slides.tex

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\usepackage{graphicx}
\usepackage{booktabs}
\usepackage{amsmath}
\usepackage{sage}

\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\renewcommand{\P}{\mathbb{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}

\newcommand{\isom}{\cong}
\newcommand{\ncisom}{\approx}   % noncanonical isomorphism
\newcommand{\union}{\cup}

\usepackage{amsthm}
\theoremstyle{plain}
\newtheorem{proposition}[theorem]{Proposition}
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\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
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\title[Modular Symbols]{Modular Symbols Statistics}

\author{William Stein\footnote{Joint work-in-progress with Barry Mazur and Karl Rubin.}}
\institute[UW]
{
University of Washington \\
\medskip
\textit{[e-mail]}
}
\date{30m talk on May 17, 2015 in Eugene, Oregon\\
Slides at \url{http://tinyurl.com/modsymdist}\\
Video at \url{http://youtu.be/mSGiSCLGug8}}

\begin{document}

\begin{frame}
\titlepage
\end{frame}

\begin{frame}
\frametitle{Overview}
\tableofcontents
\end{frame}

\section{Modular symbols and $L$-functions}


\begin{frame}
\frametitle{Modular symbols associated to an elliptic curve}
\begin{itemize}
\item {\em Elliptic curve:} $E/\Q$, modular form $f=f_E=\sum a_n q^n$.
\item {\em Period mapping:} integration defines a map $\P^1(\Q)=\Q\union\{i\infty\} \to \C$ given
by $\alpha \mapsto \int_{i\infty}^{\alpha} 2\pi i f(z) dz$.
\item {\em Homology:} $H_1(E,\Z)\isom \Lambda_E \subset \C$ is the image of all integrals of closed paths in the upper half plane, and
$E(\C)\isom \C/\Lambda_E$.
\item {\em Complex conjugation:}
$\Lambda_E^+ \oplus \Lambda_E^- \subset \Lambda_E$
has index $1$ or $2$.
Write $\Lambda_E^+ = \Z \omega^+$, where $\omega^+>0$ is well defined.
\item {\em Modular symbols:}
$
 [\alpha]^+_E:\P^1(\Q) \to \Q
$
$$
 \{\alpha\}^+_E = \frac{1}{2}\left( \int_{i\infty}^{\alpha} 2\pi i f(z) dz
    + \int_{i\infty}^{-\alpha} 2\pi i f(z) dz\right)
     = [\alpha]_E^+ \cdot \omega^+
$$
{\em WARNING:} Cannot evaluate by switching order of summation and integration!
\end{itemize}

\end{frame}

\begin{frame}[fragile]
\frametitle{Example}

We compute some modular symbols using SageMath.
Despite the numerical definitions above, the following computations
are entirely algebraic.


\begin{sagecell}
E = EllipticCurve('11a')
s = E.modular_symbol()
s(17/13)
\end{sagecell}
\begin{sageout}
-4/5
\end{sageout}

Let's compute more symbols:
\begin{sagecell}
[s(n/13) for n in [-13..13]]
\end{sagecell}
\begin{sageout}
[1/5, -4/5, 17/10, 17/10, -4/5, -4/5, -4/5, -4/5, -4/5,
-4/5, 17/10, 17/10, -4/5, 1/5, -4/5, 17/10, 17/10, -4/5,
-4/5, -4/5, -4/5, -4/5, -4/5, 17/10, 17/10, -4/5, 1/5]
\end{sageout}

Lots of random-looking rational numbers... patterns...?

Symmetry: $[a/M]^+ = [-a/M]^+$ and $[1 + (a/M)]^+ = [a/M]^+$.

\end{frame}


\begin{frame}
\frametitle{A motivation for considering modular symbols: $L$-functions}
$L$-series of $E$:
$L(E,s) = \sum a_n n^{-s}$,
where $a_p = p+1-\#E(\F_p)$.

\vspace{1ex}

For each Dirichlet character
$\chi:(\Z/M\Z)^* \to \C^*$ there is
a twisted $L$-function $L(E,\chi,s) = \sum \chi(n) a_n n^{-s}$.
Moreover,

$$
\frac{L(E,\chi,1)}{\omega_\chi} = \text{explicit sum involving $\left[\frac{a}{M}\right]^{\pm}$ and Gauss sums}
$$

\vfill

So... statistical properties of the set of numbers
$$
Z(M) = \left\{\left[\frac{a}{M}\right]^+ : a = 0,\ldots,M-1 \right\}
$$
are relevant to understanding special values of twists.

\vfill

(\small Note:
$[a/M]^+ = [1-a/M]^+$, but we
leave in this
redundant data as a double-check
on our calculations below!)
\end{frame}

\section{Statistics of modular symbols}


\begin{frame}[fragile]
\frametitle{Frequency histogram: $M=100$}
\begin{sagecell}
E = EllipticCurve('11a'); s = E.modular_symbol()
M = 100; v = [s(a/M) for a in range(M)]; print(v)
stats.TimeSeries(v).plot_histogram()
\end{sagecell}
{\small \begin{sageout}
[1/5, 1/5, 6/5, 1/5, -3/10, -4/5, 6/5, 1/5, -3/10, 1/5,
1/5, 1/5, -3/10, 1/5, 6/5, 17/10, 11/5, 27/10, 6/5, 1/5,
6/5, 27/10, 6/5, 27/10, -3/10, 7/10, 6/5, 1/5, -3/10,
27/10, 1/5, -23/10, -3/10, 1/5, -13/10, -4/5, -3/10,
-23/10, 6/5, -23/10, -13/10, -23/10, -19/5, -23/10,
-3/10, -4/5, -13/10, -23/10, -3/10, -23/10, -4/5,
-23/10, -3/10, -23/10, -13/10, -4/5, -3/10, -23/10,
-19/5, -23/10, -13/10, -23/10, 6/5, -23/10, -3/10,
-4/5, -13/10, 1/5, -3/10, -23/10, 1/5, 27/10, -3/10,
1/5, 6/5, 7/10, -3/10, 27/10, 6/5, 27/10, 6/5, 1/5,
6/5, 27/10, 11/5, 17/10, 6/5, 1/5, -3/10, 1/5, 1/5,
1/5, -3/10, 1/5, 6/5, -4/5, -3/10, 1/5, 6/5, 1/5]
\end{sageout}}
\begin{center}
\includegraphics[width=.4\textwidth]{plots/11a-Z100}
\end{center}

\end{frame}


\begin{frame}[fragile]
\frametitle{Frequency histogram: $M=1000$}
\begin{sagecell}
E = EllipticCurve('11a')
s = E.modular_symbol()
M = 1000
stats.TimeSeries([s(a/M) for a in range(M)]).plot_histogram()
\end{sagecell}
\includegraphics[width=\textwidth]{plots/11a-Z1000}


\end{frame}


\begin{frame}[fragile]
\frametitle{Frequency histogram: $M=10000$}
\begin{sagecell}
E = EllipticCurve('11a')
s = E.modular_symbol()
M = 10000
stats.TimeSeries([s(a/M) for a in range(M)]).plot_histogram()
\end{sagecell}
\includegraphics[width=\textwidth]{plots/11a-Z10000}

We quickly want {\bf\em much} larger $M$ in order to see
what might happen in the limit, and the code in Sage
is way too slow for this...

\end{frame}

\begin{frame}[fragile]
\frametitle{More frequency histograms: use Cython...}
\begin{sagecell}
%load modular_symbol_map.pyx
def ms(E, sign=1):
    g = E.modular_symbol(sign=sign)
    h = ModularSymbolMap(g)
    d = float(h.denom)  # otherwise get int division!
    return lambda a,b: h._eval1(a,b)[0]/d
s = ms(EllipticCurve('11a'))
M = 100000  # the following takes about 1 second
stats.TimeSeries([s(a, M) for a in range(M)]).plot_histogram()
\end{sagecell}
\includegraphics[width=.9\textwidth]{plots/11a-Z100000}


\end{frame}


\begin{frame}[fragile]
\frametitle{More frequency histograms (Cython)}
\begin{sagecell}
s = ms(EllipticCurve('11a'))
M = 1000000  # the following takes about 1 second
stats.TimeSeries([s(a, M) for a in range(M)]).plot_histogram()
\end{sagecell}
\includegraphics[width=\textwidth]{plots/11a-Z1000000}

Note that there are only $38$ distinct values in $Z(10^6)$
and $40$ in $Z(1500000)$.

\vfill

\end{frame}

\begin{frame}
\frametitle{Sorry...}
\Large
\begin{itemize}
\item But I can't tell you ``the answer'' yet.
\item Not sure  {\em this} is a good question.
\item So let's consider another question...
\end{itemize}
\end{frame}


\section{Random walks?}


\begin{frame}[fragile]
\frametitle{Return to $M=13$ and make a ``random walk''}
\begin{sagecell}
E = EllipticCurve('11a')
s = E.modular_symbol()
M = 13; v = [s(a/M) for a in range(M)]; print(v)
w = stats.TimeSeries(v).sums()
w.plot() + points(enumerate(w), pointsize=30, color='black')
\end{sagecell}
\begin{sageout}
[1/5, -4/5, 17/10, 17/10, -4/5, -4/5, -4/5, -4/5,
-4/5, -4/5, 17/10, 17/10, -4/5]
\end{sageout}
\includegraphics[width=\textwidth]{plots/walk-11a-Z13}
\end{frame}

\begin{frame}[fragile]
\frametitle{How about $M=20$?}
\begin{sagecell}
s = EllipticCurve('11a').modular_symbol()
M = 20; v = [s(a/M) for a in range(M)]
w = stats.TimeSeries(v).sums()
w.plot() + points(enumerate(w), pointsize=30, color='black')
\end{sagecell}
\includegraphics[width=\textwidth]{plots/walk-11a-Z20}
\end{frame}

\begin{frame}[fragile]
\frametitle{How about $M=50$?}
\includegraphics[width=\textwidth]{plots/walk-11a-Z50}
\end{frame}

\begin{frame}[fragile]
\frametitle{How about $M=100$?}
\includegraphics[width=\textwidth]{plots/walk-11a-Z100}
\end{frame}

\begin{frame}[fragile]
\frametitle{How about $M=1000$?}
\includegraphics[width=.9\textwidth]{plots/walk-11a-Z1000}
\end{frame}

\begin{frame}[fragile]
\frametitle{How about $M=10000$?}
\includegraphics[width=\textwidth]{plots/walk-11a-Z10000}
\end{frame}

\begin{frame}[fragile]
\frametitle{How about $M=100000$?}
\includegraphics[width=\textwidth]{plots/walk-11a-Z100000}
\end{frame}

\begin{frame}[fragile]
\frametitle{How about $M=100003$ next prime after $100000$?}
\includegraphics[width=\textwidth]{plots/walk-11a-Z100003}
\end{frame}

\begin{frame}
\frametitle{Notice Anything?}

\begin{itemize}
\item The pictures {\bf all look almost the same}, as if they are converging
to some limiting function.

\item Similar observation about other elliptic
curves (with a {\bf different picture}).

%\item There's a similar pattern for the
%$-1$ modular symbol $[\alpha]_E^{-}$.

\item Similar definition for modular symbols
attached to newforms with Fourier coefficients
in a number field, or of {\bf higher weight} (we
get a multi-dimensional random walk).
\begin{center}
\includegraphics[width=.5\textwidth]{plots/level7-weight3-1000}
\end{center}
\end{itemize}

\end{frame}

\begin{frame}
\vfill
{\Large Several different elliptic curves}
\vfill
\end{frame}

\begin{frame}
\frametitle{Sum for $M=10^6$ and $E=11a$ (rank 0)}
\includegraphics[width=.9\textwidth]{plots/walk-11a-Z1000000}

\end{frame}

\begin{frame}
\frametitle{Sum for $M=10^6$ and $E=37a$ (rank 1)}
\includegraphics[width=.9\textwidth]{plots/walk-37a-Z1000000}

\end{frame}

\begin{frame}
\frametitle{Sum for $M=10^6$ and $E=389a$ (rank 2)}
\includegraphics[width=.9\textwidth]{plots/walk-389a-Z1000000}
\end{frame}



\begin{frame}
\frametitle{Taking the limit}
\begin{itemize}
\item
{\em Normalize} the ``not so random walk''
so it is comparable for different values of $M$.
Consider
$f_M:[0,1] \to \Q$
given by
$$
f_M(x) = \frac{1}{M}\cdot \sum_{a=1}^{Mx} \left[\frac{a}{M}\right]^+,
\qquad\text{(write on board)}
$$
where, by
$\sum_{a=1}^{Mx}$ we mean
$\sum_{a=1}^{\lfloor M x \rfloor}$.

\item
\begin{conjecture}[-]
\begin{center}
The limit $f(x) = \lim_{m\to\infty} f_M(x)$ exists.
\end{center}
\end{conjecture}
\end{itemize}

\begin{center}
\small
(all conjectures in this talk are by Mazur-Rubin-Stein)
\end{center}

\end{frame}


\begin{frame}
\frametitle{What is the limit?}
\begin{itemize}
\item
Let $\omega^+$ be the least real period as before.  (NOTE: This need
not be the $\Omega_E$ in the BSD conjecture, since
when the period lattice is rectangular then $\Omega_E=2\omega^+$.)

\item Let $\sum a_n q^n$ be the newform attached to the elliptic
curve $E$.  Then:

\begin{conjecture}[-]
$$
f(x) =
\frac{1}{2\pi \omega^+} \cdot
\sum_{n=1}^{\infty} \frac{a_n \sin(2\pi n x)}{n^2}.
$$
\end{conjecture}
\end{itemize}

\end{frame}


\begin{frame}
\frametitle{Mazur's {\bf Heuristic} Argument}

\begin{itemize}
\item Define $\{\alpha\}^+$ exactly as before, but for {\em any} $\alpha\in \mathfrak{h}^*$:
$$
\{\alpha\}^+ = \frac{1}{2}\left( \int_{i\infty}^{\alpha} 2\pi i f(z) dz
    + \int_{i\infty}^{-\overline{\alpha}} 2\pi i f(z) dz\right)
    \in \R.
$$
\item When $\alpha=x+i\eta$, with $x\in\R$ and $\eta > 0$, evaluate $\{\alpha\}^+$
by switching summation
and integration (can since $\alpha\not \in\Q$!):
$$
\{\alpha\}^+ = \{x+i\eta \}^+ =
\sum_{n=1}^{\infty} \frac{a_n e^{-2\pi \eta n}}{n} \cos(2\pi n x)
\in \R.
$$
\item Fix $\eta>0$ and $b\in [0,1]$ and integrate the real function $x\mapsto \{x + i \eta\}^+$
above from $0$ to $b$:
$$
\int_{0}^b
\{x+i\eta\}^+ dx =
\frac{1}{2\pi} \cdot
\sum_{n=1}^{\infty} \frac{a_n e^{-2\pi \eta n}}{n^2}
\cdot \sin(2\pi n b).
$$
\end{itemize}

\end{frame}


\begin{frame}
\frametitle{Mazur's {\bf Heuristic} Argument (continued)}

Previous slide:
$$
\int_{0}^b
\{x+i\eta\}^+ dx =
\frac{1}{2\pi} \cdot
\sum_{n=1}^{\infty} \frac{a_n e^{-2\pi \eta n}}{n^2}
\cdot \sin(2\pi n b).
$$


Riemann sum approximation to this integral
at points $a/M$, and divide by
$\omega^+$ to get (heuristic!):
$$
 f_M(x) = \frac{1}{M}\cdot \sum_{a=1}^{Mx} \left[\frac{a}{M}\right]^+
   \sim
\frac{1}{2\pi \omega^+} \cdot
\sum_{n=1}^{\infty} \frac{a_n e^{-2\pi \eta n}}{n^2}
\cdot \sin(2\pi n x).
$$

Take the limit as $\eta\to 0$ and $M\to\infty$ to
``deduce'' our conjecture that
$
f(x) =
\frac{1}{2\pi \omega^+} \cdot
\sum_{n=1}^{\infty} \frac{a_n \sin(2\pi n x)}{n^2}.
$

\vfill
(Show worksheet and plots if time permits...)
\end{frame}



\begin{frame}

\Huge{\centerline{The End}}
\end{frame}

\end{document}


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