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\author{\rd{William Stein}\\
Associate Professor of Mathematics\\
University of California, San Diego}
\date{\rd{Math 168a: 2005-09-23}}
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\title{\blue \bf Explicit Approaches to 
Elliptic Curves and Modular Forms}

\begin{document}
\page{
\maketitle
}

\page{
\heading{The Pythagorean Theorem}
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\rput[cb](6.0,-0.35){Pythagoras}
\rput[cb](6.0,-0.6){Approx 569--475BC}
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\page{
\heading{Pythagorean Triples}
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\rput[lb](5,0.7){Triples of integers $a,b,c$ such that}
\rput[lb](9,-0.5){{\Large $a^2+b^2=c^2$}}
\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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} % end page


\page{
\heading{Enumerating Pythagorean Triples}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Graph: param
%% (Contact: William Stein, http://modular.ucsd.edu)
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\ds{\rm Slope} \,=\, t&\ds=\quad\! \frac{y}{x+1}\vspace{2.5ex}\\
  \ds x &=\quad\! \ds\frac{1-t^2}{1+t^2}\vspace{2.5ex}\\
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unordered triples}
\rput[lb](-1.3,-1.9){arise in this way.}

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\page{
\heading{Fermat's ``Last Theorem''\hspace{3em}\mbox{}}
No ``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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\rput[lb](5,2){{\dgreen EXAMPLES}}
\rput[lb](4,1){\Large $y^2+y = x^3-x$}
\rput[lb](4,0){{\Large $x^3 + y^3 = 1$} (Fermat cubic)}
\rput[lb](4,-1){{\Large $y^2 = x^3+ax+b$}}
\rput[lb](4,-2){{\Large $3x^3+4y^3+5=0$}}
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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{Counting Solutions 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}}
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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}
{
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{\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}
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\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}
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\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}

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\end{center}
} % 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.}  

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}


\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:

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\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}:\\
{\tt http://magma.maths.usyd.edu.au/magma/}.

\item Unfortunately, MAGMA is expensive and closed source, so I'm
  reimplementing everything 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).
\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}



