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\author{\rd{William Stein}\\
Associate Professor of Mathematics\\
University of California, San Diego}
\date{\rd{Math 168a: 2005-09-26}}
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\title{\blue \bf Explicit Approaches to 
Elliptic Curves and Modular Forms}

\begin{document}
\page{
\maketitle
}

\page{
\heading{Outline of Course and this Lecture}
\begin{enumerate}
\item Pythagoras and Fermat
\item Mordell-Weil Groups and the BSD Conjecture
\item Modularity of Elliptic Curves
\item Computing Modular Forms
\end{enumerate}

}

\page{
\heading{The Pythagorean Theorem}
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\heading{Pythagorean Triples}
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\rput[lb](5,0.7){Triples of integers $a,b,c$ such that}
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\rput[lb](0,0){{$\begin{array}{|c|}\hline
\vspace{-2ex}\\
( 3, 4, 5 )\\
( 5, 12, 13 )\\
( 7, 24, 25 )\\
( 9, 40, 41 )\\
( 11, 60, 61 )\\
( 13, 84, 85 )\\
( 15, 8, 17 )\\
( 21, 20, 29 )\\
( 33, 56, 65 )\\
( 35, 12, 37 )\\
( 39, 80, 89 )\\
( 45, 28, 53 )\\
( 55, 48, 73 )\\
( 63, 16, 65 )\\
( 65, 72, 97 )\\
( 77, 36, 85 )
\vspace{-1ex}\\\vdots \\
\hline
\end{array}
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\page{
\heading{Enumerating Pythagorean Triples}

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\ds{\rm Slope} \,=\, t&\ds=\quad\! \frac{y}{x+1}\vspace{2.5ex}\\
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unordered triples}
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\page{
\heading{Fermat's ``Last Theorem''\hspace{3em}\mbox{}}
No analogue of ``Pythagorean triples'' with exponent $3$ or higher.

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\page{
\heading{\large Wiles's Proof of FLT Uses Elliptic Curves}
\vspace{-3ex}
{\large An {\dred elliptic curve} is a nonsingular plane cubic curve with
a rational point (possibly ``at infinity'').}
\vspace{1ex}

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\page{\heading{\mbox{}\hspace{2em}The Frey Elliptic Curve}
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\par\noindent{}Suppose Fermat's conjecture is \rd{FALSE}. 
Then there is a prime $\ell\geq 5$ and coprime
positive integers $a,b,c$ with
$
a^\ell + b^\ell = c^\ell.
$

Consider the corresponding Frey elliptic curve:
$$
y^2 = x(x-a^\ell)(x+b^\ell).
$$

\begin{center}
{\dblue{Ribet's Theorem:}} This elliptic curve is not {\em modular}.

{\gr{Wiles's Theorem:}} This elliptic curve is {\em modular}.

{\rd{Conclusion:}} Fermat's conjecture is true. 
\end{center}
}

\page{
\heading{The Group Operation}

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\rput[lb](4,-3){\bf The set of points }
\rput[lb](4,-3.7){\bf on $E$ forms an \dred abelian group.}
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} % end page


\page{
\heading{The First $150$ Multiples of $(0,0)$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Graph: group2
%% (Contact: William Stein, http://modular.fas.harvard.edu)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\rput[lb](4,2){(The bluer the point, the}
\rput[lb](4,1.2){bigger the multiple.)}
\rput[lb](4,-1){{\dgreen Fact:} The group $E(\Q)$ is infinite}
\rput[lb](4,-1.8){cylic, generated by $(0,0)$.}
\rput[lb](4,-3){In contrast, $y^2+y=x^3-x^2$ has}
\rput[lb](4,-3.8){only $5$ rational points!}
\rput[lb](4,-5){\dred What is going on here?}

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%"$y^2+y = x^3-x$" at position (-1, -6.5)
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\page{
\heading{Mordell's Theorem}
\vspace{2ex}
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{\dred Theorem (Mordell).} The group $E(\Q)$ of rational points on an elliptic
curve is a {\dgreen finitely generated abelian group}, so 
$$
  E(\Q) \cong \Z^r \oplus T,
$$
with $T=E(\Q)_{\rm tor}$ finite. 

\vspace{2ex}


Mazur classified the
possibilities for $T$.  It is conjectured that 
$r$ can be arbitrary, but the biggest $r$ ever
found is (probably) $24$.
} % end page

\page{
\heading{The Simplest Solution\hspace{2in}\mbox{}\\Can Be Huge\hspace{2in}\mbox{}}
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Simplest solution to $y^2=x^3+7823$:
\begin{align*}
x &= \frac{2263582143321421502100209233517777}{143560497706190989485475151904721}\\
\\
y &= \frac{186398152584623305624837551485596770028144776655756}{1720094998106353355821008525938727950159777043481}
\end{align*}
\mbox{}
\par
(Found by Michael Stoll in 2002.)

} % end page

\page{
\heading{The Central Question\hspace{2em}\mbox{}}

{\dgreen\mbox{}\hspace{1em}\noindent{}Given an elliptic curve $E$,\\
\mbox{}\hspace{1em}what is the rank of $E(\Q)$?}\\
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} % end page


\page{
\heading{Idea!: Consider the Group Modulo $p$}
\vspace{-5ex}

$$N(p) = \text{\# of solutions }\,(\text{mod }p)$$
$$y^2 + y = x^3 - x \pmod{7}\vspace{1.1ex}$$

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} % end page


\page{
\heading{Counting Points\hspace{2em}\mbox{}}

{\dgreen\mbox{}\hspace{1em}\noindent{}Cambridge \rd{EDSAC:} The first\\
\mbox{}\hspace{1em}point counting supercomputer...}\\
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} % end page

\page{
\heading{Hecke \dred{Eigenvalues}}
\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{18}{-5}{0.3}{pics/hasse}
\put(19.5,-6){Hasse}
\endpspicture
Let
{\LARGE
$$
a_p = p+1 - N(p).
$$
}
Hasse proved that 
{\Huge\dblue
$$
 |a_p| \leq 2\sqrt{p}.
$$}
For $y^2+y=x^3-x$:
$$
a_2 = -2,\quad a_3 = -3,\quad a_5 = -2,\quad a_7 = -1,
\quad a_{11} = -5,\quad a_{13} = -2,$$
$$a_{17}=0,\quad a_{19} = 0,\quad a_{23}=2,\quad a_{29}=6,\quad \ldots $$
} % end page



\page{
\heading{Birch and Swinnerton-Dyer}
\begin{center}
\includegraphics[height=0.75\textheight]{pics/bsd1}
\end{center}
}


\page{
\heading{The $L$-Function}
{
\psset{unit=3.0}
\pspicture(0,0)(0.1,0.1)
\rput[lb](6,0){\includegraphics[width=7em]{pics/wiles1}}
\rput[lb](0,0){\includegraphics[width=7em]{pics/hecke_in_front}}
\endpspicture

{\dred Theorem (Wiles et al., Hecke)} The following
function extends to a holomorphic function on the
whole complex plane:
\Large $$
  L^*(E,s) = \prod_{p\nmid \Delta} 
  \left(\frac{1}{1 - a_p \cdot p^{-s} + p \cdot p^{-2s}}\right). 
$$}
Here
$  a_p = p+1-\#E(\F_p)$ for all $p\nmid \Delta_E$.
Note that formally,
$$
  L^*(E,1) = 
\prod_{p\nmid \Delta} 
  \left(\frac{1}{1-a_p\cdot p^{-1} + p \cdot p^{-2}}\right)
 = 
\prod_{p\nmid \Delta} 
  \left(\frac{p}{p-a_p  + 1}\right)
= \prod_{p\nmid \Delta} 
\frac{p}{N_p}
$$

Standard extension to $L(E,s)$ at bad primes.
} % end page

\page{
\heading{Real Graph of the $L$-Series of $y^2+y=x^3-x$}
\begin{center}
\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{-8}{-12}{0.8}{pics/lser}
\endpspicture
\end{center}

} % end page

\page{
\heading{More Graphs of Elliptic Curve $L$-functions}
\vspace{6ex}

\begin{center}
\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{-8}{-12}{0.8}{pics/many_lser}
\endpspicture
\end{center}
} % end page

\page{
\heading{Absolute Value of $L$-series on Complex Plane for $y^2+y=x^3-x$}
\vspace{6ex}

\begin{center}
\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{-10}{-12}{0.9}{pics/abs_elseries-37A}
\endpspicture
\end{center}
} % end page



\page{
\heading{Conjectures Proliferated}
\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{0}{-1.3}{0.2}{pics/birch1}
\endpspicture

``The subject of this lecture is rather a special one.  I want to
describe some computations undertaken by myself and Swinnerton-Dyer on
EDSAC, by which we have calculated the zeta-functions of certain
elliptic curves.  As a result of these computations we have found an
analogue for an elliptic curve of the Tamagawa number of an algebraic
group; and conjectures have proliferated.  [...] though the associated
theory is both abstract and technically complicated, the objects about
which I intend to talk are usually simply defined and often machine
computable; {\dblue experimentally we have detected certain relations between
different invariants}, but we have been unable to approach proofs of
these relations, which must lie very deep.''
\hfill -- Birch 1965

} % end page


\page{
\heading{The Birch and Swinnerton-Dyer Conjecture}
\begin{center}
\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{-7}{-12}{0.7}{pics/birch_and_swinnerton-dyer}
\endpspicture
\end{center}
\vspace{-4ex}

{\dred Conjecture:}
Let $E$ be any elliptic curve over~$\Q$.
The order of vanishing of $L(E,s)$ as $s=1$
equals the rank of $E(\Q)$.
} % end page

\page{
\heading{The Kolyvagin and Gross-Zagier Theorem}

\begin{center}
\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{-11}{-12}{0.3}{pics/koly}
\eps{-2}{-12}{0.25}{pics/gross}
\eps{6}{-12}{0.2}{pics/zagier}
\endpspicture
\end{center}
\vspace{-4ex}


{\dred Theorem:} If the ordering of vanishing $\ord_{s=1} L(E,s)$ is $\leq 1$,
then the conjecture is true for $E$.


} % end page




\page{
\heading{Elliptic Curves are ``Modular''}
An elliptic curve is {\em\rd{modular}} if the numbers
$a_p$ are coefficients of a ``modular form''.   
Equivalently, if $L(E,s)$ extends to a complex analytic function
on $\C$ (with functional equation).

{\bf Theorem (Wiles et al.):} {\em Every elliptic curve over the rational
numbers is modular.}  

\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{7}{-6}{0.4}{pics/wiles-princeton}
\put(6.4,-7){{\tiny Wiles at the Institute for Advanced Study}}
\endpspicture
}


\page{
\heading{Modular Forms}
The definition of  modular 
forms as holomorphic functions satisfying 
a certain equation is very abstract. 

For today, I will skip the abstract definition, and instead give you
an explicit ``engineer's recipe'' for producing modular forms.  In the
meantime, here's a picture:

\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{5}{-9}{0.6}{pics/modform37a}
\endpspicture


}

\page{
\heading{Computing Modular Forms: Motivation}

\vfill

\rd{Motivation:} Data about modular forms is \rd{extremely} useful to
many research mathematicians (e.g., number theorists, cryptographers).  This data is like the astronomer's telescope images.

\vfill

One of my longterm research goals is to compute modular forms on a
\rd{\Huge huge} scale, and make the resulting database widely
available.  I have done this on a smaller scale during the last 5 years
--- see {\tt http://modular.ucsd.edu/Tables/} \vfill

}

\page{
\heading{What to Compute: Newforms}

For each positive integer $N$ there is a finite list of \rd{newforms}
of level $N$.  E.g., for $N=37$ the newforms are
\begin{align*}
  f_1 &= q - 2q^2 - 3q^3 + 2q^4 - 2q^5 + 6q^6 - q^7 + \cdots\\
  f_2 &= q + q^3 - 2q^4 - q^7 + \cdots,
\end{align*}
where $q=e^{2\pi i z}$.

The newforms of level~$N$ determine all the modular forms of level~$N$
(like a basis in linear algebra).  The coefficients are algebraic integers.
{\em Goal: compute these newforms.}

{\small Bad idea -- write down many elliptic curves and compute the numbers
$a_p$ by counting points over finite fields.  No good -- this misses
most of the interesting newforms, and gets newforms of all kinds of
random levels, but you don't know if you get everything of a given
level.}

}

\page{
\heading{An Engineer's Recipe for Newforms}
Fix our positive integer $N$.  For simplicity assume that $N$ is prime.
{\small
\begin{enumerate}
\item Form the $N+1$ dimensional $\Q$-vector space $V$ with basis the
  symbols $[0], \ldots, [N-1], [\infty]$.
\item Let $R$ be the suspace of $V$ spanned by the following
vectors, for\\ $x=0,\ldots, N\!-\!1, \infty$:
\begin{align*}
&  [x] - [N-x]  \\
&  [x] + [x.S] \\
&  [x] + [x.T] + [x.T^2] \\
\end{align*}  
$S=\abcd{0}{-1}{1}{\hfill 0}$, $T=\abcd{0}{-1}{1}{-1}$,
and $x.\abcd{a}{b}{c}{d} = (ax + c)/(bx+d)$.

\item Compute the quotient vector space $M = V/R$.  This involves
``intelligent'' {\dblue sparse Gauss elimination} on a matrix with
$N+1$ columns.

\newpage
\mbox{}
\vfill
\item Compute the matrix $T_2$ on $M$ given by
$$ [x]\mapsto [x.\abcd{1}{0}{0}{2}] + [x.\abcd{2}{0}{0}{1}] + [x.\abcd{2}{1}{0}{1}]
      + [x.\abcd{1}{0}{1}{2}].
$$
This matrix is unfortunately not sparse.
Similar recipe for matrices $T_n$ for any $n$.

\item Compute the {\dblue characteristic polynomial} $f$ of $T_2$. 

\item {\dblue Factor} $f = \prod g_i^{e_i}$.  Assume all $e_i=1$ (if not,
use a random  linear combination of the $T_n$.)

\item Compute the {\dblue kernels} $K_i=\ker(g_i(T_2))$.  The {\dblue eigenvalues}
of $T_3$, $T_5$, etc., acting on an {\dblue eigenvector} in $K_i$
give the coefficients $a_p$ of the newforms of level~$N$.
\end{enumerate}
}
\vfill
}


\page{
\heading{Implementation}
\begin{itemize}
\item I implemented code for computing modular forms that's
included with \rd{MAGMA} (non-free, closed source):\\
{\tt http://magma.maths.usyd.edu.au/magma/}.

\item I want something better, so I'm
implementing modular symbols algorithms as part of \rd{SAGE}:\\
   {\tt http://modular.ucsd.edu/sage/}.

\item I'm finishing a \rd{book} on these algorithms that will be
  published by the American Mathematical Society.

\end{itemize}
}

\page{ \heading{The Modular Forms Database Project} 
\vfill
{\small\begin{itemize}
\item  Create a database of all newforms of level $N$ for each $N<100000$.
This will require many gigabytes to store.  (50GB?)
\item So far this has only been done for $N<7000$ (and is incomplete),
  so $100000$ is a \rd{major challenge}.

\item  Involves sparse linear algebra over $\Q$ on spaces of
  dimension up to $200000$ and dense linear algebra on spaces
  of dimension up to $25000$.   

\item Easy to parallelize -- run one process for 
  each $N$.

\item Will be very useful to number theorists and cryptographers.

\item John Cremona has done something similar but only for the
  newforms corresponding to elliptic curves (he's at around 120000
  right now), so this should be do-able.
\end{itemize}
}
\vfill
}


\page{ \heading{Goals for Math 168}
{
\begin{itemize}
\vfill
\item{}{[\bf Elliptic Curves]} Definition, group structure,
  applications to cryptography, $L$-series, the Birch and
  Swinnerton-Dyer conjecture (a million dollar Clay Math prize
  problem).

\vfill
\item{}{[\bf Modular Forms]} Definition (of modular forms of weight
  $2$), connection with elliptic curves and Andrew
  Wiles's celebrated proof of Fermat's Last Theorem,
  how to use modular symbols to compute modular forms.

\vfill 
\item{}{[\bf Research]} Get everyone in 168a involved in some aspect
  of my research program: algorithms needed for SAGE, making data available
  online, efficient linear algebra, etc.

\end{itemize}
}
}




\end{document}
