hom(\T, \C)$ induced by the bilinear pairing mentioned % before, then $\phi_f$ is just $\alhpa(f)$. % Then we have: \\ Again, let $\alpha$ denote the map $V \rightarrow Hom(\T, \C)$ induced by the bilinear pairing mentioned earlier. Then if $f \in V$, we have the map $\alpha(f): \T \rightarrow \C$. \begin{prop}. Let $f$ be an eigenform. Then $f$ is normalized $\Leftrightarrow$ $\alpha(f)$ is a ring homomorphism. \end{prop} \pf If $f=0$ then the statement is trivial. So assume $f\neq 0$. Then as discussed above, $a_1(f)\neq 0$. Also recall from the same discussion that if $f\neq 0$ is an eigenform, then $$f\mid T_n = \frac{a_n(f)}{a_1(f)}f.$$ For ease of notation, let $\alpha_f$ denote the map $\alpha(f)$. So $$\begin{array}{l} \alpha_f(T_nT_m) \\ = a_1(f\mid T_nT_m) \\ = a_1((f\mid T_n)T_m) \\ = a_m(f\mid T_n) \\ = a_m((a_n(f)/a_1(f))f) \\ = a_m(f)a_n(f)/a_1(f). \end{array}$$ We have $$\alpha_f(T_n)\alpha_f(T_m) = a_1(f\mid T_n) a_1(f\mid T_m) = a_n(f) a_m(f).$$ The following are equivalent: \begin{tabular}{l} $\alpha_f$ is a ring homomorphism, \\ $\alpha_f(T_nT_m) = \alpha_f(T_n)\alpha_f(T_m) \ \forall T_n,T_m$, \\ $a_1(f) = 1$, and \\ $f$ is normalized. \end{tabular} \\ The first implication follows because $\T$ is generated by the $T_n$'s. % We will only prove the ``$\Rightarrow$'' part and leave the reverse % implication as an exercise. \\ % Assuming that $f$ is normalized, we have to show that % $(\alpha(f))(T_nT_m) = (\alpha(f))(T_n)\phi_f(T_m) \ \forall n,m \geq 1$. % Now $(\alpha(f))(T_nT_m) = a_1(f\mid T_nT_m) = a_1((f\mid T_n)T_m) % = a_m(f\mid T_n) = a_m(a_n(f)f) = a_m(f)a_n(f)$. \\ % While $(\alpha(f))(T_n)\phi_f(T_m) = a_1(f\mid T_n) a_1(f\mid T_m) % = a_n(f) a_m(f)$. \\ % Thus the two are equal and we are done. \section*{January 29, 1996} \noindent{Scribe: J\'anos Csirik, \tt } \bigskip Today we'll consider questions of rationality and integrality. (References: Serre: {\em A Course in Arithmetic} and Lang: {\em Introduction to Modular Forms}.) Let $S=S_k$ be the space of cusp forms of weight $k$. Let \[ S(\Q) = S_k \cap \Q [[ q ]] \] and \[ S(\Q) \supseteq S(\Z) = S_k \cap \Z [[ q ]]. \] The following fact is easy to prove using explicit formul\ae: $S_k$ has a $\C$-basis consisting of forms with integral coefficients (see Victor Miller's construction below). Recall that for all even $k\geq4$, there is an Eisenstein series \[ G_k = \frac{-b_k}{2k} + \sum^\infty_{n=1} \left ( \sum_{d|n} d^{k-1} \right) q^n, \] which is a modular form of weight $k$. Renormalize this to obtain \[ E_k = \frac{2k}{-b_k}\cdot G_k = 1 + \cdots. \] The first few Bernoulli numbers of even positive index are $b_2=1/6$, $b_4=-1/30$, $b_6=1/42$, $b_8=-1/30$, $b_{10}=5/66$, $b_{12}=-691/2730$. The fact that the first four of these have numerator 1 is closely related to the arithmetic of cyclotomic fields. The modular forms $E_4$ and $E_6$ have $q$-expansions with constant terms equal to $1$, and all coefficients in $\Z$. The functions $E_4^aE_6^b$ with $4a+6b=k$ form a basis for $M_k$. It is easy to see that they are modular forms. From the formula that the (weighted) number of zeros of any modular form of weight $k$ is $k/12$, we deduce that $E_4$ has a simple zero at $\rho$, and $E_6$ hasn't got a zero at $\rho$. Hence $E_4^aE_6^b$ has a simple zero of order $a$ at $\rho$ so these expressions are linearly independent over $\C$. To show that they span $M_k$, consider the following modular form of weight $12$: \[ \Delta = (E_4^3 - E_6^2)/1728. \] Here the coefficient of $q$ is a simple number: $1$. $\Delta$ has a simple zero at $\infty$. Since a cusp form of weight $k$ has $k/12$ zeros, it follows that (since the weighting $e_\infty$ is $1$), that $\Delta$ does not vanish anywhere on $\H$. Therefore $S_{k+12}=\Delta\cdot M_k$. Since $E_k(i\infty)=1\neq0$, it follows that $M_k=E_k\cdot\C\oplus S_k=E_k\cdot\C\oplus\Delta\cdot M_{k-12}$. Hence ${\rm dim}\,M_k={\rm dim}\,M_{k-12}+1$. Again using the fact that $f\in M_k$ has $k/12$ zeros, we quickly deduce that the dimensions of $M_0$, $M_2$, $M_4$, $M_6$, $M_8$, $M_{10}$ are $1$,$0$,$1$,$1$,$1$,$1$ respectively. (e.g., for $k=4$ any modular form must have just a simple zero at $\rho$. So for any $f\in M_4$, it is the case that $f(\tau)-f(i\infty)E_4(\tau)$ vanishes at $\tau=i\infty$ and hence is identically zero. So $E_4$ spans $M_4$.) Thus we have determined ${\rm dim}\,M_k$ for all $k\geq0$, and it is easy to see that this number is equal to the number of solutions to $4a+6b=k$ for $a,b\geq0$. Hence the $E_4^aE_6^b$ span $M_k$ and therefore we have proved that they form a basis. The following construction comes from the first page of Victor Miller's thesis. Let $d={\rm dim}_\C\, S_k$. Then there exist $f_1,\ldots,f_d\in S(\Z)$ such that $a_i(f_j)=\delta_{ij}$ for $1\leq i,j\leq d$. To show this, recall that $E_4\in M_4$ and $E_6\in M_6$ have $q$-expansions with coefficients in $\Z$. $\Delta\in S_{12}$ has constant coefficient $0$, and the coefficient of $q$ is $1$. Also $\Delta\in S_{12}(\Z)$, as can be seen for example from the formula \[ \Delta = q\prod_{n=1}^\infty (1-q^n)^{24}. \] Now pick $a,b\geq0$ so that $14\geq 4a+6b\equiv k \pmod{12}$, with $a=b=0$ when $k\equiv0\pmod{12}$. Note that then $12d+6a+4b=k$ by our previous result on the dimension of $M_k$ (and the fact that the dimension of $S_k$ is one less than that for $k\geq12$). Hence the functions \[ g_j = \Delta^j E_6^{2(d-j)+a}E_4^b \] for $1\leq j\leq d$ will be cusp forms of weight $k$. By our previous remarks on the coefficients of $\Delta$, $E_6$, $E_4$, we have $g_j\in S_k(\Z)$ and \[ a_i(g_j) = \delta_{ij} \] for $i\leq j$. A straightforward elimination now yields the $f_1,\ldots,f_d$ with the stated properties. It is clear that these $f_1,\ldots,f_d$ are linearly independent over $\C$, hence they form a basis of $S_k$. If you take $T_1,\ldots,T_d\in\T=\T(S_k)$, they are also linearly independent: for given any linear relation \[ \sum_{i=1}^d c_iT_i=0, \] apply this to $f_j$ and look at the first coefficient \[ 0 = a_1\left(f_j\left|\sum_{i=1}^d c_iT_i\right.\right) = \sum_{i=1}^d c_ia_i(f_j) = \sum_{i=1}^d c_i\delta_{ij} = c_j, \] hence the linear relation given in the first place was trivial, so $T_1,\ldots,T_d$ form a basis for $\T(S_k)$, since they are linearly independent, and ${\rm dim}_\C\,\T={\rm dim}_\C\, V=d$. Let ${\cal R}=\Z[\ldots T_n\ldots]\subseteq {\rm End}(S_k)$. \begin{claim} \[ {\cal R}=\bigoplus_{i=1}^d\Z T_i. \] \end{claim} \pf Since the $T_i$ form a basis of $\T$, we have $T_n =\sum_{i=1}^d c_{n_i}T_i$ with $c_{n_i}\in\C$. We need to check that $c_{n_i}\in\Z$. With the $f_j$ as above, consider $$ \begin{array}{lll} a_n(f_j) & = a_1(f_j|T_n) \\ & = a_1\left(f_j\left|\sum_{i=1}^d c_{n_i}T_i\right.\right) \\ & = \sum_{i=1}^d c_{n_i}a_1(f_j|T_i) \\ & = \sum_{i=1}^d c_{n_i}a_i(f_j) \\ & = \sum_{i=1}^d c_{n_i}\delta_{ij} & = c_{n_j}. \end{array}$$ Hence $c_{n_i}\in\Z$. \qed ${\cal R}$ is called the {\em integral Hecke algebra}. It is a finite $\Z$-module of rank $d$. We still have (from the last lecture) a pairing \begin{eqnarray*} S(\Z)\times{\cal R} & \rightarrow & \Z \\ (f,T) & \mapsto & a_1(f|T). \end{eqnarray*} Now $S(\Z)\hookrightarrow{\rm Hom}({\cal R},\Z)\cong\Z^d$ by the argument given before. Therefore $S(\Z)$ is a free $\Z$-module of finite rank. But it also contains the $f_i$, so $S(\Z)\cong\Z^d$. What is $S(\Z)$ as an ${\cal R}$-module? \begin{exercise} The map $S(\Z)\hookrightarrow{\rm Hom}({\cal R},\Z)$ is in fact an isomorphism of $\T$-modules. \end{exercise} {\sc Hint.\ } The cokernel is a torsion (in fact finite) group. So if we show it torsion free, we are done. \begin{thm} The $T_n$ are all diagonalizable on $S_k$. \end{thm} $S_n$ supports a Hermitian non-degenerate inner product, the Petersson inner product \[ (f,g)\mapsto \langle f,g\rangle\in\C. \] We have $\langle f,f\rangle\geq0$, with equality iff $f=0$. Furthermore \[ \langle f|T_n,g\rangle = \langle f,g|T_n\rangle, \] i.e., $T_n$ is self-adjoint with respect to the given inner product. An operator $T$ is {\em normal} if it commutes with $T^*$ (which denotes its Hermitian transpose). Normal operators are diagonalizable (for a proof, refer to Math H110). In our case, $T_n^*=T_n$, so this fact applies. It is also true (same proof) that a commuting family of semi-simple (i.e., diagonalizable) operators is simultaneously diagonalizable. \pf Put together the above facts. \qed We can also prove that the eigenvalues are real. This depends on the following trick. For $f\neq0$ consider \[ a_n\langle f,f\rangle = \langle a_nf,f\rangle = \langle f|T_n,f\rangle = \langle f,f|T_n\rangle = \langle f,a_nf\rangle = \bar{a_n}\langle f,f\rangle. \] $a_n\in\R$ now follows since $f\neq0$ implies $\langle f,f\rangle\neq0$. \begin{exercise} The $a_n$ are totally real algebraic integers. \end{exercise} {\sc Hint.\ } The space $S_k$ is stable under the action of ${\rm Aut}(\C)$ ``on the coefficients''. Given a cusp form \[f=\sum_{n=1}^\infty c_nq^n\] and some $\sigma\in{\rm Aut}(\C)$, define $\sigmaonf=\sum_{n=1}^\infty \sigma(c_n)q^n$. This function is in $S_k$, since $S_k$ has a basis in $S(\Q)$, which is fixed by $\sigma$. Then $f$ is an eigenform iff $\sigmaonf$ is an eigenform. We'll use a lame definition of the {\em Petersson inner product} for this section. Let $z=x+iy\in\H$. Then we have a volume form $y^{-2}{dx\,dy}$ which is invariant under $GL^+_2(\R)$ (the subgroup of the general linear group of matrices of positive determinant). To prove this, note that $dz\wedge d\bar z = (dx+i\,dy)\wedge(dx-i\,dy) = -2i(dx\wedge dy)$, and hence \[ dx\,dy = dx\wedge dy = \frac{-1}{2i} dz\wedge d\bar z \] Then for any $\alpha = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in GL^+_2(\R)$ we can consider the usual action \[ \alpha: z \mapsto \frac{az+b}{cz+d} = \frac{(az+b)(c\bar z+d)}{|cz+d|^2} \] where the imaginary part of the result is $|cz+d|^{-2}(ad-bc){\rm Im}\, z = y|cz+d|^{-2}\det(\alpha).$ As for differentials, \[ d\left(\frac{az+b}{cz+d}\right) = \frac{a(cz+d)\,dz-(az+b)c\,dz}{(cz+d)^2} = \frac{ad-bc}{(cz+d)^2}\,dz \] hence under application of $\alpha$, $dz\wedge d\bar z$ takes on a factor of $$\frac{\det(\alpha)}{(cz+d)^2}\frac{\det(\alpha)}{(c\bar z+d)^2} = \left( \frac{\det(\alpha)}{|cz+d|^2} \right)^2.$$ This finally proves that the differential $y^{-2}{dx\,dy}$ is invariant under the action of $GL^+_2(\R)$. The formula for the {\em Petersson inner product} is \[ \langle f,g\rangle= \int_{SL_2(\Z)\setminus\H}\left(f(z)\overline{g(z)} y^k\right)\,\frac{dx\,dy}{y^2}.\] This could be considered either an integral over the fundamental region or, noting that the integrand is invariant under $SL_2(\Z)$, an integral over the quotient space. One then checks that for $f,g$ cusp forms, the integral will converge, since they go down exponentially as $z$ tends to infinity. It is then clear that this inner product is Hermitian. It is not immediately clear that the Hecke operators are self-adjoint. \section*{January 31, 1996} \noindent{Scribe: William Stein, \tt } \bigskip {\bf \large Modular Curves}\smallskip \subsection{Cusp Forms} Recall that if $N$ is a positive integer we define the congruence subgroups $\Gamma(N)\subset\Gamma_1(N)\subset\Gamma_0(N)$ by $$ \begin{array}{cl} \Gamma_0(N) & = \{\abcd \in \modgp : c\equiv 0 \pmod{N}\}\\ \Gamma_1(N) & = \{\abcd \in \modgp : a\equiv d\equiv 1, c\equiv 0 \pmod{N}\}\\ \Gamma(N) & = \{\abcd \in \modgp : \abcd \equiv \bigl(\begin{array}{cc}1&0\\0&1\end{array}\bigr) \pmod{N}\} \end{array} $$ Let $\Gamma$ be one of the above subgroups. One can give a construction of the space $S_k(\Gamma)$ of cusp forms of weight $k$ for the action of $\Gamma$ using the language of algebraic geometry. Let $X_{\Gamma}=\Gamma\backslash\H^{*}$ be the upper half plane (union the cusps) modulo the action of $\Gamma$. Then $X_{\Gamma}$ can be given the structure of Riemann surface. Furthermore, $S_2(\Gamma)=H^0(X_{\Gamma},\Omega^1)$ where $\Omega^1$ is the sheaf of differential 1-forms on $X_{\Gamma}$. This works since an element of $H^0(X_{\Gamma},\Omega^1)$ is a differential form $f(z)dz$, holomorphic on $\H$ and the cusps, which is invariant with respect to the action of $\Gamma$. If $\gamma=\abcd\in\Gamma$ then $$d(\gamma(z))/dz=(cz+d)^{-2}$$ so $$f(\gamma(z))d(\gamma(z))=f(z)dz$$ iff $f$ satisfies the modular condition $$f(\gamma(z))=(cz+d)^{2}f(z).$$ There is a similiar construction when $k>2$. \subsection{Modular Curves} $\modgp\backslash\H$ parametrizes isomorphism classes of elliptic curves. The other congruence subgroups also give rise to similiar parametrizations. $\Gamma_0(N)\backslash\H$ parametrizes pairs $(E,C)$ where $E$ is an elliptic curve and $C$ is a cyclic subgroup of order $N$, and $\Gamma_1(N)\backslash\H$ parametrizes pairs $(E,P)$ where $E$ is an elliptic curve and $P$ is a point of exact order $N$. Note that one can also give a point of exact order $N$ by giving an injection $\Z/N\Z\hookrightarrow E[N]$ or equivalently an injection $\mu_N\hookrightarrow E[N]$ where $\mu_N$ denotes the $N$th roots of unity. $\Gamma(N)\backslash\H$ parametrizes pairs $(E,\{\alpha,\beta\})$ where $\{\alpha,\beta\}$ is a basis for $E[N]\isom(\Z/N\Z)^2$. The above quotients spaces are {\em moduli spaces} for the {\em moduli problem} of determining equivalence classes of pairs ($E + $ extra structure). \subsection{Classifying $\Gamma(N)$-structures} \begin{dfn} Let $S$ be an arbitrary scheme. An {\bfseries elliptic curve} $E/S$ is a proper smooth curve $$\begin{array}{c} E \\ \bigl| \\ S\end{array}$$ with geometrically connected fibers all of genus one, together with a section ``0''. \end{dfn} Loosely speaking, proper generalizes the notion of projective, and smooth generalizes nonsingularity. See Chapter III, section 10 of Hartshorne's {\em Algebraic Geometry} for the precise definitions. \begin{dfn} Let $S$ be any scheme and $E/S$ an elliptic curve. A {\bfseries $\Gamma(N)$-structure} on $E/S$ is a group homomorphism $$\varphi:(\Z/N\Z)^2\into E[N](S)$$ whose image ``generates'' $E[N](S)$. \end{dfn} See Katz and Mazur, {\em Arithmetic Moduli of Elliptic Curves}, 1985, Princeton University Press, especially chapter 3. Define a functor from the category of $\Q$-schemes to the category of sets by sending a scheme $S$ to the set of isomorphism classes of pairs $$(E, \Gamma(N)\mbox{\rm -structure})$$ where $E$ is an elliptic curve defined over $S$ and isomorphisms (preserving the $\Gamma(N)$-structure) are taken over $S$. An isomorphism preserves the $\Gamma(N)$-structure if it takes the two distinguished generators to the two distinguished generators in the image (in the correct order). \begin{thm} For $N\geq 4$ the functor defined above is representable and the object representing it is the modular curve $X(N)$ corresponding to $\Gamma(N)$. \end{thm} What this means is that given a $\Q$-scheme $S$, the set $X(S)=Mor_{\Q\mbox{\rm -schemes}}(S,X)$ is isomorphic to the image of the functor's value on $S$. There is a natural way to map a pair $(E,\Gamma(N)\mbox{\rm -structure})$ to an $N$th root of unity. If $P,Q$ are the distinguished basis of $E[N]$ we send the pair $(E,\Gamma(N)\mbox{\rm -structure})$ to $$e_N(P,Q)\in\mu_N$$ where $$e_N:E[N]\times E[N]\into \mu_N$$ is the Weil pairing. For the definition of this pairing see chapter III, section 8 of Silverman's {\em The Arithmetic of Elliptic Curves}. The Weil pairing is bilinear, alternating, non-degenerate, galois invariant, and maps surjectively onto $\mu_N$. \section*{February 2, 1996} \noindent{Scribe: Lawren Smithline, \tt } (Note. These were done about a month after the fact, since the assigned person dropped the course. So they are terse.) \bigskip Earlier, we looked at V. Miller's construction for eigenforms. (See, for instance, Lang X \S4 in the course references. This is a special miracle for $\SL_2(\Z)$.) Over $\Z$, $\T \cong \Z^d$ and $S(\Z)$ are dual, where $d = \dim S_k(\C)$. Here is Shimura's explanation (Lang III \S5, VIII). The Hecke operator $\T_n$ maps $S_k$ to itself. Let $A\subset \C$ be a subring and $$\T_A = A[\T_n : n > 0] \subseteq {\rm End}_\C S_k.$$ Denote by $\T$, $\T_\Z$. There is a natural tensor product $\T_A \otimes_A \C \twoheadrightarrow \T_\C$. There is also a complex conjugation automorphism of $S_k$ by $f \mapsto \overline{f(-\bar\tau)}$. This map is conjugate linear. The map $\tau \mapsto \exp(2\pi i \tau)$ becomes $\tau \mapsto \overline{\exp(-2\pi i \bar\tau)} = \exp(2 \pi i \tau)$. Say $f = \sum a_n q^n.$ Its conjugate $g = \sum \bar a_n q^n.$ If you know $S_k(\C) = \C \otimes_\Q S_k(\Q)$, then you know that modular forms can be conjugated in this sense. There is an isomorphism $\T_\R \otimes_\R \C \stackrel\sim\longrightarrow \T_\C$ since the map is surjective and the complex dimensions on each side are equal. Shimura (1959) exhibited the (Eichler-)Shimura isomorphism $$S_k(\C) \cong H^1(X_\Gamma,\R).$$ We have $S_k(\C) = H^0(X_\Gamma, \Omega^1)$ and a map $H_1(X_\Gamma,\Z) \times S_k(\C) \rightarrow \C$. Now, $H_1(X,\Z) \cong \Z^{2d}$ embeds in $\Hom_\C(S_k(\C),\C) \cong \C^{2d}$ as a lattice. So we have $S_k(\C) \rightarrow \Hom(H_1(X,\Z),\C))$ and $S_k(\C) \stackrel\sim\rightarrow \Hom(H_1(X,\Z),\R))$ as real vector spaces. By the Shimura isomorphism, this is isomorphic to $H^1(X,\R) \sim H^1_p(\Gamma,\R)$, the parabolic cohomology of $\Gamma$. So $S_k(\C) \cong H^1_p(\Gamma, V_k)$, for a certain $d-1$ dimensional subspace $V_k$. (Let $W = \R \oplus \R$. $\Gamma$ acts by linear fractional transformation. Let $V_k = {\rm Sym}^{k-2} W$. There is a lattice in $S_k(\C)$ corresponding to $H^1_p(\Gamma, {\rm Sym}^{k-2} \Z^2)$) We have an action of $\Gamma$ by $$f\cdot \c \mapsto \int_{\tau_0}^{\c(\tau_0)} f(\tau)\tau^{k-1}d\tau.$$ Recall $\T = \T_\Z$ is a set of endomorphisms of a lattice $L$, and $\T$ has finite rank over $\Z$. We have the inclusion $S_k(\Z) \hookrightarrow S_k(\C)$, or equivalently, $$\Hom_\Z(\T,\Z) \hookrightarrow \Hom_\Z(\T,\C) = \Hom(\T_\C,\C) = S_k(\C) = S_k(\Z) \otimes_\Z \C.$$ Here is a nifty inner product (the Petersson innner product) on $S_k(\C)$. For $f,g \in S_k(\C)$, let $$\lan f,g \ran = \int_{\Gamma \backslash \H} f(\tau)g(\tau) y^k \frac{dx\,dy}{y^2}.$$ The Hecke operators are self-adjoint for $(p, N) = 1$: $$\lan f | T_p, g \ran = \lan f, g | T_p \ran.$$ Indeed, for $\a \in \GL_2^+ (\R)$, $$\lan f | \a , g | \a \ran = \lan f,g \ran.$$ \section*{February 5, 1996} \noindent{Scribe: Shuzo Takahashi, \tt } \bigskip We have studied actions of $\T$ on $S_k(\C)$, $S_k(\Q)$ and $S_k(\Z)$ where $\Gamma = SL_2(\Z)$. What we know so far is $$S_k(\Z) \simeq {\rm Hom}_{\Z}(\T,\Z).$$ Also we have studied the Peterson product. It is Hermitian, i.e., $$\lan f|T_n, g\ran = \lan f,g|T_n\ran $$ for $T_n \in \T$ and for $f,g \in S_k(\C)$. \noindent Note: $T_n$ defined on $S_k(\C)$ preserves $S_k(\Z)$. Today we study when $\Gamma = \G_1(N)$ or $\G_0(N)$ for $N \geq 1$. \medbreak \noindent {\bf\large 1. The Diamond Operator and the Decomposition of $S_k(\G_1(N))$} \begin{thm} $\G_1(N)$ is a normal subgroup of $\G_0(N)$ and we have $\G_0(N)/\G_1(N) \simeq (\Z/n\Z)^*$. \end{thm} \begin{dfn} The diamond operator $< > $ is defined as follows: for $\pmatrix{a & b \cr c & d \cr} \in \G_0(N)$, the map on $S_k(\G_1(N))$ $$f \rightarrow f | \pmatrix{a & b \cr c & d \cr} $$ defines an endmorphism $< d> $ of $S_k(\G_1(N))$ which depends only on $d \ {\rm mod}\ N$. Thus we get an action $< > $ of $(\Z/n\Z)^*$ on $S_k(\G_1(N))$. \end{dfn} Since $(\Z/n\Z)^*$ is a finite group, we have the following decomposition theorem: \begin{thm} $$S_k(\G_1(N)) = \bigoplus_{\e} S_k(\G_0(N),\e)$$ where $\e$ runs over the set of characters $(\Z/n\Z)^* \rightarrow \C^*$ and $S_k(\G_0(N),\e)$ is defined as: $$S_k(\G_0(N),\e) = \{ f \in S_k(\G_1(N)) : f | < d> = \e(d) f\}.$$ We have $S_k(\G_0(N),\e) = 0$ unless $\e(-1) = (-1)^k$. \end{thm} \medbreak \noindent {\bf\large 2. The Hecke Operators on $S_k(\G_1(N))$} For $n \geq 1$, we have the operation on $S_k(\G_1(N))$ of the $n$th Hecke operator $T_n$. The following are basic properties: \begin{thm} (1) $T_n$'s commute each other and with $< d> $. (2) $T_n$'s preserve $S_k(\G_0(N),\e)$. (3) if $(n,N) = 1$, then $\lan f | T_n,g\ran = \lan f,g|{< n>}^{-1}T_n\ran $. (4) $\lan f | < d> ,g\ran = \lan f,g|{< d>}^{-1}\ran $. (5) if $(n,N) = 1$, then $T_n$ is diagonalizable. (6) if $(n,N) \neq 1$, then $T_n$ is not diagonalizable. \end{thm} The action of $T_n$ is described in the following theorem: \begin{thm} Let $f = \sum_{n = 1}^\infty a_n q^n \in S_k(\G_0(N),\e)$. Then %$$f | T_p = %\cases %\sum_{n=1}^{\infty} a_{pn} q^n + p^{k-1}\e(p) \sum_{n=1}^{\infty} a_n %q^{pn} %&\text{if $p \not| N$;} \\ %\sum_{n=1}^{\infty} a_{pn} q^n %&\text{if $p | N$.} %\endcases$$ (1) $f | T_p = \sum_{n=1}^{\infty} a_{pn} q^n + p^{k-1}\e(p) \sum_{n=1}^{\infty} a_n q^{pn}$. (Note: when $p|N$, $\e(p)$ is considered to be $0$ and $U_p$ is used instead of $T_p$ which is called Atkin-Lehner operator.) (2) if $(n,m)=1$, then $T_{nm} = T_n T_m$. (3) if $p \not| N$, then $T_{p^l} = T_{p^{l-1}}T_p - p^{k-1}< p> T_{p^{l-2}}$. (4) if $p | N$, then $T_{p^l} = (T_p)^l$. \end{thm} The last formula in the theorem can be proved by comparing the coefficients of $q^{p^{l-1}}$ in both sides of the following formal identity: $$\left(\sum_{n=1}^{\infty} T_n q^n\right)|T_p = \sum_{n=1}^{\infty} T_{pn} q^n + p^{k-1}< p> \sum_{n=1}^{\infty} T_n q^{pn}.$$ For example, the coefficient of $q^p$ in the LHS is $T_p T_p$. On the other hand, the coefficient of $q^p$ in the RHS is $T_{p^2} + p^{k-1}< p> T_1$. Thus, we have $$T_{p^2} = (T_p)^2 - p^{k-1}< p> Id$$ where $< p> $ should be considered to be a null map if $p | N$. \medbreak \noindent {\bf\large 3. The Old Forms} Suppose $M|N$. Let $f \in S_k(\Gamma_1(M))$. Then for $d$ such that $d | \frac{N}{M}$, $f(d\tau) \in S_k(\G_1(N))$. Thus we have a map $$\phi_M : \bigoplus_{d | \frac{N}{M}} S_k(\Gamma_1(M)) \rightarrow S_k(\G_1(N)).$$ The old part of $S_k(\G_1(N))$ is defined as the subspace generated by the images of $\phi_M$ for $M | N$, $M \neq N$. \begin{eg} $\phi_M$ is not injective. Consider the case that $k = 12$, $M = p$ and $N = p^2$. $S_k(\Gamma_1(p))$ contains $\Delta(\tau)$ and $\Delta(p\tau)$. But $\phi_p$ maps both of them to $\Delta(p\tau)$ in $S_k(\Gamma_1(p^2))$. \end{eg} \begin{thm} Suppose $p \nmid N$. Consider $f,g \in S_k(\G_1(N))$. Then $f$ and $g(p\tau)$ are both in $S_k(\Gamma_1(Np))$. Then we have $$f | U_p = (f | T_p) - p^{k-1}\e(p)(f(p\tau))$$ and $$g(p\tau) | U_p = g(\tau)$$ where $f | T_p$ is considered in $S_k(\G_1(N))$. \end{thm} \noindent {\bf Proof.} Let $f = \sum_{n=1}^{\infty} a_n q^n$. Then, considering in $S_k(\G_1(N))$, we have $$ f | T_p = \sum_{n=1}^{\infty} a_{pn} q^n + p^{k-1}\e(p) \sum_{n=1}^{\infty} a_n q^{pn}.$$ Also, considering in $S_k(\Gamma_1(Np))$, we have $$f | U_p = \sum_{n=1}^{\infty} a_{pn} q^n.$$ Thus, we have $$f | U_p = (f | T_p) - p^{k-1}\e(p)(f(p\tau)).$$ Now, let $g = \sum_{n=1}^{\infty} b_n q^n$. Then $$g(p\tau) | U_p = \left(\sum_{n=1}^{\infty} b_{n/p} q^n \right) | U_p = g(\tau)$$ where $b_{n/p} = 0$ unless $p | n$. \section*{February 7, 1996} \noindent{Scribe: Amod Agashe, \tt } \bigskip We are in the process of showing that the Hecke operators $T_p$ acting on the space of cusp forms $S_k(\Gamma_1(N))$ are not necessarily semisimple if $p\mid N$. Recall from last time that if $M \mid N$ then for every divisor $d$ of $M/N$, we had a map $S_k(\Gamma_1(M)) \rightarrow S_k(\Gamma_1(N))$ given by $f(\tau) \mapsto f(d\tau)$. Note that the various $f(d\tau)$'s are linearly independent over $\C$, because the Fourier expansion of $f(d\tau)$ starts with $q^d$. Let $f$ be an eigenfuntion for all the Hecke operators $T_n$ in $S_k(\Gamma_1(M))$. Let $p$ be a prime not dividing $M$. So $f\mid T_p = af$ where $a=a_p(f)$ and $f \mid

=\epsilon (p)f$ where $\epsilon (p)$ is the character associated to the modular form $f$. Note that one can prove that if $f$ is an eigenfunction for the $T_n$'s then it is an eigenfunction for the diamond operators also (or alternatively, make it part of the definition of eigenform). Let $N=p^\alpha M$ with $\alpha \geq 1$. We will look at the action of the $p^{th}$ Hecke operator $U_p$ in $S_k(\Gamma_1(N))$ on the images of $f$ under the maps described above. Let $f_i(\tau) = f(p^i \tau)$ for $0\leq i\leq \alpha$. As we showed earlier, \\ $$f\mid T_p = \sum a_{np}q^n + \epsilon (p)p^{k-1}\sum a_nq^{pn}.$$ So $$af = f_0\mid U_p + \epsilon (p)p^{k-1}f_1.$$ Thus, $$f_0\mid U_p = af_0 -\epsilon(p)p^{k-1}f_1.$$ From last time, we have $f_1\mid U_p = f_0.$ In fact, in general, one can see easily that $f_i\mid U_p=f_{i-1}$ for $i\geq 1$. So $U_p$ preserves the 2-dimensional space spanned by $f_0$ and $f_1$. The matrix of $U_p$ (acting on the right) with this basis is given (from the equations above) by: \smallskip \[ \left( \begin{array}{cc} a & 1 \\ -\epsilon(p)p^{k-1} & 0 \end{array} \right) \] The characteristic polynomial of this matrix is $x^2-ax+p^{k-1}\epsilon(p)$. There is the following striking coincidence: Let $E$ be the number field generated over $\Q$ by the coefficients of the Fourier series expansion of $f$ and let $\la$ be a prime ideal of $\O_E$ lying over some rational prime $l$. Then we have a Galois representation \smallskip $$\rho_\la: Gal(\overline{\Q}/\Q) \rightarrow \GL_2(E_\la)$$ If $p\not|Nl$ then $\rho_\la$ is unramified and also $det\ \rho_\la(Frob_p)=\epsilon(p)p^{k-1}$ and $tr\ \rho_\la(Frob_p)=a_p(f)=a$. Thus the characteristic polynomial of $\rho_\la(Frob_p)$ is $x^2-ax+p^{k-1}\epsilon(p)$, the same as that of the matrix of $U_p$! A question one can ask is: Is $U_p$ semisimple on the space spanned by $f_0$ and $f_1$? The answer is yes if the eigenvalues of $U_p$ are different. Now, the eigenvalues are the same iff the discriminant of the characteristic polynomial is zero i.e. $a^2=4\epsilon(p)p^{k-1}$ i.e. $a=2p^{\frac{k-1}{2}}\zeta$ where $\zeta$ is some square root of $\epsilon(p)$. Here is a curious fact: the Ramanujan-Petersson conjecture proved by Deligne says $|a|\leq 2p^{\frac{k-1}{2}}$; thus the above equality is allowed by it, so we do not get any conclusion about the semisimplicity of $U_p$. Let us now specialize to $k=2$. Weil has shown that $\rho_\la(Frob_p)$ is semisimple. Thus if the eigenvalues of $U_p$ are equal, then $\rho_\la(Frob_p)$ is a scalar. Edixhoven proved that it is not. So the eigenvalues of $U_p$ are different and hence $U_p$ is semisimple in this case. So this example (for k=2) does not give us an example of $U_p$ being not semisimple. There is the following example given by Shimura which shows that the Hecke operator $U_p$ need not be semisimple. Let $W$ denote the space spanned by $f_0, f_1$ and $V$ denote the space spanned by $f_0,f_1,f_2,f_3$. $U_p$ preserves both spaces $W$ and $V$, so it acts on $V/W$. The action is given by $\overline{f_2}\mapsto \overline{f_1}=0$ and $\overline{f_3}\mapsto \overline{f_2}$ where the bar denotes the image in $V/W$. Thus the matrix of $U_p$ on the space $V/W$ is \[ \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right) \] which is nilpotent, and in particular not semisimple. If $U_p$ were semisimple on $V$ then it would be semisimple on $V/W$ also; but we have just shown that it is not. Thus $U_p$ is not semisimple on $V$, and hence not on $S_2(\Gamma_1(M))$ (because $V$ is invariant under $U_p$). \bigskip We next discuss the structure of the $\C$-algebra $\T =\T_\C$ generated by the Hecke and diamond operators and the structure of $S_k(\Gamma_1(N))$ as a $\T$-module. First we consider the case of level $1$ i.e. $N=1$. Then $\Gamma_1(1)=SL_2(\Z)$. All the $T_n$'s are diagonalizable. $S_k=S_k(\Gamma_1(N))$ has a basis of $f_1,....,f_d$ of normalized eigenforms where $d=dim(S_k)$. Thus $S_k\cong \C^d$ as a $\C$-vector space. Then we have the $\C$-algebra homomorphism $\T\rightarrow \C^d$ given by $T\mapsto (\la_1,....,\la_d$) where $f_i\mid T=\la_i f_i$. It is injective because if the image of $T$ is zero, then it kills all $f_i$ i.e. all of $S_k$ i.e. it is the zero operator. The map is surjective because $\T$ has dimension $d$. %and hence it is surjective %when we consider both sides as $\C$-vector spaces. Thus as a $\C$-algbebra, $\T\cong \C^d$. Next, we claim that the modular form $v=f_1+...+f_d$ generates $S_k$ as a $\T$-module. This follows because under the map $S_k\cong \C^d, v\mapsto (1,....,1)$ and our statement is just the trivial fact that $(1,....,1)$ generates $\C^d$ as a $\C^d$-module (acting component-wise). Thus $S_k$ is free of rank $1$ as a $\T$-module. We already know that $S_k\cong Hom(\T,\C)$ as $\T$-modules. Thus $\T\cong Hom(\T,\C)$ as $\T$-modules. In fact the isomorphism is canonical since the $f_i$'s are normalized. We remark that $v$ in fact lies in $S_k(\Q)$. Next, we deal with the general case where the level is not necessarily $1$. First we need to talk about newforms. Recall the maps $S_k(\Gamma_1(M)) \rightarrow S_k(\Gamma_1(N))$ for every divisor $d$ of $M/N$ mentioned at the beginning of this lecture. The old part of $S_k(\Gamma_1(N))$ is defined as the space generated by all the images of $S_k(\Gamma_1(M))$ for all $M\mid N, M\neq N$ under these maps. The new part of $S_k(\Gamma_1(N))$ can be defined in two different ways. Firstly we can define it as the orthogonal complement of the old part with respect to the Petersson inner product. There is also an algbraic definition as follows. There are certain maps going the other way: $S_k(\Gamma_1(N)) \rightarrow S_k(\Gamma_1(M))$ for $M\mid N, M\neq N$. The new part is the space killed under all these maps. The space of newforms, denoted $S_k(\Gamma_1(N))_{new}$ is like $S_k(\Gamma(1))$ in the sense that all the $T_n$'s (including $U_p$) are semisimple and there is a basis consisting of newforms. A form of level $N$ is said to be new of level $N$ if it is in $S_k(\Gamma_1(N))_{new}$. Next, one can show that the map $\bigoplus_{M\mid N,M\leq N} S_k(\Gamma_1(M))_{new} \rightarrow S_k(\Gamma_1(N))$ given by $f(\tau)\mapsto f(d\tau)$ for $d\mid \frac{N}{M}$ is injective (See W.-C. W.Li, Newforms and functional equations, Math. Annalen, 212(1975), 285-315). Note that an eigenform in one of the subspaces of the source need not be an eigenfuntion for all the operators in the image. If $f$ is a newform, then let $M_f$ denote its level (i.e. $f$ is new of level $M_f$). Let $S$ be the set of newforms of weight $k$ and some level dividing $N$. Let $$v=\sum_{f\in S} f(\frac{N}{M_f}\tau).$$ Then one can show that $S_k(\Gamma_1(N))$ is free of rank $1$ over $\T_\C$ with $v$ as the basis element. Also one can show that $v$ has rational coefficients. \section*{February 9, 1996} \noindent{Scribe: J\'anos Csirik, \tt } \vspace{.2in} \subsubsection*{Final comments about Hecke algebras} Recall that for the case $\Gamma=SL(2,\Z)$, if we set $f_1,\ldots,f_d$ to be the normalized eigenforms (newforms of level 1), then they have possibly complex coefficients but in any case $\{f_i\}$ is finite and stable under automorphisms of $\C$ and all the coefficients $a_n(f_i)$ lie in some number field. Furthermore this field is totally real: to show this we used that since the set is stable under conjugation, it suffices to show that all the $a_n(f_i)$ are real, which followed by remarking that they are eigenvalues of the operators $T_n$ which are self-adjoint with respect to the Petersson inner product. More generally, let $f\in S(\Gamma_1(N))$ be a normalized eigenform of character $\varepsilon$. Then $a_n=\varepsilon(n)\overline{a_n}$. Note that the algebra $\T_\Q$ generated by the $T_i$ over $Q$ contains the diamond bracket operators: the formula relating $T_{p^2}$ and $(T_p)^2$ tells us that the difference is $p^{k-1}\varepsilon(p)$, so $\varepsilon(p)\in\T_\Q$. Using Dirichlet's Theorem on primes in arithmetic progressions, for any $d$ relatively prime to $N$ we can find a prime $p\equiv d\pmod{N}$, so $\varepsilon(d)=\varepsilon(p)\in\T_\Q$. If the space of modular forms has dimension 1, then it is spanned by a (normalized) eigenform with rational coefficients, so the eigenvalues are all in $\Q$. The next simplest example is $k=24$, which is the smallest weight such that the dimension is more than one. There are two eigenforms, which are conjugate to each other. If $f=\sum a_nq^n$ is an eigenform, then $\Q(\ldots a_n\ldots)=\Q(\sqrt{144169})$. In fact $S_{24}$ is spanned by $\Delta^2$ and $\Delta^2|T_2$. (Note that $\Delta^2$ is definitely not an eigenform since its $q$-coefficient is 0.) The action of $T_2$ on $S_{24}$ with respect to this basis is described by a two by two matrix of trace $1080$ and determinant $-2^{10}3^2 2221$. For high $k$, eigenforms tend to form a single orbit; however, no proof is known for this. For every newform $f$, let $E_f$ be the number field generated by its coefficients. Let $\Sigma$ be a set of representatives for $f$'s modulo $Gal(\bar\Q/\Q)$. Define \begin{eqnarray*} \T_\Q & \rightarrow & E_f \\ T & \mapsto & \lambda_T, \end{eqnarray*} with $f|T=\lambda_Tf$. Thus $T_n\mapsto a_n$ so this map is surjective. In fact the induced \[ \T_\Q \to \prod_{f\in\Sigma} E_f \] is an isomorphism of $\Q$-algebras: it is injective since if $T$ dies on the image then it acts as zero on $f\in\Sigma$ and so on all of f, by the rationality of Hecke operators: ${}^\sigma\!(g|T)={}^\sigma\!g|T$. (And if an operator acts as zero on everything, then it is zero.) Here we used the fact that there were no oldforms around. For example, consider $S_2(\Gamma(N))$ for $N$ prime. Then $S_2(\Gamma(1))$ is empty, hence we get an isomorphism \[ \T_\Q\cong E_1\times\ldots\times E_t, \] with the right hand side a product of totally real number fields. $t>1$ is possible, e.g. for $N=37$, $\T_\Q=\Q\times\Q$. In general, oldforms complicate the situation. \subsubsection*{Final comments about Hecke algebras} We'll only treat the case $k=2$. Then (for $\Gamma$ a congruence subgroup), \[ S_2(\Gamma)=H^0(X_\Gamma,\Omega^1) \] where $X_\Gamma=\Gamma\setminus\H\cup\Gamma\setminus\P^1(\Q)$, with $\Gamma\setminus\P^1(\Q)$ being the set of cusps that we need to adjoin to make it a compact Riemann surface. Therefore \[ {\rm dim}\,S_2(\Gamma)= g(X_\Gamma). \] {\sc Example.\ } $SL(2,\Z)\setminus\P^1(\Q)$ has just one point. The proof involves the Euclidean algorithm: any element of $\P^1(\Q)$ can be written as $\left( \begin{array}{cc} x\\y \end{array} \right)$ with $x$ and $y$ relatively prime integers. By the Euclidean Algorithm, we can find $a$ and $b$ integers such that $ax+by=1$. Then \[ \left(\begin{array}{cc} a & b \\ -y & x \end{array} \right) \left( \begin{array}{cc} x\\y \end{array} \right) = \left( \begin{array}{cc} 1\\0 \end{array} \right) \] To calculate $g(X_\Gamma)$, use the following covering (recall that $X_\Gamma$ is the compactification of $\Gamma\setminus\H$ we obtain by adjoining the cusps) \[ X_\Gamma \rightarrow X_{\Gamma(1)}, \] keeping in mind the isomorphism \begin{eqnarray*} j:X_{\Gamma(1)} & \rightarrow & \P^1(\C) \\ (i,\rho,\infty) & \mapsto & (1728,0,\infty) \end{eqnarray*} The only ramification in our covering occurs above the points $0,1728,\infty$. {\sc Example.\ } Let $\Gamma=\Gamma_0(N)$. The degree of the covering is $(PSL(2,\Z):\Gamma_0(N)/\pm1)$ which is the number of cyclic subgroups of order $N$ in $SL(2,\Z/N\Z)$. We have a covering $Y_0(N)\to Y_{\Gamma(1)}$ where $Y_0(N)$ parametrizes elliptic curves $E$ with a cyclic subgroup of order $N$, $C\subseteq E[N]\cong\Z/N\Z\times\Z/n\Z$ up to isomorphism; and $Y_{\Gamma(1)}$ parametrises elliptic curves. The isomorphism $(E,C_1)\cong(E,C_2)$ is an automorphism $\alpha$ of $E$ with $\alpha:C_1\mapsto C_2$. Usually $\alpha=\pm1$, unless $j=1728,0,\infty$. If we understand ramification, we can use the {\em Riemann-Hurwitz formula}. The following mnemonic way of thinking about it is due to N. Katz. The Euler characteristic (alternating sum of the dimensions of cohomology groups) should be thought of as totally additive: \[ \chi(A\coprod B) = \chi(A) + \chi(B). \] If $X$ is a Riemann surface of genus $G$, $\chi(X)=2-2G$. A single point has Euler characteristic $\chi(P)=1$. Hence \[ \chi(X\setminus\{P_1,\ldots,P_n\})=2-2G-n. \] Therefore \[ \chi(X_{\Gamma(1)}\setminus\{0,1728,\infty\})=-1 \] and \[ \chi(X_\Gamma\setminus\{\mbox{points over $1728,0,\infty$}\})=2-2g-n \] if $n$ points lie over ${0,1728,\infty}$. If the covering map $X_\Gamma\to X_{\Gamma(1)}$ has degree $d$, then we can think of the top space as $d$ copies of the bottom space, so \[ d \cdot (-1) = 2-2g-n \] and therefore \[ 2g-2 = d-n = d-n_0 - n_{1728} -n_\infty.\] {\sc Example.\ } Let $\Gamma=\Gamma(N)$. What happens over $j=0$? This corresponds to an elliptic curve $E$ with an automorphism $\alpha$ of order three. Let $N>3$. For any $(E,P,Q)$, we have $(E,\alpha P,\alpha Q)$ and $(E,\alpha^2 P,\alpha^2 Q)$ which are isomorphic to it and hence they are the same point on $X_\Gamma$. So $E$ only has one-third the usual number of points lying over it, $n_0=d/3$. (Except if the above three points are equal, i.e., $\alpha$ fixes $E[N]$. This can't happen since $N>3$.) Similarly we get $n_{1728}=d/2$. To determine the degree $d$, fix $E$ and count the points lying over it: these are all of the form $(\C/\Z\oplus\tau\Z,1/n,\tau/N)$ with Weil pairing $e^{2\pi i/N}$ and all such occur, so we need to count the number of $P,Q\in E[N]$ which form a basis of $E[N]$ and $e_N\langle P,Q\rangle=e^{2\pi i/N}$. This gives us the order of $SL(2,\Z/n\Z)$. However, (since $N\neq2$) we have to take into account that $(E,P,Q)\cong(E,-P,-Q)$ but they are not equal so the degree of the covering is $\#(SL(2,\Z/N\Z))/2$. We also have $d=(PSL(2,\Z):\Gamma(N)/(\Gamma(N)\cap\pm1))$ since $\Gamma(N)\setminus SL(2,\Z)\cong SL(2,\Z/N\Z)$. So we have established that \[ 2g-2 = d-n_0 - n_{1728} -n_\infty = d/6 - n_\infty. \] To determine $n_\infty$, note that $SL(2,\Z)$ acts on $\P^1(\Q)\ni\infty= \left( \begin{array}{cc} 1 \\ 0 \end{array} \right)$. The stabilizer $\Gamma(N)_\infty$ of $\left( \begin{array}{cc} 1 \\ 0 \end{array} \right)$ is $U=\pm\left( \begin{array}{cc} 1 & * \\ 0 & 1 \end{array} \right)$ So the index $((\modgp/\pm1)_\infty:\Gamma(N)_\infty)=N$ is the ramification degree of a point over $\infty$, so $n_\infty=N/d$. Hence $2g(X(N))-2=d/6-d/N$. \section*{February 12, 1996} \noindent{Scribe: J\'anos Csirik, \tt } \vspace{.2in} \def\T{{\bf T}} % Hecke algebra \def\qed{\hfill $\blacksquare$} Plug: A useful reference for the next lecture is Andrew Ogg: {\em Rational points on certain elliptic modular curves} (1972). From last lecture's results on easily deduces that \[ g(X(N)) = 1 + \frac{d}{12N}(N-6). \] {\sc Example.} Let $N=5,7$. If $N$ is prime then the degree of the covering is $\frac{(N^2-1)(N^2-N)}{N-1}$. Therefore $N=5$ gives $d=60$ and $g=0$ (the Galois group of the covering in this case is $A_5$.) Similarly, $N=7$ yields $g=3$ and $d=168$. (Remark: For $p\geq5$ prime, the group $SL(2,\Z/p\Z)/\pm1=L_2(p)$ is simple.) {\sc Example.} What is the genus of $X_0(N)$, for $N$ a prime? It is an exercise to show that there are two cusps: $\left( \begin{array}{cc} 1 \\ 0 \end{array} \right)=\infty$ and $\left( \begin{array}{cc} 0 \\ 1 \end{array} \right)=0$. The covering $X_0(N)\to X(1)$ is easily seen to have degree $N+1$, since an elliptic curve has $N+1$ cyclic subgroups of order $N$. Here $\infty$ is unramified and $0$ has ramification index $N$. Hence \[ 2g-2 = N+1 -n_\infty-n_{1728}-n_0 \] with $n_\infty=2$, $n_{1728}$ approximately $d/2$ and $n_0$ approximately $d/3$. To calculate $n_0$, we need the number of isomorphism classes $(E,C)$ with $E$ fixed with $End(E)=\Z[\mu_6]$. $\mu_6$ acts on the set of $C$'s. Since $\pm1$ is acting trivially, we really have an action of $\mu_3$ with $(E,C)\cong(E,\zeta C)\cong(E,\zeta^2C)$ with $\zeta$ some third root of unity. If we consider $E=\C/\O$ with $\O=\Z[(-1+i\sqrt3)/2]$, then $E[N]=\O/N\O$ as an $\O$-module. In fact \[ \O/N\O = \left\{ \begin{array}{ll} \F_N\oplus\F_N & \mbox{if N splits in $\O$} \\ \F_{N^2} & \mbox{if N does not split in $\O$.} \end{array} \right. \] The above covers all possibilities, because $N>3$ can't ramify in $\O$. By Kummer's theorem, $N$ splits in $\O$ iff $\left(\frac{-3}{N}\right)=1$, and there are exactly two $\O$-stable submodules of $\O/N\O$. In the second case, which happens iff $\left(\frac{-3}{N}\right)=-1$, $\O/N\O$ has no $\O$-stable submodules. Therefore \[ n_0 = \left\{ \begin{array}{ll} \frac{N-1}{3} + 2 & \mbox{if $N$ splits in $\O$, i.e., $N\equiv1\pmod3$.}\\ \frac{N+1}{3} & \mbox{if $N$ does not split in $\O$, i.e., $N\equiv2\pmod3$} \end{array} \right. \] In the first case, note that the two $\O$-stable submodules have to be counted separately. Similarly we obtain \[ n_{1728} = \left\{ \begin{array}{ll} \frac{N-1}{2}+2 & \mbox{if $N\equiv1\pmod4$} \\ \frac{N+1}{2} & \mbox{if $N\equiv3\pmod4$} \end{array} \right. \] depending on whether$N$ splits in $\Q(i)$. {\sc Examples.} For $N=37$ the genus is $2$. Using the formula, we get $2g-2=36-(2+18)-14=2$ (so it works). Using the formula we can also obtain $g(X_0(13))=0$ and $g(X_0(11))=1$. It is therefore clear that the genus of $X_0(N)$ is approximately $N/12$. In the article by Serre in the Lecture Notes in Mathematics 349 (Antwerp), we find the following table. Write $N=12a+b$ with $0\leq b\leq11$. Then \begin{center} \begin{tabular}{c|cccc} $b$ & $1$ & $5$ & $7$ & $11$ \\ \hline $g$ & $a-1$ & $a$ & $a$ & $a+1$. \\ \end{tabular} \end{center} Hence \[ 1 + g(X_0(p)) = {\rm dim}\,M_{p+1}(\modgp). \] It was Serre's idea to think of ``modular forms mod $p$'', for some congruence subgroup $\Gamma\ni\left(\begin{array}{cc}1&1\\0&1\end{array}\right)$, like $\Gamma_0(N)$ or $\Gamma_1(N)$. We could use our moduli theoretic interpretations, but instead we'll define \[ M_k(\Gamma,\F_p) \subseteq \F_p[[q]]. \] By Shimura's cohomology trick, we know that $M_k(\Gamma,\Z)$ is a lattice in $M_k(\Gamma,\C)$. Hence we can set \[ M_k(\Gamma,\F_p) = M_k(\Gamma,\Z) \otimes_{\Z} \F_p. \] Then {\em Serre's equality} states that for a prime $p$, \[ M_{p+1}(\modgp,\F_p) = M_2(\Gamma_0(p),\F_p) \] in $\F_p[[q]]$. The philosophy is {\em mod $p$ forms with $p$ in the level can be taken to mod $p$ formswith no $p$ in the level, but of a higher weight}. So for example $M_k(\Gamma_1(p^\alpha N),\F_p)$ is a subset of $M_?(\Gamma_1(N),\F_p)$ of forms of some higher level. Finally, consider the map from the right hand side to the left hand side in Serre's equality. Recall that \[ G_k = \frac{-B_k}{2k} + \sum^\infty_{n=1}\sigma_k(n) q^n \in M_k(\modgp). \] By Kummer, ${\rm ord}_p(B_{p-1}) = -1$, so \[ E_{p-1}=1+\frac{-2(p-1)}{B_{p-1}}\sum\sigma_{p-1}(n)q^n \equiv 1 \pmod{p}. \] Hence we got the map from the right to the left: multiply by $E_{p-1}$ to get to $M_{p+1}(\Gamma_0(p),\F_p)$. Then take the trace to get to $M_{p+1}(\modgp,\F_p)$. The trace map is dual to the inclusion and is expressed by \[ \tr(f) = \sum_{i=1}^{p+1} f | \gamma_i \qquad \gamma_i\in\Gamma_0(p)\setminus\modgp. \] \section*{February 14 and 16, 1996} \noindent{Scribe: Jessica Polito, \tt } \bigskip Our goal, for these two days, is to define the modular curves $\X$ over $\Q(\mu_N)$, and $X_1(N)$, and $X_0(N)$ over $\Q$. These notes will spell out the construction of $\X$, with some discussion of the construction of the other two types of curves. The idea comes from Shimura. Let $\Q(t)$ be the function field of $\P^1/\Q$, and pick an elliptic curve $\E/\Q(t)$ with $j$-invariant $t$: $$\E: y^2 = 4x^3 - \frac{27t}{t-1728}x - \frac{27t}{t-1728}$$ (Note that the general formula for the $j$-invariant of a curve $y^2 = 4x^3 - g_2x -g_3$ is $j = \frac{1728g_2^3}{g_2^3 - 27g_3^2}$, and that $j(E)$ determines the isomorphism class of the given curve $E$ over the algebraic closure of the field of definition.) By substituting in a given value $j$ for $t$, we would get a formula for an elliptic curve over $\Q(j)$ with $j$-invariant $j$; for $j= 0 $ or 1728, we could pick a diferent formula for $\E$ which would give an isomorphic curve over $\Q(t)$, for which that substitution would make sense. Notice that, in general, if we have an elliptic curve $E/K$, with $K$ some field of charictaristic prime to $N$, then we can consider $E[N](\K)$, the set of all $N$-torsion point of $E$ defined over $\K$, which is isomorphic to $(\Z/N\Z)^2$. Then we let $K(E[N])$ be the smallest extension of $K$ over which all the points of $E[N]$ are defined. Notice also that $E/K$ and $N$ together define a representation of the Galois groups $\gal(\K/K)$ into the automorphisms of the $N$-torsion points of $E$, as they are defined by polymial equations with coefficients in $E$. We get $$\rho_{E,N}: \gal(\K/K) \into \aut(E[N]) \isom \GL_2(\Z/N\Z)$$ with $\gal(\K/K(E[N])) = \ker(\rho_{E,N})$. Unsurprisingly, we will frequently leave out the $N$, writing simply $\rho_E$. Now, to return to our construction, where $K=\Q(t)$. We will show that $\Q(t)(E[N])$ is the function field of a curve defined over $\Q(\mu_N)$ (so $\Q(t)(E[N]) \cap \overline{\Q} = \Q(\mu_N)$), and that this curve corresponds to $X(N)$. We will actually do this, not by looking at $\rho_E$, but instead at $\rh$, where $\rh$ is given by reducing the image of $\rho_E$ mod $\pm 1$: \begin{diagram}[tight,width=4em,height=2em] \gal(\K/K) & \rTo^{\rho_E} &\GL_2(\Z/N\Z) \\ & \rdTo_{\rh} & \dTo\\ & &\GL_2(\Z/N\Z)/\{\pm 1\} \end{diagram} Note: $\rho_E$ is surjective iff $\rh$ is. \pf If $\rh$ is surjective, then $$\pm \left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) \in \im(\rho_E).$$ By squaring, we see that $$\left(\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right) \in \im(\rho_E),$$ and thus $-1 \in \im(\rho_E)$, so $\rho_E$ is surjective. (Obviously, if $\rho_E$ is surjective, so is $\rh$.) \qed Why will we use $\rh$ instead of $\rho_E$? The kernal of $\rho_E$ is the field generated over $K$ by the $X$ and $Y$ coordinates of the non-zero points of $\E[N]$, whereas the kernal of $\rh$ is the field generated by just the $X$ coordinates of those points. This follows because $X(P) = X(-P)$, while $Y(P) = -Y(-P)$, as then the $X$ but not the $Y$ coordinates are fixed by $\pm 1$. (We are assuming that we have a Weierstrass model for $\E$ here.) \begin{remark} Let $E_1$ and $E_2$ be elliptic curves over $K$ with equal $j$ invariants. (In other words, $E_1$ and $E_2$ are isomorphic over $\overline{K}$.) Then $\rho_{E_1}$ is surjective iff $\rho_{E_2}$ is. (We'll assume that either $N \neq 2, 3$ or $E$ does not have complex multiplication.) \end{remark} \pf Assume $\rho_{E_1}$ is surjective. Then $E_1$ does not have complex multiplication over $\overline{K}$. (If $E$ has CM, then the image of $\rho_E$ is abelian for some quadratic extension of $K$, but $\GL_2(\Z/N\Z)$ has no abelian subgroup of index 2, as $N>3$.) Then $\aut(E_1) = \{\pm 1\}$ Pick an isomorphism $\phi:E_1 \isom E_2$ over $\overline{K}$. Then, for any $\sigma \in \gal(\overline{K}/K)$, we have ${}^\sigma\phi:{}^\sigma E_1 \isom {}^\sigma E_2$, and ${}^\sigma E_i = E_i$. Thus ${}^\sigma\phi = \pm \phi$ for all $\sigma \in \gal(\overline{K}/K)$. So $\phi: E_1[N] \isom E_2[N]$ does not quite give us an isomorphism of representations $\rho_{E_1}$ and $\rho_{E_2}$, but it does give us an isomorphism $\overline{\rho_{E_1}}\isom \overline{\rho_{E_2}}$. We then have that $\overline{\rho_{E_2}}$, and thus $\rho_{E_2}$, is surjective. \qed From here on, let $K=\C(t)$, $\E$ and $\rho_E$, $\rh$ as above. We have $$\rho_E: \gal(\K/K) \into \GL_2(\Z/N\Z),$$ so the determinant of $\rho_E$ is a cyclotomic character. As the $N$th roots of unity are all already in $K$, this means that the determinant of $\rho_E$ is trivial, so $$\rho_E: \gal(\K/K) \into \SL_2(\Z/N\Z).$$ Our goal is to prove that $\rho_E$ is surjective onto $\SL_2(\Z/N\Z)$. (For any field $F$ of characteristic not dividing $p$, there is an algebraic proof by Igusa that a ``generic elliptic curve'' (such as $\E$ here), has the maximal possible Galois action on its division points.) We will now introduce another way of looking at $K$. It is equal to $\C(j)=\ff_1$, the field of modular functions of level 1. (A modular function of level $N$ is a $\Gamma(N)$-invariant function on $\H$ which is meromorphic on $\H$ and at the cusps.) The set of all modular functions for $\SL_2(\Z)$ are generated over $\C$ by a single function $j$. Similarly, $\ff_N$ is the field of modular functions of level $N$. We then have the tower of field extensions $$\begin{array}{ccc} \K& = &\overline{\ff_1} \\ &&| \\ &&\ff_N \\ &&| \\ K&=&\ff_1 \end{array} $$ $\SL_2(\Z)$ acts on $\ff_N$, via the map $f(\tau) \to f\left(\frac{a\tau +b}{c\tau + d}\right)$, and fixes $\ff_1$. By reviewing definitions, we see that $\SL_2(\Z)/\Gamma(N)=\SL_2(\Z/N\Z)$ acts on $\ff_N$, and that $-1$ acts trivially. We will show that $\SL_2(\Z/N\Z)/\{\pm 1\}$ is that Galois group of $\ff_N/\ff_1$, by exhibiting a set of functions on whose permutations this group acts faithfully. So, the entire picture so far is: \begin{diagram}[tight,height=2em] && & \overline{\ff_N} \\ &&\ldLine& \dLine \\ K(\E[N])/\{\pm 1\} && & \ff_N \\ &&\rdLine&\dLine \\ && &\ff_1=K \end{diagram} (Recall that $K(\E[N])/\{\pm 1\}$ is the fiexed field of the kernal of $\rh$.) We will show that $K(E[N]) = \ff_N$, at the same time that we show that the Galois group is what we want it to be. To do this, we'll use the Weierstrass $\wp$-function attached to a given lattice $L$ of $\C$ (and thus to the corresponding elliptic curve $E/\C$). So we will pick a specific $\tau \in \H$. (Then sending $t$ to $j(\tau)$ gives us a map from $\E_t$ to $E_\tau$, the elliptic curve over $\C$ with $j$-invariant $j(\tau)$, which is also given by the lattice $\Z+\Z\tau$.) To the lattice $L=\Z+\Z\tau$ we attach $\wp{z}$, which is a modular form of weight 2 satisying the differential equation $$\wp'(z)^2 = \wp(z)^2 - 4g_2(\tau)\wp(z) g_3(\tau),$$ which, replacing $\wp$ by $X$ and $\wp'$ by $Y$, is a Weierstrass equation for $E_\tau$. Thus we have $\wp(\frac{r\tau + s}{N}, \Z+\Z\tau)$, with $r,s \in \Z$ and not both divisible by N, running through the $X$-coordinates of points in $E_\tau[N]$. This form is of weight 2; we need to find modular functions, so let $$f_{(r,s)}(\tau):\tau \mapsto \frac{g_2(\tau)}{g_3(\tau)} \wp\left(\frac{r\tau+s}{N}, \Z+\Z\tau\right).$$ (The $f$ are certainly meromorphic on $\H$ and at the cusps, as $g_2, g_3$ and $\wp$ all are.) $\SL_2(\Z)$ acts on these functions $f$ by $\alpha:f_{(r,s)}(\alpha \tau) = f_{(r,s)\alpha}(\tau)$, where $\alpha$ simply multiplies the row vector $(r,s)$ on the right. Then we can see that $\{f_{(r,s)}\}$ is invariant under the action of $\Gamma[N]$, so they are permuted by $\SL_2(\Z/N\Z)$. In fact, as $\wp$ is an even function ($X(P) = X(-P)$ for any point on the eliptic curve), $f_{(r,s)}$ is also fixed by $\pm 1$, so they are permuted by $\SL_2(\Z/N\Z)/\{\pm1\}$. We have defined $f_{r,s}$ for $r,s \in \Z,$, not both divisible by N. In fact, if $(r,s) = (r', s') \pmod N$, then $f_{(r,s)} = f_{(r', s')}$, so we only need to consider $(r,s) \in \Z/N\Z^2/\{\pm 1\} \backslash (0,0)$ As $f_{(r,s)}(\tau)$ runs over all of the $X$-coordinates of points in $E_\tau[N]$ which are not zero for these $(r,s)$, we must have these $f_{(r,s)}$ all distinct. (There are $N^2 - 1$ non-zero points on $E[N]$, and $X(P) = X(Q)$ iff $P=-Q$, so there are exactly as many pairs $(r,s)$ as there are $X$ coordinates of points in $E[N]$.) Thus we have shown that the non-zero points of $\E[N]$ have $X$-coordinates (in $\overline{\ff_N}$) given by the $f_{(r,s)} \in \ff_N$, and thus $\ff_N = K(\E[N])$. (As a corollary, we have shown that the Galois group of $\ff_N$ over $\ff_1$ is $\SL_2(\Z/N\Z)/\{\pm1\}$, as this group acts faithfullly on the $f_{(r,s)}$.) Thus, finally, we have shown that for any $E/\C(t)$ with $j$-invariant $t$, the associated representation $\rho_E$ has image $\SL_2(\Z/N\Z)$. We know that, if we let $K=\Q(\mu_N)(t)$, $E$ an elliptic curve with $j$-invariant $t$, then $\rho_E$ has image contained in $SL_2(\Z/N\Z)$. (This doesn't work for $K=\Q(t)$, as we need to have $\mu_n \subset K$ in order to have the image of $\rho_E$ contained in $SL_2(\Z/N\Z)$.) Furthermore, if we replace $\C(t)$ by $\Q(\mu_N)(t)$, the image of $\rho_E$ can only get larger. In the following diagram \begin{diagram}[tight,height=2em] & &\C(t)(E[N]) & & \\ &\ldLine^{G'} & &\rdLine & \\ \C(t) & & & &\Q(\mu_N)(t)(E[N]) \\ &\rdLine & &\ldLine^{G} & \\ & &\Q(\mu_N)(t) & & \end{diagram} $G$ and $G'$ are the Galois groups (and also the images of $\rho_E$ over $\Q(\mu_N)(t)$ and $\C(t)$ respectively). Basic Galois theory tells us that $G' \subset G$, but we also know that $G' = \SL_2(\Z/N\Z)$ and $G \subset \SL_2(\Z/N\Z)$, so $G=G'=\SL_2(\Z/N\Z)$. Next, let us consider $E/\Q(t)$, as always with $j$-invariant $t$. Then $\rho_E:\gal(\Q(t)(E[N]) \into \GL_2(\Z/N\Z)$. The determinant of $\rho_E$ is again a cyclotomic character, but is not trivial this time; indeed, it is surjective and thus contains a set of coset representatives for $\SL_2(\Z/N\Z)$ in $\GL_2(\Z/N\Z)$. We see that $\rho_E$ is again surjective, although onto $\GL_2(\Z/N\Z)$ this time. Notice that, in this case, $\Q(t)(E[N])$ contains $\mu_N$, so we now have the diagram \begin{diagram}[tight,height=2em] \overline{\Q} & & & & \Q(t)(E[N]) \\ \dLine & & & &\dLine \\ \Q(\mu_N) & & & &\Q(\mu_N)(t) \\ &\rdLine(2,4)& & &\dLine \\ & & & &\Q(t) \\ & & &\ldLine& \\ & &\Q & & \end{diagram} %$$\begin{array}{ccccccl} %\overline{\Q} & & && && \Q(t)(E[N]) \\ % | & & && && \quad| \\ % \Q(\mu_N) & & && && \Q(\mu_N)(t) \\ % & \sel & && && \quad| \\ % & &\sel&&& &\Q(t) \\ % && &\sel &&\swl & \\ % &&&&\Q&& % \end{array} % $$ Basic Galois theory now tells us that $\overline{\Q} \cap \Q(t)(E[N]) = \Q(\mu_N)$: \pf Let $F$ be the intersection of these two fields. It clearly contains $\Q(\mu_N)$. If $F$ is bigger than $\Q(\mu_N$), then $F(t)$ is bigger than $\Q(\mu_N)(t)$, and thus $$\gal(\Q(t)(E[N])/F(t)) \subsetneq \SL_2(\Z/N\Z) .$$ But we know (as above) that $$\SL_2(\Z/N\Z) = \gal(\C(t)(E[N]/\C(t)) \subset \gal(\Q(t)(E[N])/F(t))$$ so we must have equality above, and thus $F = \Q(\mu_N)$. Recall that $\C(t)(E[N])/\C(t)$ is the function field for $X(N)$. We have shown that $\Q(t)(E[N])$ is a function field for the corresponding curve defined over $\Q(\mu_N)$, so we have defined $X(N)$ over $\Q(\mu_N)$. We can similarly define $X_1(N)$ and $X_0(N)$ over $\Q$. Let $L=K(E[N])$ (where $\Q(t) = K$). Then $\gal(L/K)=\GL_2(\Z/N\Z)$. Consider the subgroup $H$ of this Galois group consisting of all matrices of the form $\left(\begin{array}{cc} *&* \\ 0&*\end{array}\right)$. Then $L^H \cap \overline{\Q} = \Q(\mu_N)^H = \Q$. As $L^H$ is an extension of $\Q$ of transcendence degree 1, it is the function field of a curve defined over $\Q$. It turns out to be the curve $X_0(N)$. To get $X_1(N)$, we would use the subgroup $H= \left(\begin{array}{cc} *&* \\ 0&1\end{array}\right)$. \section*{21 February, 1996} \noindent{Scribe: David M Jones, \tt } \bigskip First of all, we start with a correction from last time. Let $L/K$ be a Galois extension. Let $E_{/L}$ be an elliptic curve. Suppose that $j(E) \in K$ and for all $g\in\gal(L/K)$, ${}^g\!E\isom_{/L} E$. Last time it was stated that we could conclude that there exists ${E_0}_{/K}$ such that $E\isom_{/L} E_0$. This is not correct. We may only conclude that $E\isom_{/{\bar L}} E_0$. The point is that $L = {\bar K}$. \bigskip Let $E_{/K}$ be an elliptic curve. Let $g\in\gal(L/K)$ for some extension $L/K$. Then let $\la_g: {}^g\!E \buildrel {\la_g} \over \isomap E$ be a family of isomorphisms. Now, $\la_{gh}$ and $\la_g \circ {}^g\!\la_h$ are both isomorphisms from ${}^{gh}\!E$ to $E$ so they differ by an element of Aut$(E) = \{\pm 1\}$. \begin{dfn} Let $g,h\in\gal(L/K)$. We define $c(g,h)\in$ Aut$(E)$ by the relation $c(g,h) \la_{gh} = \la_g \circ {}^g\!\la_h$. \end{dfn} \begin{claim} $c(g,h)$ is a 2-cocyle. \end{claim} \pf First, let's rewrite $c(g,h) = \la_g\circ {}^g\!\la_h\circ {\la_{gh}}^{-1}$. Then in order to prove that $c(g,h)$ is a 2-cocycle, we must show the following (see Serre's ``Local Fields", p.113): $${}^g\!c(g',g'')\cdot c(g,g'g'') = c(g,g')\cdot c(gg',g'')$$ Then $c(g,g')\cdot c(gg',g'') = \la_g\circ {}^g\!\la_{g'}\circ {\la_{gg'}}^{-1}\circ\la_{gg'}\circ {}^{gg'}\!\la_{g''}\circ {\la_{gg'g''}}^{-1} = \la_g\circ {}^g\!\la_{g'}\circ {}^{gg'}\!\la_{g''} \circ {\la_{gg'g''}}^{-1} = \la_g\circ {}^g\!(\la_{g'}\circ {}^{g'}\! \la_{g''})\circ{\la_{gg'g''}}^{-1} = \la_g\circ {}^g\!(c(g',g'')\cdot \la_{g'g''})\circ{\la_{gg'g''}}^{-1} = {}^g\!c(g',g'')\cdot\la_g \circ {}^g\!\la_{g'g''}\circ{\la_{gg'g''}}^{-1} = {}^g\!c(g',g'')\cdot c(g,g'g'')$. \qed We want this 2-cocyle $c(g,h)$ to be trivial. \begin{lem} Let $G = \gal(L/K)$. If $L = {\bar K}$, then $H^2(G,\{\pm 1\})\to H^2(G,L^*)$ is injective. \end{lem} \pf Consider the exact sequence $$\0 \rightarrow \{\pm 1\} \rightarrow {\bar K}^* \buildrel {\bullet^2} \over \rightarrow {\bar K}^* \rightarrow \0.$$ Looking at a piece of the long exact cohomology sequence we get $$...\rightarrow H^1(G,{\bar K}^*) \rightarrow H^2(G,\{\pm 1\}) \rightarrow H^2(G,{\bar K}^*)[2] \rightarrow \0.$$ Now $H^1(G,{\bar K}^*) = 0$ by Hilbert Theorem 90 so $H^2(G,\{\pm 1\})$ is isomorphic to the 2-torsion in $H^2(G,{\bar K}^*)$. So $H^2(G,\{\pm 1\})$ certainly injects into $H^2(G,{\bar K}^*)$. Since $L = {\bar K}$, the result follows. \qed \bigskip In order to show that the class of $c(g,h)$ in $H^2(G,\{\pm 1\})$ is trivial, by the above lemma, it is enough to show that the class of $c(g,h)$ in $H^2(G,{\bar K}^*)$ is trivial. To do this we will use \vskip .5cm Differentials: Pick a non-zero differential $\om \neq 0$ on $E$. That is, $\om\in H^0(E,\Om^1)$. Let $g\in\gal(L/K)$. Then $\la_g: {}^g\!E \isomap E$ so the pullback $\la_g^*: H^0(E,\Om^1) \isomap H^0({}^g\!E,\Om^1)$. Thus, $\la_g^*\om\in H^0({}^g\!E,\Om^1)$, which has basis ${}^g\!\om$ so we must have $\la_g^*\om = a_g\cdot {}^g\!\om$ for some $a_g\in L^*$. \begin{note} What is ${}^g\!\om$? Let $E_{/L}$ be an elliptic curve and let $L\hookrightarrow M$ be an injection of fields. Then $H^0(E,\Om^1) \tensor_L M = H^0(E_{/M},\Om^1)$. In our case, $M = L$ and the map $L\hookrightarrow M$ is given by the action of $g\in\gal(L/K)$. So we get $H^0(E,\Om^1)\tensor_L L = H^0({}^g\!E, \Om^1)$. The element of $H^0({}^g\!E,\Om^1)$ which corresponds to $\om\tensor 1$ under this map is ${}^g\!\om$. \end{note} \begin{lem} Show that $c(g,h) = ({}^g\!a_h\cdot a_g)/a_{gh}$. \end{lem} \pf First, consider the following: $a_g\cdot {}^g\!a_h\cdot {a_{gh}}^{-1}\om = a_g\cdot {}^g\!a_h\cdot ((\la_{gh}^*)^{-1} {}^{gh}\!\om) = a_g\cdot (\la_{gh}^*)^{-1}\circ {}^g\!(a_h\cdot {}^h\!\om) = a_g\cdot (\la_{gh}^*)^{-1}\circ {}^g\! (\la_h^* \om) = (\la_{gh}^*)^{-1}\circ {}^g\!(\la_h^*)\circ (a_g {}^g\!\om) = (\la_{gh}^*)^{-1}\circ {}^g\!(\la_h^*) \circ \la_g^* \om = ((\la_{gh})^{-1}\circ {}^g\!(\la_h)\circ\la_g)^* \om$. But $c(g,h) = \la_g\circ {}^g\!\la_h\circ{\la_{gh}}^{-1}$ is multiplication by a constant ($\pm 1$) so the dual is multiplication by the same constant. Thus, we have $a_g\cdot {}^g\!a_h\cdot {a_{gh}}^{-1}\om = c(g,h)\om$ and as $\om\neq 0$, we must have $c(g,h) = (a_g\cdot {}^g\!a_h)/a_{gh}$. \qed This shows that $c(g,h)$ is a coboundary, and thus its class is trivial in $H^2(G,{\bar K}^*)$. \bigbreak\bigbreak\bigbreak Let $N>3$ be an integer. Recall that we wrote down an elliptic curve $E_{/\Q(j)}$ whose $j$-invariant was $j(E) = j$. Let $\ff_N$ be the field of modular functions of level $N$. That is, $\ff_N = \Q(j)(E[N]/\{\pm1\})$. This field lies over $\ff_1 = \Q(j)$. We have $\gal(\ff_N/\ff_1) = \GL_2 (\Z/N\Z)/\{\pm 1\}$. Now define $\ff = \cup_N \ff_N$. This field, which is the compositum of all the of the $\ff_N$'s, corresponds to a projective system of modular curves. \begin{dfn} The finite adele ring of $\Q$, denoted $\A_f$, is $$\{ (x_p)\in\prod_p \Q_p :x_p\in\Z_p \mbox{{\rm { }for almost all }} p\}$$ \end{dfn} We may also describe $\A_f$ as ${\hat \Q} = {\hat \Z}\tensor\Q$. Notice that $\GL_2 ({\hat \Z}) \subset \GL_2 (\A_f)$. \begin{claim} The group $\GL_2 (\A_f)$ acts on $\ff$. \end{claim} First, recall that $\ff = \Q(f_{(r,s)} : (r,s)\in(\Z/N\Z)^2\backslash (0,0), N\geq 1)$. $\GL_2 (\Z/N\Z)$ acts on the set of $f_{(r,s)}$ via $$\abcd : f_{(r,s)}\mapsto f_{(ar + cs,br + ds)}$$ \bigskip Consider the object $\plim X(N)$. It is not a variety but does exist in some appropriate category. We have the following two notions of ``points'' on $\plim X(N)$, which are contra-dual to each other: 1. Pairs $(E,\iota)$ where $E$ is an elliptic curve and $\iota : (\Q/\Z)^2 \isomap E_{\rm tors}$ 2. Pairs $(E,\a)$ where $E$ is an elliptic curve and $\a : {\hat\Z}^2 \isomap Ta(E)$, where $Ta(E) = \plim E[N] = \prod_p Ta_p(E)$ \bigskip We will now do a warm-up to show that $\GL_2(\A_f)$ acts on $\ff$. Let $g = \abcd\in\GL_2^+ (\Q)$. Look at pairs $(E,\a)$ with $E_{/\C}$ and $\a : {\Z}^2 \isomap H_1(E(\C),\Z)$. Note that we have $$\begin{array}{cccc} {\Z}^2 & \isomap & H_1(E(\C),\Z) & \\ \bigcap | & & \bigcap | & \\ {\Q}^2 & \isomap & H_1(E,\Q) & = H_1(E(\C),\Z)\tensor\Q \\ \end{array}$$ We denote this bottom map $\a : {\Q}^2\isomap H_1(E,\Q)$ as well. Then we find $$\begin{array}{cccc} \a\circ g: & {\Q}^2 & \isomap & H_1(E,\Q) \\ & \bigcup & & \bigcup \\ & {\Z}^2 & \isomap & L' \\ \end{array}$$ where $L'$ is defined as the image of ${\Z}^2$ under $\a\circ g$. As we have a new lattice $L'$, it determines a new elliptic curve $E'$. We also get a map $\la : L'\isomap H_1(E',\Z)$. So $g : (E,\a)\mapsto (E',\a')$ where $\a'$ is the composed map $\la\circ\a\circ g : {\Z}^2 \isomap L' \isomap H_1(E',\Z)$. \bigskip Let $\t\in\H, E = E_{\t}, L_{\t} = \Z + \Z\t \isom {\Z}^2$. Let us denote the inverse to this latter isomorphism by $\a_{\t} : {\Z}^2 \isomap \Z + \Z\t$. Under $\a_{\t}$, $(1,0) \mapsto \t$ and $(0,1) \mapsto 1$. \begin{claim} Check that $g : (E_{\t},\a_{\t})\mapsto (E_{\t'},\a_{\t'})$ where $\t' = {{a\t + b}\over{c\t + d}} = g\t$. \end{claim} \pf First of all, $\a_{\t'}$ takes $(1,0)$ to $\t' = {{a\t + b}\over{c\t +d}}$ and takes $(0,1)$ to $1$. Now we look at $\la\circ\a\circ g$. $\GL_2^+(\Q)$ acts on the right on $\Z^2$ so $g$ sends $(1,0)$ to $$(1,0)\cdot\abcd = (a,b)$$ and sends $(0,1)$ to $(c,d)$. Next $\a$ sends $(a,b)$ to $a\t + b = (c\t + d)\t'$ and sends $(c,d)$ to $c\t + d$. Now $\la$ just scales the result so that it is of the form $\a_{\mbox{\rm something}}$. That is, $(0,1)$ gets sent to $1$. This is dividing by $c\t + d$ and we see that under $\la\circ\a\circ g$, $(0,1)$ is sent to $1$ and $(1,0)$ is sent to $\t'$. \qed In the end we want to show that $\GL_2(\A_f)$ acts on the left on functions and on the left on curves. \section*{23 February, 1996} \noindent{Scribe: David M Jones, \tt } \bigskip Today we start of with some philosophy about where we are headed in this course and why we are headed there. We defined modular forms over $\C$ early on in the course. We also discussed a trick by which we could define modular forms over any ring $R$. We also defined the Hecke operators and algebras for these modular forms over $\C$. Recently, we defined, following Shimura, modular curves and modular forms over more ``arithmetic" rings (such as $\Q$). The construction of the modular curves over these rings was done on 2/14 and 2/16 and then the modular form were realized at differentials on the modular curves; elements of $H^0(X,\Om^1)$ where $X$ is the modular curve. Now we are setting out to define the Hecke operators on these modular forms -- to define the Hecke operators and Hecke algebras in a more algebraic (or ``arithmetic") way. \bigskip Recall: For $N'|N$, we had a tower: $$\begin{array}{ccc} X(N) & & \ff_N \\ \downarrow & & \uparrow \\ X(N') & & \ff_{N'} \\ \downarrow & & \uparrow \\ \end{array}$$ And we also have $\ff = \dlim \ff_N$. That is, $\ff$ is the compositum of the fields $\ff_N$. Intuitively, $\ff$, which is a field, corresponds to the ``pro-curve" $\plim X(N)$, which is not an algebraic curve. \bigskip We were discussing the operation of $\GL_2 (\A_f)$, where $\A_f$ is the ring of finite adeles of $\Q$. $\A_f = {\hat \Z}\tensor\Q$ and $\A_f$ contains ${\hat \Z}$ as a subring. Consider a pair $(E,\a)$, where $E_{/\Q}$ is an elliptic curve and $$\a : {\hat \Z}^2 \isomap \prod_p Ta_p(E)$$ We refer to $\prod_p Ta_p(E)$ as $Ta(E)$, the Tate module of $E$. Now, $Ta(E)$ is free of rank $2$ over ${\hat \Z}^2$. Let $V(E) = Ta(E)\tensor_{\Z}\Q = \prod_p V_p(E)$ where $V_p(E) = Ta_P(E) \tensor_{\Z_p}\Q_p$. Since $\a$ maps ${\hat \Z}^2$ to $Ta(E)$, we may extend by $\Q$ to get a map (also called $\a$) ${\hat \Q}^2 \isomap V(E)$. If $g\in\GL_2(\A_f)$, then $g:{\hat \Q}^2\to {\hat \Q}^2$. So by composing $\a$ and $g$, we get a new map ${\hat \Q}^2 \isomap V(E)$. Look at the subring ${\hat \Z}^2$ in ${\hat \Q}^2$, and take its image under $\a\circ g$, and we get a lattice $T'$ in $V(E)$ which is different from $Ta(E)$. The following picture illustrates the idea: $$\begin{array}{ccc} & & Ta(E) \\ & & \bigcap | \\ {\hat \Q}^2 & \buildrel {\a\circ g}\over\isomap & V(E) \\ \bigcup | & & \bigcup | \\ {\hat \Z}^2 & \isomap & T' \\ \end{array}$$ Our goal, as stated earlier, is to make the Hecke operators $T_n$ and $$ into objects defined over $\Q$. The key idea is that $X_1(N)$ and $X_0(N)$ have Hecke operators acing as correspondences. \begin{dfn} A correspondence between two curves $X$ and $Y$ is a third curve $C$ and two maps $\a:C\to X$ and $\b:C\to Y$. \end{dfn} Given a correspondence $(C,\a,\b)$ of curves $X$ and $Y$, we can form the maps $$\a^* : H^0(X,\Om^1)\to H^0(C,\Om^1)$$ and $$\b_* : H^0(C,\Om^1)\to H^0(Y,\Om^1),$$ this latter map $\b_*$ being a trace map of sorts. $\a^*$ is the standard pull-back map. $\b_*$ can be defined using Serre duality. Recall that $H^0(C,\Om^1)^* = H^1(C,\O_C)$ and $H^0(Y,\Om^1)^* = H^1(Y,\O_Y)$. So the dual of $\b_*$ (which may be described as $(\b_*)^*$ if one likes) may be defined via the pull-back $$\b^* : H^1(Y,\O_Y)\to H^1(C,\O_C).$$ \bigskip We will make this explicit for $T_p$, $p$ not dividing $N$. In order to define $T_p$ (for $p$ not dividing $N$) as a correspondence from $X_0(N)$ to itself, we will say what $C$ is and what $\a$ and $\b$ are. Let $C = X_0(pN)$. View $$X_0(N) = \{ \mbox{\rm pairs } (E,C) | C\isom {\Z/N\Z}\}$$ $$X_0(pN) = \{ \mbox{\rm triples } (E,C,D) | C\isom {\Z/N\Z}, D\isom {\Z/p\Z}\}$$ Then we can consider the maps $\a$ and $\b$ as given by: $$\a : (E,C,D)\mapsto (E,C)$$ $$\b : (E,C,D)\mapsto (E/D,(C+D)/D)$$ One must also describe what happens at the cusps but we will gloss over that. We may also describe this complex analitically are follows: $\G_0(pN)\subseteq \G_0(N)$ so $\G_0(pN)\backslash\H \rightarrow \G_0(N)\backslash\H$, which corresponds to $\a: X_0(pN)\to X_0(N)$. Also, conjugation by $\left(\begin{array}{cc} p&0\\0&1\\ \end{array} \right)$ maps $\G_0(pN)$ to $\G_0(N)$ since $$\left(\begin{array}{cc} p&0\\ 0&1\\ \end{array}\right) \left(\begin{array}{cc} a&b\\ pNc&d\\ \end{array}\right) \left(\begin{array}{cc} p&0\\ 0&1\\ \end{array}\right)^{-1} = \left(\begin{array}{cc} a&pb\\ Nc&d\\ \end{array}\right)$$ So we get $\G_0(pN)\backslash \H \to \left(\begin{array}{cc} p&0\\ 0&1\\ \end{array}\right)\G_0(pN) \left(\begin{array}{cc} p&0\\ 0&1\\ \end{array}\right)^{-1}\backslash \H\to \G_0(N)\backslash\H$ and this is $$\b : X_0(pN)\to X_0(N).$$ Now we consider pulling back differentials via these maps. As always, recall that $$H^0(X_0(N),\Om^1) = S_2(\G_0(pN))$$ so as we have $\a,\b : X_0(pN)\to X_0(N)$, we get $\a^*,\b^* : H^0(X_0(N),\Om^1) \to H^0(X_0(pN),\Om^1)$ and thus $\a^*,\b^* : S_2(\G_0(N))\to S_2(\G_0(pN))$. These are given by $\a^* (f) = f$ and, if $f = \sum_{n=1}^{\infty} a_nq^n$, then $\b^* (f) = p\sum_{n=1}^{\infty} a_nq^{pn}$. \begin{exercise} Check that $\b^* (f) = p\sum_{n=1}^{\infty} a_nq^{pn}$ rather than $\b^* (f) = \sum_{n=1}^{\infty} a_nq^{pn}$. \end{exercise} \begin{exercise} Show that $H^0(X_0(N),\Om^1)\to H^0(X_0(N),\Om^1)$ given by $\b_* \circ\a^*$ is the same as that given by $\a_*\circ\b^*$, which both equal $T_p$. One must show that they both give $$T_p : \sum_{n=1}^{\infty} a_nq^n \mapsto \sum_{n=1}^{\infty} a_{pn}q^n + p\sum_{n=1}^{\infty} a_nq^{pn}$$ \end{exercise} Another way to look at this is as follows: Let $\varphi : X\to Y$. Then let $\G =$ the graph of $\varphi \subset X\times Y$. So we get $$\begin{array}{ccccc} \G & \hookrightarrow & X\times Y & \buildrel \a\over\rightarrow & X \\ & & \b \downarrow & & \\ & & Y & & \\ \end{array}$$ as correspondences. We have $\a : \G\to X$ and $\b : \G\to Y$. $\G \buildrel \a\over\rightarrow X$ has degree $d\geq 1$. So $\b(\a^{-1} (x)) \in\Div(Y)$, and deg$(\b(\a^{-1} (x))) = d$. So we have, for each $n$, $\Div^n(X)\to \Div^{dn}(Y)$. For $n = 0$, we get $\Div^0(X)\to \Div^0(Y)$. Applying this to our situation, we have $T_p$ $$\begin{array}{ccc} X_0(pN) & \buildrel \b\over\rightarrow & X_0(N) \\ \a \downarrow & & \\ X_0(N) & & \\ \end{array}$$ And we get $$(E,C) \buildrel {\a^*}\over\mapsto \sum_{D\subseteq E[p]} (E,C,D) \buildrel \b\over\mapsto \sum_{D\subseteq E[p]} (E/D,(C+D)/D)$$ \bigbreak What happens on Jacobians? Given a curve $X$, we get $Jac(X) = J(X)$ which is an abelian variety. If genus$(X) = g$, then $\dim(J(X)) = g$. Over $\C$, $J(X) = H^0(X,\Om^1)^*/H_1(X,\Z)$. $J(X)$ is covariantly associated to $X$. This is the Albanese construction of $J(X)$. There is another construction of $J(X)$ which is contravariant. If $X = Y$, you need to be careful about which construction you use so that correspondences from $X$ to $X$ give the right morphisms $J(X)$ to $J(X)$. \section*{February 26, 28 and March 1, 1996} \noindent{Scribe: Shuzo Takahashi, \tt } \\ The topics covered in this week's lectures are {\bf 1. Generality on Correspondences} {\bf 2. Hecke Operators on $J_0(N)$, $H^0(X_0(N),\Omega^1)$, and $\Cot(J_0(N))$} {\bf 3. Reduction of $J_0(N)\mod p$} {\bf 4. The Eichler-Shimura Relation} {\bf 5. Galois Action on Tate Module} {\bf 6. Shimura Construction of an Abelian Variety Associated to a New Form} {\bf 7. Galois Representations Associated to New Forms of Weight 2} \bigbreak \noindent {\bf\large 1. Generality on Correspondences} Before considering Hecke operators on $J_0(N)$, $H^0(X_0(N),\Omega^1)$, and $\Cot(J_0(N))$, let's consider correspondences in a general setting. Consider a correspondence \begin{diagram}[tight,height=2em] &\Gamma& \\ \ldTo(1,2)^{\alpha}& &\rdTo(1,2)^{\beta} \\ X & &Y \end{diagram} This induces naturally a map $$\alpha_*\circ\beta^* : H^0(Y,\Omega^1) \rightarrow H^0(X,\Omega^1).$$ Since $X \rightarrow J(X)$ induces $\Cot(J(X)) \simeq H^0(X,\Omega^1)$, the above map induces $$\Cot(J(Y)) \rightarrow \Cot(J(X)).$$ Also, the correspondence induces $$\beta_* \circ \alpha^* : J(X) \rightarrow J(Y)$$ by the Albanese functoriality of $J(.)$. Moreover , if we have $\psi : J(X) \rightarrow J(Y)$, then we have $\psi^{\vee} : J(Y)^{\vee} \rightarrow J(X)^{\vee}$. However since we have the canonical duality $J(X) \cong J(X)^{\vee}$, $\psi$ induces $J(Y) \rightarrow J(X)$. We have $(\beta_* \circ \alpha^*)^\vee = \alpha_* \circ \beta^*$. \bigbreak \noindent {\bf\large 2. $T_p$ on $J_0(N)$, $H^0(X_0(N),\Omega^1)$, and $\Cot(J_0(N))$} Assume $p \not| N$. Consider a correspondence \begin{diagram}[tight,height=2em] &X_0(pN)& \\ \ldTo(1,2)^{\alpha}& &\rdTo(1,2)^{\beta} \\ X_0(N) & &X_0(N) \end{diagram} where $\alpha$ and $\beta$ are degeneracy maps (Consider a point of $X_0(pN)$ as $(E,\Z/p\Z)$, regarding a group of order $N$ as a part of $E$.). This correspondence is by definition $T_p$. $T_p$ naturally induces $$T_p^* = \alpha_*\circ\beta^* : H^0(X_0(N),\Omega^1) \rightarrow H^0(X_0(N),\Omega^1).$$ Identifying $H^0(X_0(N),\Omega^1)$ and $S_2(\Gamma_0(N))$, this $T_p^*$ is the same as the usual $T_p$. $$ \begin{array}{cccc} H^0(X_0(N),\Omega^1) & \simeq & S_2(\Gamma_0(N)) \\ T_p^* \downarrow & & \downarrow T_p \\ H^0(X_0(N),\Omega^1) & \simeq & S_2(\Gamma_0(N)) \end{array}$$ Now consider the Jacobian of $X_0(N)$ = $J_0(N)$ which is an abelian variety over $\Q$. $\alpha : X_0(pN) \rightarrow X_0(N)$ induces $\alpha^* : J_0(N) \rightarrow J_0(pN)$ by the Picard functoriality and $\alpha_* : J_0(pN) \rightarrow J_0(N)$ by the Albanese functoriality. \begin{thm} The action of $\alpha_*\circ\alpha^*$ on $J_0(N)$ is multiplication by $p+1$. \end{thm} \begin{thm} We have $\beta_*\circ\alpha^* = \alpha_*\circ\beta^*$. Let $T_p = \beta_*\circ\alpha^*$. Then $T_p$ is an endmorphism of $J_0(N)$. ($T_p$ is called $\xi_p$ in Shimura's book.) \end{thm} $\xi_p$ induces $$\xi_p^* : \Cot(J_0(N)) \rightarrow \Cot(J_0(N))$$ Next, consider $\theta : X_0(N) \rightarrow J_0(N)$, mapping $x$ to the class of $(x) - (\infty)$. $\theta$ induces $$\theta^* : \Cot(J_0(N)) \simeq H^0(X_0(N),\Omega^1)$$ Identifying $\Cot(J_0(N))$ and $H^0(X_0(N),\Omega^1)$, $\xi_p^*$ is the same as $T_p^*$. $$ \begin{array}{cccc} \Cot(J_0(N)) & \simeq & H^0(X_0(N),\Omega^1) \\ \xi_p^* \downarrow & & \downarrow T_p^* \\ \Cot(J_0(N)) & \simeq & H^0(X_0(N),\Omega^1) \end{array}$$ So far we have considered the case $p \not| N$. If $p | N$, we need to choose between $\beta_*\circ\alpha^*$ and $\alpha_*\circ\beta^*$. The usual choice $\xi_p^*$ induces the usual $T_p$ on $S_2(\Gamma_0(N))$. Now we have $\xi_p \in \endo(J_0(N))$ for every $p$. $\T = \Z[\cdots T_p \cdots] \subseteq \endo(J_0(N))$ is the same ring as $\T_{\Z} \subseteq \endo(S_2(\Gamma_0(N)))$. \bigbreak \noindent {\bf\large 3. Reduction of $J_0(N) \mod p$} Igusa showed that for all $p \not| N$, $J_0(N)$ over $\Q \supset \Z[1/N]$ has a good reduction mod $p$ considering $X_0(N)$ over $\Z[1/N]$. On the other hand, $X_0(pN)/\F_p$ is obtained by attaching the two copies of $X_0(N)/\F_p$ at their supersingular points (those which arise from supersingular elliptic curves over $\overline{\F}_p$) a supersingular point $x$ on the first copy being identified with its Frobenius transform $x^{(p)}$ on the second. \bigbreak \noindent {\bf\large 4. The Eichler-Shimura Relation} Suppose $p \not| N$. $J_0(N)$ has a good reduction mod $p$. We have $$T_p \in \endo(J_0(N)) \hookrightarrow \endo(J_0(N)/\F_p).$$ Frobenius morphism $F: X_0(N)/\F_p \rightarrow X_0(N)/\F_p$ of degree $p$ induces $$\frob: J_0(N)/\F_p \rightarrow J_0(N)/\F_p$$ by covariant functoriality, and $$\ver: J_0(N)/\F_p \rightarrow J_0(N)/\F_p$$ by contravariant functoriality. Since the degree of $F$ is $p$, we have \begin{thm} $\ver \circ \frob = \frob \circ \ver = p$ \end{thm} The Eichler-Shimura relation is \begin{thm} $T_p = \frob + \ver$. \end{thm} \noindent {\bf Note.} If we use $X_1(N)$, then we have $T_p = \frob + \langle p \rangle \ver$. \noindent {\bf Idea of Proof.} First, by the work of Deligne-Rapport, the reduction of $X_0(pN)$ $\mod p$ can be considered as two copies of $X_0(N)/\F_p$. Consider points on $X_0(pN)$ as isogenies $(E \rightarrow E')$. One copy of $X_0(N)/\F_p$ is obtained by the map $$r : E \mapsto (E {\buildrel \frob\over\longrightarrow} E^{(p)}).$$ The other copy is obtained by the map $$s : E \mapsto (E^{(p)} {\buildrel \ver\over\longrightarrow} E).$$ Through these maps, we have $$X_0(pN)/\F_p \cong \frac{X_0(N)/\F_p \coprod X_0(N)/\F_p} {{\rm \ identifying\ supersingular\ points\ of\ } X_0(N)/\F_p {\rm \ via\ }\frob}$$ Consider $\alpha : (E \rightarrow E') \mapsto E$ and $\beta : (E \rightarrow E') \mapsto E'$. Then we have \begin{diagram}[tight,height=2em] X_0(N) & & & &X_0(N) \\ \dLine &\rdTo(2,2)_r & &\ldTo(2,2)_s &\dLine \\ id & &X_0(pN)& &id \\ \downarrow &\ldTo(2,2)^{\alpha}& &\rdTo(2,2)^{\beta}&\downarrow\\ X_0(N) & & & &X_0(N) \end{diagram} where $\alpha \circ r = id = \beta \circ s$ and $\alpha\circ s = F = \beta\circ r$. Thus the correspondence $T_p$ can be defined by two correspondences as follows: \begin{diagram}[tight,height=2em] &X_0(pN)& & & &X_0(N) & & & &X_0(N) & \\ \ldTo(1,2)^{\alpha}& &\rdTo(1,2)^{\beta}&=& \ldTo(1,2)^{id} & &\rdTo(1,2)^{F} &+& \ldTo(1,2)^{F} & &\rdTo(1,2)^{id} \\ X_0(N) & &X_0(N) & & X_0(N) & &X_0(N) & & X_0(N) & &X_0(N) \end{diagram} The first diagram of the right hand side is the graph of the Frobenius $F: X_0(N) \rightarrow X_0(N)$ and the second one is its dual. From this, one can deduce the Eichler-Shimura relation. \bigbreak \noindent {\bf\large 5. Galois Action on Tate Module} Look at $Tate_l(J_0(N)) = \plim J_0(N)[l^\nu] \tensor_{\Z_l} \Q_l$ ($l \neq p$). Let's call this $X$. This is the same in char $p$ as in char $0$. \begin{thm} We have operators $T_p$, $\frob_p$, and $\ver_p$ on $X$ satisfying the following: (a) $T_p = \frob_p + \ver_p$, (b) $\frob_p\ver_p = p$, (c) $T_p = \frob_p + p{\frob_p}^{-1}$, (d) $\frob_p^2 - T_p\frob_p + p = 0$. \noindent Moreover $X$ has actions of (1) $\gal(\overline{\Q}/\Q)$, (2) $\endo_\Q J_0(N) \tensor_\Z \Q_l$, (3) $\endo_{\F_p} J_0(N) \tensor_\Z \Q_l$, (4) $\gal(\overline{\F_p}/\F_p)$. \noindent (1) and (2) commute, and (3) and (4) commute. (4) is a special case of (1) because we have a map $D_p \onto \gal(\overline{\F_p}/\F_p)$ where $D_p \subset \gal(\overline{\Q}/\Q)$ is a decomposition group. \end{thm} \noindent {\bf Point.} There is a Frobenius element $\varphi_p \in \gal(\overline{\F_p}/\F_p)$ and a Frobenius endmorphism $\frob_p \in \endo_{\F_p} J_0(N) \tensor_\Z \Q_l$. On Tate module $X$, $\frob_p$ acts as $\varphi_p$, and we have $$\varphi_p^2 - T_p\varphi_p + p = 0$$ which reminds Cayley-Hamilton theorem with $\det \varphi_p = p$ and $\tr \varphi_p = T_p$. Also, we have $$T_p = \varphi_p + p {\varphi_p}^{-1}.$$ \bigbreak \noindent {\bf\large 6. Shimura Construction of an Abelian Variety Associated to a New Form} Consider $\T_0 = \Z[\cdots T_n \cdots]$ where $(n,N) = 1$. We have $$\T_0 \tensor \Q \simeq \prod_f E_f$$ where $f$ runs over the set of new forms of level dividing $N$ modulo conjugation by $\gal(\overline{\Q}/\Q)$. Also, we have $$\T \tensor \Q \simeq \prod_f {\cal A}_f$$ where ${\cal A}_f$ is an $E_f$ algebra. \begin{exercise} Show $\dim_{E_f} {\cal A}_f = \# {\rm \ of\ divisors\ of\ } \frac{N}{N(f)}$ where $N(f)$ is the level of $f$. \end{exercise} \begin{eg} Suppose $N = N(f)q$ where $q \not| N(f)$. Then we have $${\cal A}_f = E_f[U]/(U^2 - a_q U + q).$$ \end{eg} \begin{exercise} Write ${\cal A}_f$ in general as $$E_f[\cdots U_q \cdots]/({\rm relations})$$ where $q | \frac{N}{N(f)}$. \end{exercise} \noindent {\bf More explanation about the above example.}\\ ${\cal A}_f$ is the algebra generated by the $T_p$ acting on the $\C$ vector space generated by the forms $f(d\tau)$ where $d | \frac{N}{N(f)}$. $T_p$ acts as the scalar $a_p$ if $p \not| N$. If $q | N$, $T_q = U_q$ acts as $\left( \begin{array}{cc} a_q&1\\-q&0 \end{array} \right)$. \noindent {\bf An abelian subvariety $A_f$ of $J_0(N)$}.\\ Let $\pi_f$ be the projection of $\T_0 \tensor \Q \simeq \prod_f E_f$ over $E_f$. Then define $A_f = \pi_f(J_0(N))$ ($\pi_f$ is not exactly an endmorphism but is in $\endo(J_0(N)) \tensor \Q$). In the category of abelian varieties up to isogeny, $\prod {\cal A}_f$ acts on $J_0(N)$ and pulls apart $J_0(N)$ into $\prod_f A_f$. We have $\sum_f A_f = J_0(N)$ and $\prod_f A_f \rightarrow J_0(N)$ is an isogeny. Also, $J_0(N)/\C$ can be considered as $$J_0(N) = \Hom(S_2(\Gamma_0(N)),\C)/H_1(X_0(N),\Z)$$ (call it $V/L$). Then $V = \bigoplus V_f$ and $A_f = V_f/(V_f\cap L)$ where the basis of $V_f$ consists of $f(d\tau)$'s as well as conjugates of them. \noindent {\bf Special case when the level of $f$ is $N$.}\\ Now, let's consider a special case where $f$ is a new form ($N(f) = N$). Then we have ${\cal A}_f = E_f$. Also $d = \dim A_f = \dim_\Q {\cal A}_f = [E_f : \Q]$. Fix $f$ and let $A = A_f$ and $E = E_f$. Let $Tate_l$ be the $\Q_l$-adic Tate module of $A_f \cong \Q_l^{2d}$. Action of $E \tensor_\Q \Q_l$ on $Tate_l$ is $\prod_{\lambda | l} E_\lambda$. Thus $Tate_l = \prod_{\lambda | l} Tate_\lambda$ where $Tate_\lambda$ is an $E_\lambda$ vector space. We have \begin{lem} $\dim_{E_\lambda} Tate_\lambda = 2$ for all $\lambda | l$. That is, $Tate_l$ is free of rank 2 over $E \tensor \Q_l$. \end{lem} \noindent \pf $A_f = V_f/L_f$ where $L_f = V_f \cap L$. $L_f \tensor \Q$ affords an action of $E_f$ (a field). So $L_f \tensor \Q$ is a vector space over $E_f$ of $\dim = 2$. On the other hand, $Tate_l$ is $(L_f \tensor \Q) \tensor_\Q \Q_l$. \bigbreak \noindent {\bf\large 7. Galois Representations Associated to New Forms of Weight 2} $\gal(\overline{\Q}/\Q)$ acts on $Tate_l$ which is compatible with $E \tensor \Q_l$. That is, we have $$\gal(\overline{\Q}/\Q) {\buildrel \rho_l\over\rightarrow} Aut_{E\tensor\Q_l} Tate_l = \GL(2,E\tensor\Q_l) = \prod_\lambda \GL(2,E_\lambda).$$ Then we have $$\rho_l = \bigoplus \rho_\lambda$$ where $$\rho_\lambda : \gal(\overline{\Q}/\Q) \rightarrow Aut_{E_\lambda} Tate_\lambda$$ If $p \not| lN$, then $\rho_\lambda$ is unramified at $p$ and $\rho_\lambda(\varphi_p)$ has a well defined (trace, det) characteristic polynomial $\Phi(X)$ given as \begin{thm} $$\Phi(X) = X^2 - a_p X + p$$ where $a_p$ is the image of $T_p$ in $E_f$. \end{thm} \noindent {\bf Idea of Proof.} By Cayley-Hamilton, we have (1) $\varphi_p^2 - (\tr\rho_\lambda(\varphi_p))\varphi_p + \det\rho_\lambda(\varphi_p) = 0$ \noindent By Eichler-Shimura, we have (2) $\varphi_p^2 - a_p\varphi_p + p = 0$ \noindent Thus it suffices to prove $(\det\rho_\lambda)(\varphi_p) = p$ which is equivalent to $\det\rho_\lambda = \chi_l$ by Cebotarev density theorem where $\chi_l : \gal(\overline{\Q}/\Q) \rightarrow \Z_l^* \subseteq E_\lambda^*$ is the $l$-adic cyclotomic character. A complete proof of this theorem will be given next week. \section*{March 4, 1996} \noindent{Scribe: J\'anos Csirik, \tt } \bigskip Last time we had some weight $2$ eigenform $f$ on $\G_0(N)$ of exact level $N$. ($\G_1(N)$ would give an interesting variant which we shall consider briefly at the end of the lecture.) We found some abelian subvariety $A_f\subseteq J_0(N)$, it has an action on it by $\T\otimes\Q$ which acts through the quotient $E_f=\Q(\ldots a_n(f)\ldots)$, and $[E_f:\Q]=\dim\,A_f$. $E_f$ is a totally real algebraic number field. We have discovered that $\tate_\ell(A_f)$ is free of rank $2$ over the $\ell$-adic completion $E_f\otimes\Ql=\prod_{\la|l}E_\la$, and the Tate module breaks apart thus: \[ \tate_\ell = \prod_\la\tate_\la. \] We also have \[ \rho_\la : \GQ \to \aut_{E_\la}\tate_\la = \GL(2,E_\la). \] For $p\!\!\not|\,\ell N$, $\rho_\la(\frob_p)$ satisfies \[ X^2 - a_pX + p. \] (Note that $a_p$ is well-defined as $E_f\inj\C$ via $f$: $a_p$ is a certain eigenvalue $\in\C$.) By the Cayley-Hamilton theorem, $\rho_\la(\frob_p)$ also satisfies \[ X^2 -\tr\,\rho_\la(\frob_p) +\det\,\rho_\la(\frob_p) =0. \] \begin{thm} (1) $\tr\,\rho_\la(\frob_p)=a_p$. (2) $\det\,\rho_\la(\frob_p)=p$. \end{thm} Note that (2) implies (1) by the Cayley-Hamilton theorem: use that $\rho_\la(\frob_p)$ is an invertible matrix and subtract the above two polynomials that it is known to satisfy. To prove (2), we need the following ``true theorem'': \begin{thm} Using $\chi_\ell$ to denote the $\ell$-adic cyclotomic character, \[ \det\,\rho_\la = \chi_\ell. \] \end{thm} Once this theorem is proved, we can substitute $\frob_p$ (on both sides) and get (2) above. \bigskip Before tackling the true theorem, we need to talk about a generalization of the Weil pairing that we'll need in the proof. We have an isogeny \[ A_f \inj J_0(N) \isomap J_0(N)^\vee \to A_f^\vee. \] This is a polarisation. Generally, for a polarisation $\phi_L$ of an abelian variety $A$ defined by the line bundle $L$, we need \begin{eqnarray*} \phi_L: A & \rightarrow & {\rm Pic}^0\,A=A^\vee \\ a & \mapsto & T_a^*L\otimes L^{-1}, \end{eqnarray*} where $T_a$ stands for translation by $a$. By the theorem of the square, this is a homomorphism. We also require that $\phi_L$ be an isogeny ($L$ needs to be ample). The Weil pairing has a natural generalization to $A=A_f$: For each $n$, it is a natural map \[ A[n] \times A^\vee[n] \to \mu_n \] which is a bilinear perfect pairing. Given $\phi:A\to A^\vee$, we get \[ A[n] \times A[n] \to \mu_n \] which is an alternating, but not perfect, pairing. We can make it perfect by setting $n=\ell^r$ and taking $\plimr$: \[ (.,.) : \tate_\ell A \times \tate_\ell A \to \Ql(1) = (\plimr\mu_{\ell^r})\otimes_{\Z_\ell}\Ql. \] In fact for a map of abelian varieties $t:A\to B$, we have for $\b\in B^\vee$ and $a\in A$, \[ (ta,\b)_B = (a,t^\vee\b)_A. \] Now take $A=B$ and recall our isogeny $\phi:A\to A^\vee$ above. Then taking $t\in\T_0\otimes\Q$, we get \[ (ta,a')=(a,ta'). \] To put this into context, note that since $\phi$ is an isogeny, it has an inverse if we throw in denominators, so we can take $t\in\endo\,A$ into $t^*\in\endo\,A\otimes\Q$. This is called the {\em Rosati involution} associated to the polarisation $\phi$. It follows easily that in this case \[ (ta,a') = (a,t^*a'). \] Looking at $J_0(N)$ and its natural polarisation, if $t\in\T$ comes from a correspondence then $t^*$ comes from its dual. In our case, $\phi=w$ is an involution, so $t^*=wtw$. We have \[ \T_0\otimes\Q\subseteq\T\otimes\Q\subseteq\endo(J_0(N)) \] and ${}^*$ acts as the identity on $\T_0\otimes\Q$. In general, $wtw$ is not always in $\T\otimes\Q$: if $t=T_p$ for $p|N$, $t^*$ does not commute with $t$. Note that \[ \dim_\Q\T\otimes\Q = \dim_\C S_2(\G_0(N)) \] and \[ \dim_\Q\T_0\otimes\Q = \sum_{d|N}\dim_\C S_2(\G_0(d))^{\rm new}. \] \bigskip Let's try to prove our determinant assertion now. Restrict \[ (.,.) : \tate_\ell A \times \tate_\ell A \to \Ql(1) \] to \[ \langle.,.\rangle : \tate_\la A \times \tate_\la A \to \Ql(1). \] This is an alternating pairing. Since the projectors come from elements of $E$, which are their own inverses with respect to the pairing, this pairing is also non-degenerate. The pairing in Galois-invariant so for $\si\in\GQ$ and $x,y\in\tate_\la$, $\langle\si x,\si y\rangle = {}^\si\!\langle x,y\rangle$, and since $\si$ acts on $\Ql(1)$ by multiplication by $\chi_\ell(\si)$, we get \[ {}^\si\!\langle x,y\rangle = \chi_\ell(\si)\langle x,y\rangle. \] {\sc Fancy proof.} $\langle tx,y\rangle=\langle x,ty\rangle$ for $t\in E_\la$, so $\langle.,.\rangle$ factors through $D=\bigwedge^2_{E_\la}\tate_\la$: \[ \langle.,.\rangle : D \to \Ql(1). \] The left hand side is one dimensional over $E_\la$. So we get \[ \langle.,.\rangle\in\Hom_\Ql(D,\Ql(1)) =\Hom_\Ql(D,E_\la(1)), \] where the equality is through composition with the trace. So we got $D\cong E_\la(1)$, the left hand side the determinant of our representation, the right hand side acted on by $\chi_\ell$. So we proved \[ \det\,\rho_\la = \chi_\l. \] {\sc Non-fancy proof.} Let $\si\in\GQ$, and take $E_\la$-linearly independent vectors $x,y\in\tate_\ell$. Set \begin{eqnarray*} \si x & = & a(\si)x + b(\si)y \\ \si y & = & c(\si)x + d(\si)y. \end{eqnarray*} We want to show that \[ \d=\d(\si)=a(\si)d(\si)-b(\si)c(\si)-\chi_\ell(\si)=0. \] Observe that \[ \langle ax,dy \rangle = \langle adx,y \rangle, \] and \[ \langle by,cx \rangle = \langle bcy,x \rangle = -\langle x,bcy \rangle = -\langle bcx,y \rangle. \] Then since \[ \langle bdy,y \rangle = \langle by,dy \rangle = -\langle dy,by \rangle = -\langle bdy,y \rangle, \] if follows that \[ \langle by,dy \rangle = 0, \] and similarly \[ \langle ax,cx \rangle = 0. \] Therefore \[ \chi(\si)\langle x,y \rangle = \langle \si x,\si y \rangle = \langle ax+by,cx+dy \rangle = \langle (ad-bc)x,y \rangle, \] So $\langle \d x,y \rangle = 0$ for any choice of linearly independent vectors $x,y$. So we proved once again that \[ \d = 0. \] \subsubsection*{The case of $\G_1(N)$} For a newform $f$, we have $f|T_n=a_n(f)$ for all $n\geq1$, and $f|\langle d\rangle=\ve(d)f$ for $d\in(\Z/N\Z)^*$. Repeating the construction we get an abelian variety $A_f$ with an action of a number field $E_f$. However in this case $E_f$ is not necessarily real: but if it isn't then it is equipped with a canonical complex conjugation: $\overline{a_n}=\ve(n)^{-1}a_n$. Also we have $\ve(d)\in E_f^\times$. In this case, we have \begin{thm} (1) $\tr\,\rho_\la(\frob_p)=a_p$. (2) $\det\,\rho_\la(\frob_p)=\ve(p)p$. \end{thm} This can be proved from the following: \begin{thm} $\ve$ can be considered representation of the Galois group. Still using $\chi_\ell$ to denote the $\ell$-adic cyclotomic character, \[ \det\,\rho_\la = \ve\chi_\ell. \] \end{thm} {\sc Signs.} There is less gratuitous self-duality going on here than in the case of $\G_0(N)$, so we shall have two occasions to make choices. For every $p$, $T_p$ is defined as a correspondence on $X_1(N)$, with the same $\a$ and $\b$ that we have used before. Do we want to do $(T_p)^*$ or $(T_p)_*$? We have another choice in Eichler-Shimura. What does $X_1(N)$ classify? {\sc Either} an elliptic curve $E$ and a point $P$ of order $N$ (which amount to a pair $(E,\Z/n\Z\inj E)$); {\sc Or} a pair $(E,\mu_n\inj E)$. The second one is more suitable for us (and for use with the Tate curve), and it forces us to use the Picard functoriality of the Jacobians. Therefore we shall make $T_p:J_1(N)\to J_1(N)$ so that $(T_p^\vee)^*$ on $H^0(X_1(N),\Om^1)=S_2(\G_1(N))$ is the classical $T_p$. (Here regard $T_p^\vee$ as an element of $\endo(J_1(N))$ by the autoduality of $J_1(N)$.) \section*{6 March, 1996} \noindent{Scribe: David M Jones, \tt } \bigskip Today we will discuss the process of reducing a $\la$-adic representation mod $\la$. Recall: Let $f = \sum_{n=1}^{\infty} a_nq^n$ be a weight 2 newform on $\G_0(N)$. That is, $f$ has trivial Nebentypus and is new of level $N$. We looked at the field $E = \Q(...a_n...)$. For each prime $\la$ of $E$ lying above a fixed rational prime $\l$, let $E_{\la}$ be the completion of $E$ at the valuation corresponding to $\la$. Then we created a representation $$\rho_{\la} : \gal({\bar\Q}/\Q)\to \GL_2(E_{\la})$$ such that $\det\rho_{\la} = \chi_{\l}$, the $\l$-adic cyclotomic character, and $\tr\rho_{\la}(\frob_p) = a_p$ for $p\nmid \l N$. \bigskip \begin{lem} $\rho_{\la}$ is equivalent to a representation with values in $\GL_2 (\O)$, where $\O =$ the ring of integers in $E_{\la}$. \end{lem} \pf An equivalent statement to the lemma is: there is an $\O$-lattice $L$ in $V$ (the vector space on which $\gal({\bar\Q}/\Q)$ acts via $\rho_{\la}$) which is $\gal({\bar\Q}/\Q)$-stable. Pick any lattice $L_0$ in $V$. The set of lattices $gL_0$ with $g\in \gal({\bar\Q}/\Q)$ is finite. This is because $\gal({\bar\Q}/\Q)$ is compact and $L_0$ is discrete. So we take $L = \sum_g gL_0$, which is stable under Galois. \qed So we are led to ask the following Question: Can we define ${\bar{\rho_{\la}}} = \rho_{\la} \mod\la$ by taking an equivalent representation as in the lemma and reducing it $\mod\la$? More specifically, let $\F = \O/\la$, a finite field. Then the map $\O\to\F$ induces $\GL_2(\O)\to\GL_2(\F)$. Can we define ${\bar{\rho_{\la}}}$ by the composition $$\si : \gal({\bar\Q}/\Q)\to\GL_2(\O)\to\GL_2(\F)?$$ The answer is no, because this map depends on the choice of $L$ in the lemma. So we do the following: \begin{dfn} ${\bar{\rho_{\la}}} = (\si : \gal({\bar\Q}/\Q)\to\GL_2(\F))^{\rm ss}$, the semisimplification of the map described above. That is, 1. If $\si$ is irreducible, ${\bar{\rho_{\la}}} = \si$. 2. If $\si\sim \left(\begin{array}{cc} \a&*\\ 0&\b \\ \end{array}\right)$, then $\si^{\rm ss} = \a\oplus\b$. \end{dfn} \noindent Now we list the key properties of ${\bar{\rho_{\la}}}$: (a) ${\bar{\rho_{\la}}}$ is semisimple. (b) $\det{\bar{\rho_{\la}}} = (\chi_{\l} \mod\l)$. (c) $\tr{\bar{\rho_{\la}}}(\frob_p) = (a_p\mod\la)\in\F, \forall p\nmid \l N$. \begin{thm} (Brauer-Nesbitt Theorem) Suppose $\rho, \rho'$ are two representations as above, both semisimple, with the same traces and determinants, then $\rho\isom\rho'$. \end{thm} So ${\bar{\rho_{\la}}}$ does not depend on the lattice $L$; it is well-defined. Now let's view this situation in a new way: $T_n\in \endo(S_2(\G_0(N))), \T = \Z[...T_n...]\subseteq\endo(S_2(\G_0(N)))$ Take $\M\subseteq\T$, a maximal ideal, and $\T/\M = k = \F_{\l^{\nu}}$. \begin{prop} There exists a (unique) semisimple representation $$\rho_{\M} : \gal({\bar\Q}/\Q)\to\GL_2(k)$$ such that: 1. $\rho_{\M}$ is unramified outside $\l N$. 2. $\tr\rho_{\M} (\frob_p) = T_p \forall p\not| \l N$. 3. $\det\rho_{\M} (\frob_p) = p \forall p\nmid \l N$. \end{prop} \pf First of all, it is enough to prove this for $\T_0 = \Z[...T_n...] \subseteq\T$, where $\T_0$ is formed using only those $T_n$ with $(n,N) = 1$. Then $\T_0/\M_0 = k_0\subseteq k$. $$\begin{array}{cccc} \T_0\tensor\Q & = & E_1\times ...\times E_t & \\ \bigcup | & & \bigcup | & \\ \T_0 & \subseteq & \O_1\times ...\times \O_t & = \O \\ \bigcup | & & & \bigcup | \\ \M_0 & = & & \la \\ \end{array}$$ We can make ${\bar{\rho_{\la}}} : \gal({\bar\Q}/\Q)\to\GL_2(\O/\la)$. This has the required properties of $\rho_{\M}$ except that it takes values in $\GL_2(\O/\la)\supseteq \GL_2(k_0)$. $\O/\la$ being bigger than $k_0$ is a potential obstruction to defining $\rho_{\M} = {\bar{\rho_{\la}}}$. But, the characteristic polynomials of ${\bar{\rho_{\la}}}(g)$ have coefficients in $k_0$, and since the Brauer group of a finite field is trivial, there is a model for ${\bar{\rho_{\la}}}$ over $k_0$. We take $\rho_{\M}$ to be this model. \qed Let us digress in order to explain in a different way what we've just done. Recall: $\T_0\tensor\Q = E_1\times ...\times E_t$ and each factor $E_i$ corresponds to a newform of level $M$ for some $M|N$. Suppose $E_1$ corresponds to a newform of level $N$. We get $J_0(N)\sim A_1\times ...\times A_t$, where we have $\dim A_1 = [E_1:\Q]$. Also, $Ta_{\l} A_1$ is free of rank 2 over $E_1\tensor_{\Q} \Q_{\l}$. But if $E_i$ corresponds to a form which is new of level $M, M|N, M\neq N$, then this doesn't work. We must go back to $\G_0(M)$. \bigskip Question: Let $A = A_1 \subseteq J_0(N)$. Then we have $\T\to\endo(A_1)$. Let ${\bar\T}$ be the image of this map. So ${\bar\T}\subseteq\O_1$, but ${\bar\T}$ may not be integrally closed. But there exists a $\Z_{\l}$-adic Tate module: $H_{\l}(A) = \prod_{\M} H_{\M}(A)$ which is isomorphic as a group to $\Z_{\l}^{2\dim A}$. $H_{\l}(A)$ has an action of ${\bar\T}\tensor\Z_{\l} = \prod_{\M} {\bar{\T}}_{\M}$. For $\M |\l$, we have $H_{\M}$ free of rank 2 over ${\bar{\T}}_{\M}$. \bigskip Let $J = J_0(N), \T\subseteq\endo J$. Let $\M\subseteq\T$ be a maximal ideal. Then define $J[\M] = \{ t\in J({\bar\Q}) | xt = 0 \forall x\in\M\}$. Then $J[\M]\subseteq J[\l]$, and $J[\l]$ is isomorphic as a group to $(\Z/\l\Z)^{2g}$. $J[\l]$ is a $\T/\l\T$-module. But $J[\M]$ is ``better" because it is a $\T/\M$-module and $\T/\M$ is a field, so $J[\M]$ is a vector space. $J[\M]$ is a representation of $k[G] = (\T/\M)[\gal({\bar\Q}/\Q)]$. Naive Idea: $J[\M]$ is a model for $\rho_{\M}$, at least when $\rho_{\M}$ is irreducible. \begin{thm} If $\l\neq 2$ and $\l\nmid N$, then $J[\M]$ is a model for $\rho_{\M}$. \end{thm} Note: $J[\M]$ may indeed be a model for $\rho_{\M}$ in other cases. Let $V = k\oplus k$ with the $\rho_{\M}$-action of $\gal({\bar\Q}/\Q)$. Fact: $\dim_{\T/\M} H^0(X_0(N)_{/\F_{\l}},\Om^1)[\M]\leq 1$. \section*{8 March, 1996} \noindent{Scribe: David M Jones, \tt } $\T = \Z[...T_n...]\subseteq \endo J_0(N)$. Let $\M\subset\T$ be a maximal ideal with $\M |\l$. As we showed last time, there exists a representation $$\rho_{\M} : \gal({\bar\Q}/\Q)\to\GL_2(\T/\M)$$ which is semisimple. \begin{dfn} Let $R$ be a $\Z_{\l}$-algebra. Then we say that $R$ is Gorenstein if $\Hom_{\Z_{\l}}(R,\Z_{\l})\isom R$. \end{dfn} Goal: To prove that if $\M\nmid 2N$ and $\rho_{\M}$ is irreducible, then $\T_{\M}$ is Gorenstein. \begin{thm} If $J = J_0(N), J[\M]\subseteq J[\l]$, then $\dim_{\T/\M}J[\M] = 2$. That is, $\rho_{\M}$ arrises from $J[\M]$. \end{thm} Remarks: Let $M = \Hom(\T,\Z) = (S_2(\G_0(N)),\Z)$, a $\T$-module. Saying that $M_{\M}$ is Gorenstein amounts to $M_{\M}$ being free of rank 1 over $\T_{\M}$. Let $W = J[\M]$. We will compare $W$ and $V$, where $V$ is the Galois module given by $\rho_{\M} : \gal({\bar\Q}/\Q)\to\aut_{\T/\M}V$. Let $W^{\rm ss}$ be the direct sum of the Jordan-Holder quotients of $W$. \begin{thm} (Mazur) $W^{\rm ss}$ is isomoprhic as a group to $V\times ...\times V$, with $t$ copies of $V$. (This is true $\forall N$ as long as $\rho$ is irreducible.) \end{thm} Remark: A variant of this is the following \begin{dfn} A representation $\rho_{\M} : \gal({\bar\Q}/\Q)\to\GL_2(\T/\M)$ is absolutely irreducible if the composition $\gal({\bar\Q}/\Q)\to \GL_2(\T/\M)\hookrightarrow\GL_2({\overline{\T/\M}})$ is irreducible, where ${\overline{\T/\M}}$ is the algebraic closure of $\T/\M$. \end{dfn} \begin{thm} (Boston-Lenstra-Ribet) If $\rho_{\M}$ is absolutely irreducible, then $W$ is isomorphic as a group to $V\times ...\times V$. \end{thm} \pf (Of Mazur's theorem) Suppose $d = \dim W$. Consider $$\Hom_{\F-{\l}}(W,\mu_{\l}) = \Hom_{\T/\M}(W,\T/\M (1)) = W^*.$$ We will show that $$W^{\rm ss}\oplus (W^*)^{\rm ss} = V\times ...\times V$$ where the product has $2d$ copies of $V$. We will show that the two representations above have the same characteristic polynomials. Then the Brauer-Nesbitt theorem will prove that they are isomorphic. Look at $\frob_p$. It has the following characteristic polynomial on $V$: $X^2 - T_pX + p = (X - r)(X - pr^{-1})\in(\T/\M)[X]$. So the characteristic polynomial of $\frob_p$ on $V^d$ is $(X - r)^d(X - pr^{-1})^d$. And the characteristic polynomial of $\frob_p$ on $W$ is $(X - \a_1)\cdot ...\cdot(X - \a_d)$, where $\a_i$ is $r$ or $pr^{-1}$. The $\a_i$'s are of this form because $\frob_p^2 - T_p\frob_p + p = 0$. So the characteristic polynomial of $\frob_p$ on $W\oplus W^*$ is $(X - \a_1)(X - p\a_1^{-1})\cdot...\cdot(X - \a_d)(X - p\a_d^{-1}) = (X - r)^d(X - pr^{-1})^d$. \qed \begin{thm} $J[\M]\neq 0$. ie. $t\geq 1$. \end{thm} \pf Suppose $J[\M] = 0$. Then $J[\M^i] = 0 \forall i\geq 1$. This follows from Nakayama's lemma. So now look at the $\l$-divisible group $J_{\M} = \cup_i J[\M^i]$. So in order to show $J[\M]\neq 0$, equivalently, we will show that $J_{\M}\neq 0$. From the discussion to follow, we will see that $J_{\M}$ is dual to $Ta_{\M}^* J$. But $Ta_{\M}^* J$ is non-zero by the theorem of Weil stated below, and thus $J_{\M}$ is non-zero; hence, $J[\M]$ is non-zero. \qed \begin{thm} (Weil) $\T\tensor\Z_{\l}\subseteq \endo J\tensor\Z_{\l}$ injects into $\endo (Ta_{\l} J)$. \end{thm} \begin{cor} $Ta_{\l} J \neq 0$. \end{cor} \pf Since $\T\tensor\Z_{\l}\neq 0$ and $\T\tensor\Z_{\l}$ injects into $\endo (Ta_{\l}^* J)$, thus $\endo (Ta_{\l}^* J)\neq 0$ and so we must have $Ta_{\l}^* J \neq 0$. \qed Now we will discuss the above duality between $J_{\M}$ and $Ta_{\M}^* J$. $Ta_{\l} J = \plim J[\l^i]$, which is isomorphic as a group to $\Z_{\l}^{2\dim J}$. On the other hand, $J_{\l} = \cup_{i=1}^{\infty} J[\l^i]$ which is isomorphic as a group to $(\Q_{\l}/\Z_{\l})^{2\dim J}$. \begin{dfn} The covariant Tate module $Ta_{\l} J$ is $\Hom (\Q_{\l}/\Z_{\l}, J_{\l})$. \end{dfn} \begin{dfn} The contravariant Tate module $Ta_{\l}^* J$ is $\Hom(J_{\l}, \Q_{\l}/\Z_{\l})$. \end{dfn} The contravariant Tate module is the Pntryajin dual of $J_{\l}$. Note that $\Hom (J_{\l},\Q_{\l}/\Z_{\l}) = \Hom_{\Z_{\l}} (Ta_{\l} J, \Z_{\l})$ so $Ta_{\l}^* J = \Hom (Ta_{\l} J, \Z_{\l})$. Now, we have the Weil pairing, $J[\l^i]\times J[\l^i]\to \mu_{\l^i}$. Passing to the limit induces a pairing $< , > : Ta_{\l} J\times Ta_{\l} J\to\Z_{\l}(1) = \plim \mu_{\l^i}$. This pairing is perfect and gives us that $Ta_{\l} J \isomap \Hom (Ta_{\l} J,\Z_{\l}(1)) = (Ta_{\l}^* J)(1)$. Now if $t\in\T$, then $\< tx,y\> = \< x,t^{\vee}y\>$ where $t^{\vee} = wtw$ and $w = w_N\in \endo J_0(N)$ is the Atkin-Lehner involution. So this pairing is not $\T$-compatible. So we define a new pairing, which is $\T$-compatible, by $$[{\mbox{ }},{\mbox{ }}] = \< x,wy\>.$$ Check for $\T$-compatibility: $[tx,y] = \< tx,wy\> = \< x,t^{\vee} wy\> = \< x,wty\> = [x,ty]$. As a reminder: $w$ is an involution on $X_0(N)$ given by $(E,C)\mapsto (E/C,E[N]/C)$. $[{\mbox{ }},{\mbox{ }}]$ defines an isomorphism of $\T\tensor\Z_{\l}$-modules: $$Ta_{\l} J \isomap Ta_{\l}^* J(1) = \Hom_{\Z_{\l}} (Ta_{\l} J, \Z_{\l}(1))$$ which is isomorphic as a group to $\Hom (Ta_{\l} J, \Z_{\l}) = Ta_{\l}^* J$. 1\section*{March 11, 1996} \noindent{Scribe: J\'anos Csirik, \tt } \bigskip Our basic situation is that we have $J_0(N)$ (or $J_1(N)$) and a maximal ideal $\M\subseteq\T\subseteq\endo(J_0(N))$ lying over $\l$. We have $\T\otimes_\Z\Zl=\prod_{\M|\l}\Tm$, where $\Tm$ is the $\M$-adic completion of $\T$. This ring acts on $\tatel$, breaking it up as $\prod_{\M|\l}\tatem$. The last time we have confirmed that $\tatem\ne0$, and that $\tatel$ is $\T\otimes\Zl$-autodual, hence each $\Tm$ is $\Zl$-autodual. \bigskip On a finite level, take $J[\M]=W$, a representation of $\GQ$ over the field $\T/\M$. Also \[ V = \hbox{2-dimensional representation of } \GQ \hbox{ over } \Tm \] which gives the representation $\rho_\M$ (which is given by its trace, determinant and the fact that it is semisimple.) Now assume that $V$ is irreducible. Then \[ W^{ss} \cong V \times \cdots \times V \] (with $t$ copies of $V$ on the right hand side). This was proved in the last lecture using an argument of Mazur. We want to show that $t=1$. We already know that $t>0$, since by Nakayama's Lemma, $W=0$ implies $\tatem=0$ which we know is false. How is one supposed to think about $J[\l]$? Think of it as $J[\l]=\tatel/\l\tatel$. Since $\tatels$ is the isomorphic linear dual of $\tatel$, you can think of $J[\l]$ as $\tatels/\l\tatels$. Then \[ J[\M] = \tatems/\M\tatems \] by Pontrjagin duality. \bigskip More on Pontrjagin duality: $J_\l=J[\l^\infty]$ is Pontrjagin dual to $\tatels=\Hom_\Zl(\tatel,\Zl)= \Hom_\Zl(J[\l^\infty],\Ql/\Zl)$. In general, Pontrjagin duality assigns to each discrete abelian group $M$ a compact abelian group $M^*=\Hom(M,\Q/\Z)$; and to each compact abelian group $N$ a discrete $\Hom(N,\Q/\Z)$. Therefore the groups $\Z$ and $\Q/\Z$ correspond to one another. We are using a kind of local version under which the group $\Ql/\Zl$ is sent to $\Zl$ and the group $\Zl$ is sent to $\Ql/\Zl$. Given the group \[ J[\l^\infty] = \Hom(\tatels,\Ql/\Zl), \] it has the subgroup \[ J[\M^\infty] = \Hom(\tatems,\Ql/\Zl), \] which in turn contains \[ J[m] = \Hom_{\Z/\l\Z}(\tatems/\M\tatems,\Ql/\Zl). \] Since the part of $\Ql/\Zl$ killed by $\l$ is just $\Z/\l\Z$, this can be written as \[ J[m] = \Hom_{\Z/\l\Z}(\tatems/\M\tatems,\Z/\l\Z). \] Now if $J[m]$ is zero, then so must be $\tatems/\M\tatems$, but then by Nakayama's Lemma $\tatems$ should be zero, which it isn't. \bigskip We have the following to goals: {\sc First goal.} To show that $t=1$, i.e., that $J[\M]$ is generated by two elements over $\T/\M$. (Since $J[\M]$ is killed by $\M$, this is equivalent to showing that $J[\M]$ is generated over $\T$ by two elements.) {\sc Second goal.} To show that $\Tm$ is Gorenstein, i.e., that \[ \Tm \cong \Hom_\Zl(\Tm,\Zl). \] (Why would we want this? Well, recall that $S(\Z)\cong\Hom_\Z(\T,\Z)$ as $\T$-modules. Our goal just says that $S(\Z)$ is locally free at $\M$.) We are assuming that $\rho_\M$ is irreducible, and that $\l\not|2N$. $\l\not|N$ means that $J$ has good reduction at $\l$, and $2\not|N$ means that we can apply Raynaud's results on group schemes. \bigskip Postpone proving the first goal now, and show instead why the first goal implies the second one. By Nakayama's Lemma, $\tatem$ is generated by $2$ elements as a module over $\Tm$. We need that $\tatems/\M\tatems$ is also generated by $2$ elements: but for $\T/\M$ vector spaces $R$, we have \[ \Hom_{\Z/\l\Z}(R,\Z/\l\Z) \cong \Hom_{\T/\M}(R,\Z/\l\Z) \] by considering the trace. So we can find a map \[ \Tm \times \Tm \onto \tatem. \] \begin{claim} \[ 2\rank_\Zl\Tm = \rank_\Zl\tatem.\] \end{claim} Recall that $\rank_\Zl\tatel=2\dim\, J=2\dim\,\T=2\rank_\Zl\T\otimes\Zl$. This makes the claim pausible, but isn't a proof. The proof is postponed for now. Accepting the Claim, it follows by basic graduate algebra (Math 250) that $\Tm\times\Tm\cong\tatem$. Recall that $\tatem$ is auto-dual in the following sense: \[ \tatem \cong \Hom_\Zl (\tatem,\Zl). \] Hence $\Tm\times\Tm$ is self-dual in the same sense. Does it follow that $\Tm$ is self-dual? Yes. To do this, use a stupid thing: call two copies of $\Tm$ $F_1=\Tm$ and $F_2=\Tm$. Then we have \[ F_1^* \times F_2^* \isomap F_1 \times F_2, \] where ${}^*$ denotes taking linear duals ($\Hom$ of the thing to $\Zl$). Then $F_1^* \times F_2^*$ is free, so $F_1^*$ is projective, so it's free (over a local ring, all projective modules are). \bigskip Now going back to our claim, note that $\rank_\Zl\Tm =\dim_\Ql(\Tm\otimes_\Zl\Ql)$. Also $\rank_\Zl\tatem =\dim_\Ql(\tatem\otimes_\Zl\Ql)$. $\tatel$ is gotten by considering the $\l$-power torsion points in the Jacobian, or else it is just those complex points in $J(\C)$. But using the Albanese covariant interpretation, (and recalling that $H^1(X_0(N),\C) = S_2(\G_0(N),\C)$), \[ J(\C) = \frac{\Hom_\C( S_2(\G_0(N),\C), \C)} {H_1(X_0(N),\Z)} .\] The lattice $L=H_1(X_0(N),\Z)$ is a $\T$-module, and $\tatel=L\otimes_\Z\Zl$. Instead, we'll look at \begin{eqnarray*} L\otimes_\Z\R & = & \Hom_\C(S_2(\G_0(N),\C),\C) = \Hom_\R(S_2(\G_0(N),\R),\C) \\ & = & \Hom_\R(S_2(\G_0(N),\R),\R) \otimes_\R \C = (\T \otimes_\Z \R) \otimes_\R \C, \end{eqnarray*} hence $ L\otimes_\Z\R = (\T\otimes_\Z\R)\otimes_\R\C$, so $L\otimes_\Z\R$ is free of rank $2$ over $\T \otimes_\Z \R$, therefore $L\otimes_\Z\C$ is free of rank $2$ over $\T \otimes_\Z \C$. \bigskip Now {\em choose} an embedding $\Ql\inj\C$. $\tatel=\prod_{\M}\tatem$ is a module over $\T\otimes\Zl=\prod_{\M}\T_\M$. Consider $\tatel\otimes\Z\Ql = \prod_\M(\tatem\otimes_\Zl\Ql$. $\prod_{\M}(\tatem\otimes_\Zl\Ql)=\tatel\otimes_\Zl\C =L\otimes_\Z\C$ is free of rank $2$ over $\T\otimes_\Z\C=(\T\otimes_\Z\Zl)\otimes_\Zl\C= \prod_\M(\tatem\otimes_\Zl\C)$. Hence $\tatem\otimes_\Zl\C$ is free of rank $2$ over $\T_\M\otimes_\Zl\C$, therefore $\dim_\C\tatem\otimes_\Zl\C=2\dim_\C\T_\M\otimes_\Zl\C$, so $\dim_\Ql\tatem\otimes_\Zl\Ql=2\dim_\Ql\T_\M\otimes_\Zl\Ql$, as the dimension of a vector space does not change if we tensor up to a bigger field. This settles our claim that the second goal implies the first one. The first goal shall be proved in the next lecture. \section*{March 13, 1996} \noindent{Scribe: Amod Agashe, \tt } \bigskip Let $S$ be a scheme. Then a group scheme over $S$ is a group object in the category of $S$-schemes i.e. it is an $S$-scheme G with $S$-morhphisms $m:G \times G \rightarrow G, i: G \rightarrow G, e: S \rightarrow G$ satisfying the usual group axioms (See [1] for details). We will be interested in finite flat group schemes over $\F_p$ where $p$ is a prime and $\F_p$ is the finite field with $p$ elements. Let $A$ be a $\F_p$-vector space and let $G = Spec(A)$. Then the order of $G$, $\# G = dim_{\F_p}A$. If $R$ is an $\F_p$-algebra, then $G(R) = Mor(SpecR,G)$ is the group of R-valued points of $G$. Here are some examples: \\ 1)The additive group scheme $\G_a =Spec(\F_p[T])$. \\ 2) The multiplicative group scheme $\G_m =Spec(\F_p[T,T^{-1}])$. \\ 3) The $p$th roots of unity ${\bf \mu}_p$ is the kernel of the $p$-power map on $\G_m$. \\ 4) The group $\alpha_p$ is the kernel of the Frobenius map on $\G_a$. \\ 5) Let $E/\F_p$ be an elliptic curve thought of as a group scheme. Then $E[p]$ is the kernel of the multiplication by $p$ map on $E$. It has order $p^2$ and \\ $E[p](\overline{\F_p}) = \left \{ \begin{array}{ll} 1 & \mbox{if E is supersingular}\\ p & \mbox{if E is ordinary} \end{array} \right. $ The primary reference for what follows is [2]. Let $J=J_0(N)$ be the Jacobian of $X_0(N)$. Then $J$ is defined over $\Q$ and has good reduction at primes not dividing $N$. Let $l$ be a prime not dividing $N$. Then $J[l]$ extends to a finite flat group scheme over $\Z[\frac{1}{N}]$ (Ref: [3, exp. IX]). This in turn gives rise to a group scheme over $\F_l$ which can be thought of as $J_{/\F_l}[l]$. Let $V$ be the two dimensional vector space over $\T /\M$ which gives rise to the representation $\rho_\M$ constructed last week. We want to show that this is isomorphic to the naturally defined Galois representation $W=J[\M]$. We have $0 \subset V \subset W \subset J[l]$. We make the following assumptions: $l \not| N, l \not= 2$ and $V$ is irreducible. This is because we will be invoking theorems of Raynaud. Let $\underline{J}$ denote $J$ considered as a group scheme. Then $\underline{J}[l]$ is a finite flat group scheme over $\Z_l$; in fact, it is the Spec of a free finite rank module over $\Z_l$ of rank $l^{2g}$. Raynaud showed (See [4] or [1]) that if $l\not=2$ (so that the absolute ramification index $\rho=1 < l-1$), almost everything about the group scheme $\underline{J}[l]$ can be seen in terms of its Galois module $J[l](\overline{\Q_l})$. Let $\underline{V}$ and $\underline{W}$ be the group schemes given by the Zariski closure of $V$ and $W$ in $\underline{J}[l]_{/ \Z_l}$ respectively. Then we have: \begin{center} $\underline{V} \subset \underline{W} \subset \underline{J}[l]$. \end{center} Our goal is to show that $V \hookrightarrow W$ is an isomorphism. Raynaud showed that the category of finite flat group schemes over $\Z_l$ is an abelian category, so it makes sense to talk about the quotient $\underline{Q}= \underline{W}/\underline{V}$, which is again a finite flat group scheme over $\Z_l$. Note that $\underline{Q}_{/ \Q_l}$ corresponds to $W/V$. Thus, $V = W \Leftrightarrow \underline{Q}_{/ \Q_l} =0$. Since $\underline{Q}$ is flat, it has the same rank over $\Q_l$ as over $\F_l$. Passing to characteristic $l$ yields the following exact sequence: \begin{center} $0 \rightarrow \underline{V}_{/\F_l} \rightarrow \underline{W}_{/\F_l} \rightarrow \underline{Q}_{/\F_l} \rightarrow 0$. \end{center} Again, since the ranks remain constant, \begin{center} $V \hookrightarrow W$ is an isomorphism $\Leftrightarrow \underline{V}_{/\F_l} \hookrightarrow \underline{W}_{/\F_l}$ is an isomorphism $\Leftrightarrow \underline{Q}_{\F_l}=0$. \end{center} Following Raynaud, $\underline{V}$, $\underline{W}$ and $\underline{Q}$ inherit the action of $k=\T/\M$ (i.e. they are $k$-vector space schemes). We will show that $\underline{Q}_{\F_l}=0$. For that, we need Dieudonn\'{e} module theory. Let $G/k$ be a finite $k$-vector scheme where $k$ is a finite field of order $q$. Suppose the order of $G$ is $q^n$. Then the Dieudonn\'{e} module functor $D$ (See [5]) is a contravariant functor which maps $G$ to a $k$-vector space $D(G)$ of dimension $n$, called the Dieudonn\'{e} module of $G$. Let $Frob: G \rightarrow G$ be the morphism on $G$ induced by the $p$th power map on the underlying sets and let $Ver$ be the dual of $Frob$. Then the maps $\phi$ and $\nu$ induced on $D(G)$ by $Frob$ and $Ver$ respectively satisfy $\phi \circ \nu = \nu \circ \phi = 0$. $D$ is a fully faithful functor i.e. we can tell everything about the group scheme from its Dieudonn\'{e} module. For example, let $k=\F_l=\F_p$. Let $G=\mu_p$, $\alpha_p$ or $\Z/p\Z$. Then in all cases, $dim_k(G) =1$. For $\alpha_p$, $\phi=\nu=0$; for $\mu_p$, $\phi=0, \nu=id$ and for $\Z/p\Z$, $\phi=id, \nu=0$. Let $G^\vee=Hom(G,{\bf \mu}_p)$ denote the Cartier dual of G. Then $D(G^\vee)=Hom_k(D(G),k)$ with $\phi$ and $\nu$ interchanged. Here is a sophisticated example of the above theory: Let $A$ be an abelian variety over $\F_l$ and $G=A[l]$. Then, $G$ is an $\F_l$-vector space scheme of order $l^{2g}$ and thus $D(G)$ has dimension $2g$. Furthermore, $D(G)=H^1_{DR}(A_{/\F_l})$. Consider the Hodge filtration: \begin{center} $0 \rightarrow H^0(A,\Omega^1) \rightarrow H^1_{DR}(A_{/\F_l}) \rightarrow H^1(A,{\cal O}_A) \rightarrow 0$. \end{center} Now $H^1(A,{\cal O}_A) = Tan(A^\vee_{\F_l}) = Hom(H^0(A^\vee,\Omega^1),\F_l)$ and it corresponds to the subgroup $D(A_{\F_l}[Ver])$ of $D(A_{/\F_l})$ where $Ver$ is the dual to $Frob$. Consider the exact sequence \begin{center} $0\rightarrow W_{/\F_l} \rightarrow J_{/\F_l}[l] \stackrel{\M}{\rightarrow} J_{/\F_l}[l]$. \end{center} Since $D$ is an exact functor, applying $D$ to the above, we get the exact sequence \begin{center} $D(J_{/\F_l}[l])\stackrel{\M}{\rightarrow} D(J_{/\F_l}[l]) \rightarrow D(W_{/\F_l}) \rightarrow 0$. \end{center} In other words, $D(W_{/\F_l})=D(J_{/\F_l}[l])/\M D(J_{/\F_l}[l])$. Next time, following Fontaine, we shall consider $D(W_{/\F_l}[Ver]) = H^1(J,{\cal O})/\M H^1(J,{\cal O})$. The latter is dual (in the sense of $Hom$ of a vector space into $\F_l$) to $H^0(J^\vee,\Omega^1)[\M] = H^0(X_0(N)_{/\F_l},\Omega^1)[\M]$. For the duality above, we need to use $J^\vee = Alb(X_0(N))$ i.e. $J=Pic^0(X_0(N))$. We will be using the q-expansion principle to show that $D(W_{/\F_l}[Ver])$ is small. \vspace{1 in} {\bf References:} [1] S. Shatz, Group Schemes, Formal Groups, and p-Divisible Groups. In {\em Arithmetic Geometry}, G.Cornell and J.Silverman, eds., Springer-Verlag, New York, 1986, pg. 29-78. [2] B.Mazur, Modular Curves and the Eisenstein Ideal, {\em Publ.Math. I.H.E.S.}, {\bf 47} (1977), 33-186. [3] S\'{e}minaire de G\'{e}om\'etrie alg\'ebrique du Bois-Marie, 67-69. P.Deligne and N.Katz, {\em Lecture Notes in Mathematics}, nos. {\bf 288,340}, Berlin-Heidelberg-New York, Springer, 1972, 1973. [4] M. Raynaud, Sch\'{e}mas en groupes de type (p,p,...,p), {\em Bull. Soc. Math. Fr.}, {\bf 102} (1974), 241-280. [5] T.Oda, The first De Rham cohomology group and Dieudonn\'{e} modules, {\em Ann. scient. \'{E}.Norm.Sup.}, $4^e$ s\'{e}ie, t. {\bf 2} (1969), 63-135. \section*{April 1, 1996} \noindent{Scribe: J\'anos Csirik, \tt } \bigskip \subsubsection*{Local properties of $\rhla$} Summary of all we know, without proofs: $\rhla$ is associated to $f$, a newform of weight $2$, level $N$, character $\ve:(\Z/n\Z)^*\to\C^*$ (i.e., we don't necessarily only consider $\G_0(N)$ any more), conductor of $\ve$ dividing $N$ (not necessarily equal to $N$, could be trivial). Let $\la|\l$, and look locally at $p$ (with $p\neq\l$). The real story is that $f$ gives rise to some irreducible representation of $\GL(2,\A)$, which corresponds to a family of representations $(\pi_v)$, with $\pi_v$ being a representation of $\GL(2,\Q_v)$. Here $v$ stood for any valuation of $\Q$, but for the archimedean prime in the weight $2$ case we always get a well-known discrete series representation on $\GL(2,\R)$, so we really only care about the representations $\pi_p$ on $\GL(2,\Qp)$. It is a theoretical result of Carayol (building on the work of Deligne and Langlands) that $\rhla|_{D_p}$ up to isomorphism only depends on $\pi_p$. In our case, we are most interested in the case where $p^2\!\!\not|N$. A reference is R. P. Langlands: {\em Modular Forms and $\l$-adic representations}, pp. 361--500 in Antwerp II. For a warmup, consider the case of $p\!\!\not|N$. Then we have the following ``characteristic polynomial'' floating about: $X^2-a_pX+p\ve(p)$. Factor it as $(X-r)(X-s)$ with $r,s\in\C$. From the Riemann Hypothesis for abelian varieties (proved by Andr\'e Weil) it follows that $|r|=|s|=\sqrt{p}$. $\rhla|_{D_p}$ is an unramified representation so $I_p$ is killed by it, so $\frob_p$ makes sense up to conjugacy. So $\rhla(\frob_p)$ has a well-defined characteristic polynomial $X^2-a_pX+p\ve(p)\in E[X]$, where $E$ is the field generated over $\Q$ by the coefficients of our newform $f$. So since $p$ does not divide $N$, $\rhla(\frob_p)$ is semisimple, so it follows that $\rhla(\frob_p)$ is conjugate to $\left(\begin{array}{cc}r&0\\0&s\end{array}\right)$. What is $\pi_p$ supposed to be? Let \[ \a,\b: \Qp^* \to \C^* \] be Gr\"ossencharacters (i.e., charaters with values that don't necessarily have absolute value 1), such that \begin{enumerate} \item $\a,\b$ are unramified: $\a|_{\Zp^*}=\b|_{\Zp^*}=1$ (here we used $1$ to denote the unit character). Note that this condition is called being unramified because by local class field theory, $\Zp^*$ corresponds to $I_p$ in $\GQp$; \item $\a(p^{-1})=r$ and $\b(p^{-1})=s$. \end{enumerate} By the relation between Gr\"ossencharacters and characters of $\GQ$ which is explained in Weil (1950), or J.-P. Serre and J. Tate, {\em Good reduction of abelian varieties}, Ann. of Math., {\bf 2}, 1968, pp. 492--517, or Serre (1972). $\a$ and $\b$ correspond to characters \[ \a_\la, \b_\la : \GQ \to \overline{E_\la}^*, \] which satisfy \begin{enumerate} \item $\a_\la, \b_\la$ are unramified; \item $\a_\la(\frob_p)=r$, $\b_\la(\frob_p)=s$. \end{enumerate} Then $\rhla=\a_\la\oplus\b_\la$. Define a character of the Borel subgroup $B=\left(\begin{array}{cc}*&*\\0&*\end{array}\right)\in\GL(2,\Qp)$, \[ \th : \left(\begin{array}{cc}x&y\\0&z\end{array}\right) \mapsto \a(x)\b(z). \] This is a character $\th:B\to\C^*$. Then $\pi_p={\rm Ind}^{\GL(2,\Qp)}_B\th$ is irreducible since $r/s$ is neither $p$ nor $p^{-1}$. This is called the (unramified) principal series representation attached to $\a$ and $\b$. (Unramified in this case since $\a$ and $\b$ are unramified.) This is denoted \[ \pi_p = {\rm PS}(\a,\b). \] Here is the end of the warmup. \bigskip Now look at the case where $p$ exactly divides $N$. There are two very different cases, according as $\ve$ is ramified or unramified at $p$. (It is ramified iff $p$ divides the conductor of $\ve$.) \bigskip {\sc The case of $\ve$ ramified at $p$. } Here $\pi_p={\rm PS}(\a,\b)$, and $\rhla=\a_\la\oplus\b_\la$. $\a$ (say) is unramified and $\b$ is ramified (with conductor $=p$), i.e. $\b|_{\Zp^*}$ is nontrivial. But it is not too bad since $\b|_{1+p\Zp^*}=1$ for $p\ne2$. To see what's going on (what is $\a$ and $\b$?), think about $L$-functions. Let $f=\sum a_nq^n$ and fix attention to $a_p$ (which is non-zero by the analytic point of view). By definition, \[ L(f,s) = \sum_{n=1}^\infty a_n/n^s = \prod_{p\not|N}(1-a_pp^{-s}+p\ve(p)p^{-2s})^{-1} \prod_{p|N}(1-a_pp^{-s})^{-1},\] the $L$-factor corresponding to $p$ being $(1-a_pp^{-s})^{-1}$. Now $V_\la$ has a one-dimensional quotient on which $\frob_p$ acts by multiplication by the number $a_p$. So the unramified character $\a$ can only be the one with $\a(p^{-1})=a_p$. But we know that $\a_\la\b_\la=\det\rhla=\ve\chi_\l$ (with $\ve$ corresponding to the Gr\"ossencharacter taking $p^{-1}$ to $p$). Hence we must have $\b(p^{-1})=\ve(p)pa_p^{-1}$. Now use the formula $\overline{f}=\sum\overline{a_n}q^n=f\otimes\ve^{-1}$. For $(n,N)=1$, we have $a_n=\overline{a_n}\ve(n)$, so in particular this holds for almost all primes, so by the Chebotarev density theorem $\rho_{\la,f}=\rho_{\la,\overline{f}}\otimes\ve$, hence if we let $\a_\la,\b_\la$ to correspond to $f$ and $\d_\la,\g_\la$ to correspond to $\overline{f}$, we have \[ \left(\begin{array}{cc}\a_\la&0\\0&\b_\la\end{array}\right) = \left(\begin{array}{cc}\g_\la\ve&0\\0&\d_\la\ve\end{array}\right), \] so up to permutation we must have $a_\la=\g_\la\ve$ and $b_\la=\d_\la\ve$ (*). Now exactly one of $\g_\la$ and $\d_\la$ is ramified (exactly as in the case of $\a$ and $\b$). Hence by the (*) above, $\d_\la$ is unramified while $\g_\la$ is ramified. So using $\det=\chi_\l\ve$, we get \[ \b_\la = \d_\la\ve = \ve\chi_\l\a_\la^{-1}. \] Looking at the Frobenii, $\overline{a_p}=pa_p^{-1}$, hence $\overline{a_p}a_p=p$, so \[ |a_p| = \sqrt p . \] \bigskip {\sc The case of $\ve$ unramified at $p$. } So here we can talk of $\ve(p)$, and it is going to be a root of $1$ in $E^*$. It is supposed to remind you of the Tate curve (elliptic curve with multiplicative reduction at $p$). Now \[ \rhla|_{D_p} = \left(\begin{array}{cc}\a_\la&*\\0&\b_\la\end{array}\right). \] Here $\a_\la=\chi_\l\b_\la$ and $\a_\la=\b_\la=\det=\ve\chi_\l$, so plugging in we get $\b_\la^2=\ve$. So $\b_\la$ is unramified, \[ \b_\la(\frob_p) = a_p \qquad \a_\la(\frob_p) = pa_p .\] $\rhla|_{D_p} $ is an indecomposable representation: the $*$ in the above formula is non-zero. Since $p\ne\l$, this property characterizes it. We can say that \[ \rhla = \b_\la \otimes \left(\begin{array}{cc}\chi_\l&*\\0&1\end{array}\right), \] where the second term in the tensor product is the unique indecomposable $\la$-adic representation of $\GQp$ with semisimplification $1\oplus\chi_\l$. Correspondingly \[ \pi_p = \b \otimes st, \] where $st$ is a pun for both the ``standard representation'' and the ``Steinberg representation''. Morally, a PS representation attached to $(1,||.||)$ is not irreducible, but it has a unique irreducible quotient. {\sc Remarks.} A {\em special representation} is any abelian character tensored with the Steinberg representation. Langlands invented a dictionary between representations of $\GL(2,\Qp)$ and representations of $\GQp$, which goes through complex representations of a smaller group (the {\em Weil group}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% 04/08/96, Scribe notes by William Stein %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The weight and fundamental characters} Let $$\rho:\mbox{\rm Gal}(\overline{\Q}/\Q) \rightarrow\mbox{\rm GL}(2,\F_{\ell^{\nu}})$$ be an irreducible modular mod $\ell$ representation. Let $D=D_p=\mbox{\rm Gal}(\overline{\Q}_p/\Q_p)$ be the decomposition group at $p$. Today we will see how studying the local properties of $\rho$ help us to understand the following theorem. \begin{thm} If $\ell>2$ and $\rho$ is modular then $\rho$ is modular of a certain weight $k(\rho)$ and level $N(\rho)$. \end{thm} Since $\rho$ is modular, $\rho\sim\overline{\rho}_{\lambda,f}$ for some modular form $f$. The theorem asserts that we can actually take $f\in S_{k(\rho)}(\Gamma_1(N(\rho)))$. As we have seen before $N(\rho)$ is the prime to $\ell$ Artin conductor of $\rho$. Today we attack the question of what $k(\rho)$ should be. To do this we restrict to the inertia group $I_p$. Before doing this we introduce a certain pair of characters. First we prove a lemma. Let $\sigma$ be the semisimplification of $\rho|_D$. Thus $\sigma$ is either a direct sum of two characters or $\rho|_D$ depending on whether or not $\rho|_D$ is irreducible. \begin{lem} The representation $\sigma$ is tame, i.e, $\sigma$ is trivial on the $\ell$-Sylow subgroup of $I_{\ell}$. \end{lem} {\em Proof. } Let $P$ be the $\ell$-Sylow subgroup of $I_{\ell}$. Since $I_{\ell}$ is normal in $D$ and $P$ is unique it follows that $P$ is normal in $D$. Let $W=\F_{\ell^{\nu}}\times\F_{\ell^{\nu}}$ be the representation space of $\sigma$. Then $$W^P=\{w\in W : \sigma(\tau)w = w \quad\mbox{\rm for all} \quad\! \tau\in P\}$$ is a subspace of $W$ invariant under the action of $D$. To see this let $\alpha\in D$ and suppose $w\in W^P$. Since $P$ is normal in $D$, $\alpha^{-1}\tau\alpha=\tau'$ for some $\tau'\in P$. Therefore $$\sigma(\alpha)^{-1}\sigma(\tau)\sigma(\alpha)w = \sigma(\tau') w = w$$ so $\sigma(\tau)\sigma(\alpha)w = \sigma(\alpha)w$ hence $\sigma(\alpha)w\in W^P$. But $W^P\neq 0$. To see this write $W$ as a disjoint union of its orbits under the action of $P$. Since $P$ is an $\ell$-Sylow group and $W$ is finite we see that the size of each orbit is either $1$ or a positive power of $\ell$. Now $\{0\}$ is a singleton orbit, $W$ has $\ell$-power order, and all non-singleton orbits have order a positive power of $\ell$ so there must be at least $\ell-1$ other singleton orbits. Each of these other singleton orbits gives a nonzero element of $W^P$. If $W^P=W$ then $P$ acts trivially so we are done. If $W^P\neq W$ then $W^P$ is a one dimensional subspace invariant under $D$ so by semisimplicity $\sigma$ is a diagonal representation. Suppose $\tau\in P$ then $\tau$ has order $\ell^n$ for some $n$. Write $$\sigma(\tau)=\left(\begin{array}{cc}\alpha&0\\0&\beta\end{array}\right)$$ then $\alpha^{\ell^n}=1$ and $\beta^{\ell^n}=1$. Since $\alpha, \beta\in\F_{\ell^{\nu}}$ it follows that $\alpha=\beta=1$. \qed The lemma implies $\sigma|_I$ factors through the tame quotient $I_t=I/P$. We now describe $I_t$ explicitly. Let $\Q_p^{tame}$ be the maximal tame extension of $\Q_p$ and let $K=\Q_p^{ur}$ be the maximal unramified extension. Since $P$ is the part of inertia fixing $\Q_p^{tame}$, $$I_t=I/P=\mbox{\rm Gal}(\Q_p^{tame}/\Q_p^{ur}).$$ For each $n$ prime to $p$ there is a tower of fields $$\begin{array}{c} \Q_p^{tame}\\ |\\ K(p^{\frac{1}{n}})\\ |\\ K=\Q_p^{ur}\\ |\\ \Q_p\end{array}$$ By Kummer theory $\mbox{\rm Gal}(K(p^{\frac{1}{n}})/K)=\mu_n(K)$ where $\mu_n(K)$ denotes the group of $n$th roots of unity in $K$. Thus for each $n$ prime to $p$ we obtain by restriction a map $I_t\rightarrow\mu_n(K)$. They are compatible so passing to the limit we obtain a map $$I_t\rightarrow\lim_{\leftarrow n} \mu_n(K)=\prod_{r\neq p}\lim_{\leftarrow a}\mu_{r^a}(K)=\prod_{r\neq p}\Z_r(1).$$ Viewed more cleverly mod $p$ we obtain a map $$I_t\rightarrow\lim_{\leftarrow n} \mu_n(\overline{\F}_p)=\lim_{\leftarrow n}\F_{p^n}^*.$$ We now fix attention on $n=p^i-1$. The point is, that for every such $i$ we have a standard map $$I_t\rightarrow\F_{p^i}^*.$$ We call this map the {\em fundamental character of level $i$}. This is unnatural because we had to take an embedding $\F_{{\ell}^{\nu}}\hookrightarrow \overline{\F}_{\ell}$. Instead Serre begins with some disembodied field $F$ of order $p^i$. There are $i$ different maps $\F_{p^i}\rightarrow F$ corresponding to the $i$ automorphisms of $\F_{p^i}$. Restricting these maps to $\F_{p^i}^*$ and composing with the fundamental character $I_t\rightarrow\F_{p^i}^*$ defined above gives Serre's fundamental characters of $I_t$ with values in $F$. We have thus defined some characters on $I_t$ which can be viewed as standard. Recall that we have a mod $\ell^{\nu}$ representation $\rho$ which gives rise to $$\sigma:I_t\rightarrow \mbox{\rm GL}(2,\F_{\ell^{\nu}}).$$ Since the elements of $I_t$ have order prime to the characteristic $\ell$ this representation is semisimple so it can be diagonalized upon passing to an algebraic closure of $\F_{\ell^{\nu}}$. Thus $\sigma$ corresponds to a pair of characters $$\alpha,\beta:I_t\rightarrow\F_{\ell^{2\nu}}^*.$$ These characters have some stability properties since $\sigma$ is the restriction of a homomorphism from the full decomposition group. Consider the tower of fields $$K(p^{\frac{1}{n}})\supset K=\Q_p^{ur}\supset \Q_p.$$ Let $G=\mbox{\rm Gal}(K(p^{\frac{1}{n}})/\Q_p)$. Recall that $\mbox{\rm Gal}(K(p^{\frac{1}{n}})/K)=\mu_n(K)$ and $\mbox{\rm Gal}(K/\Q_p)$ is generated by $\mbox{\rm Frob}_p$. What happens is that if $h\in\mu_n(K)$ and $g\in G$ is such that $g$ restricts to $\mbox{\rm Frob}_p$ then we have a conjugation formula: $ghg^{-1}=h^p$. Applying this reasoning to the $\sigma$ situation with $h\in I_t$ and $g$ restricting to $\mbox{\rm Frob}_p$ we find that $$\sigma(ghg^{-1})=\sigma(h^p)=\sigma(h)^p.$$ Thus the pair of characters $\{\alpha,\beta\}$ is stable under $p$th powering, i.e., $\{\alpha,\beta\}=\{\alpha^p,\beta^p\}$. The point is that $$\sigma(g)\sigma(h)\sigma(g^{-1})=\sigma(ghg^{-1})=\sigma(h)^p.$$ Thus the representation $h\mapsto\sigma(h)^p$ is isomorphic to $h\mapsto\sigma(h)$ via conjugation by $\sigma(g)$. Thus $\alpha$, $\beta$ as a pair are the same as $\alpha^p$, $\beta^p$ as a pair since they came from isomorphic representations. What does this mean? One possibility is that $\alpha=\alpha^p$ and $\beta=\beta^p$. This means $\alpha, \beta$ take values in $\F_p^*$. The other possibility is that $\alpha^p=\beta$ and $\beta^p=\alpha$. Then $\alpha^{p^2}=\alpha$ and $\beta^{p^2}=\beta$ so $\alpha, \beta$ take values in $\F_{p^2}^*$. The first situation in which $\alpha$ and $\beta$ take values in $\F_p^*$ is called the {\em level 1} situation. The second situation is called the {\em level 2} situation. We will play a carnival game, ``guess your weight.'' First we consider the level 2 case. Our strategy is to try and express $\alpha$ and $\beta$ in terms of the two fundamental characters of level 2. First some background. Before stating the general result we talk about a special case. Suppose $E$ is an elliptic curve over $\Q$ and $p$ is a prime (2 is allowed). Assume $E$ has good supersingular reduction at $p$. Then there is a representation $$\mbox{\rm Gal}(\overline{\Q}/\Q)\rightarrow\mbox{\rm Aut}(E[p])$$ which may or may not be irreducible. It gives rise via restriction to two characters $\alpha,\beta:I_t\rightarrow\F_p^*$. Serre [{\em Groupes de Galois attach\'{e}s aux points d'ordre fini des courbes elliptiques sur un corps de nombres}, 1972] proves that $\alpha, \beta$ are the two fundamental characters of level 1 and that $I_t$ acts tamely and irreducibly. He also obtained a map $$I_t\rightarrow\F^*_{p^2}\subset\mbox{\rm GL}(2,\F_p)$$ where $\F^*_{p^2}$ sits inside $\mbox{\rm GL}(2,\F_p)$ via the action of the multiplicative group of a field on itself after choice of a basis. The action of $I_t$ on $\F^*_{p^2}$ is through the fundamental character. This was the world's first view of fundamental characters. Serre next asked Fontaine the corresponding question for more general weight. The first published proof appears in [Edixhoven, {\em The weight in Serre's conjectures on modular forms}, Invent. Math., 1992]. Suppose $f$ is a newform of weight $k$ such that $2\leq k\leq p$ and that the level of $f$ is prime to $p=\ell$. Take $\overline{\rho}_{f,\lambda}$ where $E_f=\Q(\mbox{\rm coefficients of }f)$ and $\lambda$ is a prime of $\O_{E_f}$ dividing $p$. Suppose we are in the supersingular case so $a_p\equiv 0 \mod \lambda$. Semisimplifying and restricting as before gives a pair of characters $\alpha,\beta:I_t\rightarrow\F_{p^{\nu}}^*$. \begin{thm}Under the above hypothesis, $\{\alpha,\beta\}$ equals $\{\psi^{k-1},(\psi')^{k-1}\}$ where $\psi$ and $\psi'$ are the two fundamental characters of level $2$, i.e., $\psi=(\psi')^p$ and $\psi'=\psi^p$. \end{thm} Edixhoven's method is to reduce from weight $k$ to weight $2$. Given a modular form mod $p$ on $\Gamma_1(N)$ of weight $k$ and character $\varepsilon$ he shows that it is enough to look at a corresponding modular form on $\Gamma_1(Np)$ of weight $2$ and character $\varepsilon\omega^{k-2}$ where $\omega$ is a Teichmuller character on $(\Z/p\Z)^*$. This is a method which helps us understand what happens at primes $p$ where a certain abelian variety does not have good reduction but has potentially good reduction. In this direction Ogus suggests reading the paper [Faltings and Jordan, {\em Crystalline cohomology and ${\rm GL}(2,{\bf Q})$}, Israel J. Math. 90 (1995), no. 1-3, 1--66.] What can we guess about the arbitrary situation? The situation is as follows. We begin with a representation $\rho$ which gives rise to a representation $\sigma$ which in turn gives rise to a pair of fundamental characters $\psi^a$ and $(\psi')^a=\psi^{pa}$. (Here $a$ should be thought of as a number mod $p^2-1$.) The condition that we are not in level $1$ means that $a$ is not divisible by $p+1$ since $$\psi\psi'=\psi^{p+1}:I_t\rightarrow\F^*_p$$ is the unique fundamental character of level $1$ (namely the mod $p$ cyclotomic character $\chi$). We next do a Euclidean division. Take $0\leq a} \bigskip In last lecture, the following theorem was stated: \begin{thm} Suppose $\rho : G \rightarrow \GL(2,\F_{l^\nu})$ satisfies the usual hypotheses, that is, $l > 2$, irreducible, modular, $\det \rho = \chi$, and $\rho$ is semistable. Suppose $A$ be a complete local Noetherian ring with residue field $\F_{l^\nu}$; $0 \rightarrow \M \rightarrow A \rightarrow \F_{l^\nu} \rightarrow 0$. Then, $\tilde{\rho} : G \rightarrow \GL(2,A)$ is modular if it satisfies the following: (1) $\tilde{\rho}$ lifts $\rho$, (2) $\det \tilde{\rho} = \tilde{\chi}$ (the $\l$-adic cyclotomic character), (3) $\tilde{\rho}$ is ramified at only a finite number of primes, (4) $\tilde{\rho}$ satisfies the following condition (*) which is defined according to two cases: Case 1. $\rho$ is finite flat at $l$. In this case, $k(\rho) = 2$ and $\rho | I_l$ is given by two fundamental characters $I_l \onto \F_{l^2}^*$ of level 2. In this case, (*) requires that $\tilde{\rho}$ is also finite flat at $l$ which means that for every $n \geq 1$, $\tilde{\rho} \mod \M^n : G \rightarrow \GL(2,A/\M^n)$ comes from a finite flat group scheme over $\Z_l$ provided with an action of $A/\M^n$. Case 2. $\rho | D_l \sim {\left(\begin{array}{cc}\alpha&*\\0&\beta\end{array}\right)} $, $\beta$ is unramified, and $\alpha | I_l = \chi$. In this case, (*) requires that $\tilde{\rho} | D_l \sim {\left(\begin{array}{cc}\tilde{\alpha}&*\\0&\tilde{\beta}\end{array}\right)} $ and $\tilde{\beta}$ is unramified ($\tilde{\beta}$ is a lift of $\beta$). \end{thm} \begin{remark} In case 2, it is possible that $\rho | D_l$ can be finite flat without $\tilde{\rho}$ being finite flat. \end{remark} \begin{dfn} Let $\Sigma$ be a finite set of prime numbers. A class of liftings of $\rho$ of type $\Sigma$ consist of liftings $\tilde{\rho}$ with $\det \tilde{\rho} = \tilde{\chi}$ which satisfy the following properties (which make $\tilde{\rho}$ like $\rho$): (1) $p \neq l$. \ \ (1-1) $p \in \Sigma$. No condition. \ \ (1-2) $p \notin \Sigma$. \ \ \ \ (a) If $\rho$ is unramified at $p$, then $\tilde{\rho}$ is unramified at $p$. \ \ \ \ (b) If $\rho | I_p \sim {\left(\begin{array}{cc}1&*\\0&1\end{array}\right)} $ is ramified but unipotent, then $\tilde{\rho} | I_p \sim {\left(\begin{array}{cc}1&*\\0&1\end{array}\right)} $. (2) $p = l$. \ \ (2-1) $l \in \Sigma$. No condition. \ \ (2-2) $l \notin \Sigma$. If $\rho$ is finite flat, then $\tilde{\rho}$ is finite flat. \end{dfn} \begin{eg} Let $\tilde{\rho} = \rho_{f,\lambda}$ ($\lambda | l$). Then $\ord_p N(f) = \ord_p N(\rho) + \dim(\rho)^{I_p} - \dim(\tilde{\rho})^{I_p}$. We have $\ord_p N(\rho) + \dim(\rho)^{I_p} = 2$, and $\dim(\rho)^{I_p} - \dim(\tilde{\rho})^{I_p} \geq 0$. So, $\ord_p N(f) \leq 2$. If $p \notin \Sigma$, then $\ord_p N(f) = \ord_p N(\rho)$. If $\rho$ is ramified at $p$, $\ord_p N(\rho) = 1$ and hence $\ord_p N(f) = 1$. \end{eg} \begin{dfn} $$N_\Sigma = \prod_{p \in \Sigma - \{l\}} p^2 \cdot \prod_{p \notin \Sigma \cup \{l\}} p^{\ord_p N(\rho)} \cdot l^\delta$$ where $\delta = 0$ or $1$, and $\delta = 1$ if and only if (a) $k(\rho) = l+1$ or (b) $l \in \Sigma$ and $\rho | D_l \sim {\left(\begin{array}{cc}\alpha&*\\0&\beta\end{array}\right)} $. \end{dfn} \noindent{\bf Guess.} Suppose $\tilde{\rho} = \rho_{f,\lambda}$ belongs to the class defined by $\Sigma$. Then $N(f) | N_\Sigma$. \begin{exercise} Justify, a priori, the definition of $\delta$. Suppose, for example, that $\rho_{f,\lambda}$ satisfies (*). Prove $l^2 \not| N(f)$. \end{exercise} Our goal is to prove \begin{thm} Every $\tilde{\rho}$ in the class defined by $\Sigma$ comes from $S_2(\Gamma_0(N_\Sigma))$. \end{thm} \begin{remark} (1) $\rho$ comes by reduction from a $\rho_{f,\lambda}$ with $f$ in $S_2(\Gamma_0(N_\Sigma))$ because $N(\rho) | N_\Sigma$ and $k(\rho) = 2$. %or $l+1$(if $k(\rho) = l + 1$, then $\delta = 1$ and so $l | N_\Sigma$. (2) Let $\T = \Z[\cdots T_n \cdots] \subseteq \endo(S_2(\Gamma_0(N_\Sigma)))$ where $(n,lN_\Sigma) = 1$. Then $\exists \M \subseteq \T$ such that \ \ (a) $\exists \tilde{\rho} : G \rightarrow \GL(2,\T_\M)$ such that $\tr \tilde{\rho}(\frob_r) = T_r$. \ \ (b) $\tilde{\rho}$ is universal for liftings of type $\Sigma$. \end{remark} We have $\rho : G \rightarrow \GL(2,\F_{l^\nu})$, $l > 2$, irreducible, modular, $\det\rho = \chi$, semistable. Let $\T = \Z[\cdots T_n \cdots]$ where $(n,l N_\Sigma) = 1$, and let $R = \Z[\cdots T_n \cdots] = \T[T_l;\cdots U_p\cdots]$ where $p|N_\Sigma$. (Note that $\rank_\Z \T = \sum_{M|N_\Sigma} \dim S_2(\Gamma_0(N_\Sigma))$ and $\rank_\Z R = \dim S_2(\Gamma_0(N_\Sigma))$.) Since $N(\rho) | N_\Sigma$, there is a new form $f$ of level $M | N_\Sigma$ whose coefficients are in $\O_\lambda$ and there is a map \begin{diagram}[tight,height=2em] \T &\rTo(2,0) &\O_\lambda \\ &\rdTo(2,2) &\dTo(0,2) \\ & &\F_{l^\nu} \end{diagram} such that $$T_r \mapsto \tr\rho(\frob_r) \in \F_{l^\nu}.$$ Let $\M = \ker(\T \rightarrow \overline{\F})$. We need to show that $\T_\M$ is Gorenstein. We want to find $\M_R \subseteq R$ such that $\T_\M = R_{\M_R}$. To do this, find an eigenform $h$ of level $N_\Sigma$ satisfying $h|U_l = ({\rm unit})h, {\rm unit\ } \in \O_\lambda$, $h|U_p = 0$ for$p \notin \Sigma - \{l\}$, $h|T_p = a_p(f)h$ for $p \in \Sigma$. \noindent Then $h$ gives us the following diagram: \begin{diagram}[tight,height=2em] \M_R &\subseteq&R & & & \\ & & &\rdTo(2,2)\rdTo(4,2)& & \\ \bigcup& &\bigcup& &\O_\lambda&\rTo(0.5,0)&\overline{\F}\\ & & &\ruTo(2,2) & &\ruTo(4,2) \\ \M &\subseteq&\T & & & \end{diagram} where $\M_R = \ker(R \rightarrow \overline{\F})$. \noindent {\bf Construction of $h$.} To construct an $h$, we make the following two changes to $f$. (The order the changes are made in doesn't matter.) \noindent (1) Let $f = \sum a_n q^n$. Then change $f$ to $g = \sum a_n q^n$ where the sum is over $n$ prime to each $p \notin \Sigma$. Then $g$ is an eigenform for $T_p$ for $p \notin \Sigma$. We can do this because $g = (f \tensor \epsilon) \tensor \epsilon$ where $\epsilon$ is a Dirichlet character ramified at $\Sigma$. In terms of L-functions, roughly if $L(f,s) = \prod_p (1- a_p p^{-s} + p^{1 - 2s})^{-1}$, then $L(g,s) = \prod_{p \notin \Sigma} (1- a_p p^{-s} + p^{1 - 2s})^{-1}$. \noindent (2) Suppose $l | N_\Sigma$ but $l \not| N(f)$. This happens if $l \in \Sigma$ and $\rho$ is good ordinary at $l$. We have $f |T_l = a_l(f) f$, but $f|U_l = {\rm \ junk}$. In this case, change $f$ to $g = f + (*) f(q^l)$. If we choose (*) correctly, then $g | U_l = Cg$ where $C$ is a root of $X^2 - a_l X + l = 0$ where $a_l = a_l(f)$. $X^2 - a_l X + l = 0$ has exactly one unit root in $\O_\lambda$ because $a_l \notin \lambda$. So take this unit root and then $g|U_l = ({\rm unit})g$. \begin{remark} When $p || N_\Sigma$, then $p \notin \Sigma$ and $p || N(\rho)$. We have $N(f) = N(\rho)$ or $N(\rho)l$. In this case, we have $f|U_p = a_p(f)f$. So nothing has to be done. \end{remark} The following lemma will be used to prove $\T_\M = R_\M$. \begin{lem} $R = \T[\cdots U_p\cdots]$ if $l \not| N_\Sigma$. \end{lem} \noindent{\bf Proof.\ } It is sufficient to show that $(R : \T[\cdots U_p\cdots])$ is prime to $l$ (since then we cannot tell the difference if we tensor both sides with $\Q_l$). We show that $\T[\cdots U_p\cdots] \rightarrow R/lR$ is surjective. Let $A$ be the image of this map, that is, $A = \F_l[\cdots T_n \cdots]$ where $(n,l) = 1$. We have the duality: $$R/lR \times S_2(\Gamma_0(N_\Sigma);\F_l) \rightarrow \F_l.$$ To show $R/lR = A$ is equivalent to show that $A^\perp = 0$ in this paring. Suppose $f \in A^\perp$ and $f \neq 0$. $f \in A^\perp$ means that $a_n(f) = 0$ for all $n$ such that $(n,l) = 1$. That is, $f$ can be written as $f = \sum a_{nl} q^{nl}$. Then $\theta(f) = 0$. This is a contradiction because $w(f) = 2$ and $w(\theta f) = w(f) + l + 1$ if $l \not| w(f)$. \begin{eg} Suppose $l = 2$ and consider $S_2(\Gamma_0(23))$. Then $R = \Z[\frac{1+\sqrt{5}}{2}]$ and $\Z[\cdots T_n\cdots] = \Z[\sqrt{5}]$ where $(n,l) = 1$. This shows that the lemma does not work for $l = 2$. \end{eg} \end{document}