1

数理解析研究所講究録
第2012巻 2016年 1-11

Overconvergent Modular Symbols in Sage
David Roe(with Rob Pollack, Rob Harron, Ander Steele and others)
Department of Mathematics
University of Pittsburgh

Towards new development of mathematics via computational

algebra systems
RIMS, Kyoto

OutlineModular Forms

Modular Symbols\mathrm{Q} Overconvergent Modular Symbols$\Phi$ Positive Slope Families

2Classical Modular Forms

H- upper half plane: complex numbers z=x+iy with y>0.

$\Gamma$_{0}(N)\subseteq \mathrm{S}\mathrm{L}_{2}(\mathbb{Z}) consists of $\gam a$=\left(\begin{ar ay}{l ba\ cd \end{ar ay}\right) with N|c.
Acts on (H byk‐an$\gamma$\displaystyle \cdot z=\frac{az+b}{cz+d} Example of level $\Gamma$.
a.

integer, the weight.

M_{k}( $\Gamma$)- holomorphic functions f : H\rightarrow \mathbb{C} withf( $\gamma$\cdot z)=(cz+d)^{k}f(z) for $\gamma$\in $\Gamma$ These are modular forms.
.

Since (_{01}^{11} ) \in$\Gamma$_{0}(N),f(z+1)=f(z)around i\infty : if q=e^{2 $\pi$ iz},.

Get a Fourier expansionf(z)=\displaystyle\sum_{n=0}^{\infty}a_{n}q^{n}Note: a_{n}=0 for n<0 is an additional condition on f.

Examples: Eisenstein Series
For k>2 even,

G_{k}(z)=$\Sigma$_{(m,n)\neq(0,0)}\displaystyle \frac{1}{(mz+n)^{k} \in M_{k}(\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})

G_{k}(z)=2 $\zeta$(k)(1+\displaystyle \frac{2}{ $\zeta$(1-k)}\sum_{n=1}^{\infty}$\sigma$_{k-1}(n)q^{n})where.

,$\sigma$_{k-1}(n)=$\Sigma$_{0<d|n}d^{k-1}0For other $\Gamma$

0

Basis for Eisenstein series of forms that take value 1 on one cusp

,a cusp is a $\Gamma$ ‐orbit on\mathbb{Q}\cup\{\infty\}.

and zero on others.
0Cusp forms S_{k}( $\Gamma$)\subset M_{k}( $\Gamma$) are those vanishing on all cusps.

3

Examples: Modular Forms from Elliptic Curves
If E is an elliptic curve y^{2}=x^{3}+ax+b over \mathb {Q} with discriminant-16(4a^{3}+27b^{2}) and conductor N (same prime factors),a_{p}=(p+1)-\# E(\mathrm{F}_{p}) if p { Na_{p}=0 if E has additive reductiona_{p}=1 if E has split multiplicative reductiona_{p}=-1 if E has non‐split multiplicative reductiona_{p^{r}}=a_{p^{r-1}}\cdot a_{p}-p\cdot a_{p^{r-2}} if pfN

a_{p^{r}}=a_{p}^{r} if p|Na_{mn}=a_{m}\cdot a_{n} if(m, n)=1.

f_{E}=\displaystyle \sum_{n=1}^{\infty}a_{n}q^{n}\in M_{2}($\Gam a$_{0}(N).

Hecke Operators

0For fixed k and $\Gamma$ the space M_{k}( $\Gamma$) is finite dimensional (with,

explicit dimensions via Riemann‐Roch).
\mathrm{Q}For each n\geq 1 there is a linear operator T_{n} on M_{k}( $\Gamma$) and the T_{n},commute with each other.
0An eigenform is a simultaneous eigenvector for these operators(e.g. f_{E} ).

4LMrVB\mathrm{L} ‐functions and Modular Forms Database

Break for demo of http: / \mathrm{b}\mathrm{e}\mathrm{t}\mathrm{a}.

lmfdb. org.

Modular Symbols

For k>1,

computation made possible by modular symbols.

\triangle 0-\mathrm{D}\mathrm{i}\mathrm{v}^{0}(\mathbb{P}^{1}(\mathbb{Q}) : formal sums \displaystyle \sum_{ $\alpha$\in \mathb {Q}\cup\{\infty\} a_{ $\alpha$} $\alpha$ with \displaystyle \sum_{ $\alpha$}a_{ $\alpha$}=0.

S_{0}(p)-\{\left(\begin{ar ay}{l } a & b\\ c & d \end{ar ay}\right)|(a,p)=1,p|c and ad‐bc \neq 0\}.
V-\mathrm{a}\mathbb{Z} ‐module (e.g. \mathb {C} or

S_{0}(p)\mathrm{S}\mathrm{y}\mathrm{m}^{k-2}(\mathb {C}) ) with right actions of $\Gamma$ and.

$\Gamma$- acts on \mathrm{H}\mathrm{o}\mathrm{m}(\triangle 0, V) by ( $\varphi$| $\gamma$)(D)= $\varphi$( $\gamma$ D)| $\gamma$.

\mathrm{S}\mathrm{m}\mathrm{b}_{ $\Gamma$}(V)-\{ $\varphi$\in \mathrm{H}\mathrm{o}\mathrm{m}(\triangle_{0}, V)| $\varphi$= $\varphi$| $\gamma$\}.

$\varphi$|T_{t'}= $\varphi$|(_{01}^{t0})+\displaystyle \sum_{a=0}^{\el -1} $\varphi$|\left(\begin{ar ay}{l} 1a\ 0\el \end{ar ay}\right) for \ell $\dager$ N.
U_{q}- acts by $\varphi$|U_{q}=\displaystyle \sum_{a=0}^{q-1} $\varphi$|(0_{q}1a) for q|N.
Tp‐ acts by5Manin Relations
G‐PSL2 ( \mathb {Z} )[ $\gamma$]-\displaystyle \frac{b}{d}-\frac{a}{c}\in\triangle 0 when $\gamma$=\left(\begin{ar ay}{l} ba\ cd \end{ar ay}\right)\in G.

o^{-}-(_{10}^{0-1}) a two‐torsion element.
$\tau$-(_{1-1}^{0-1}) a three‐torsion element.
I- the left ideal \mathbb{Z}[G](1+ $\sigma$)+\mathbb{Z}[G](1+ $\tau$+$\tau$^{2}),,.

\{g_{i}\}- right coset reps for $\Gamma$\backslash G generate \mathbb{Z}[G] as a free \mathb {Z}[ $\Gamma$] ‐module.
,

Using continued fractions, every element of \triangle 0 is the sum ofelements [ $\gamma$] so get surjective map,\mathbb{Z}[G]\rightarrow\triangle_{0}.
Manin showed that the kernel is I Therefore \triangle 0 is generated by the.

g_{i} with relations given by I For instance,.

,g_{i}(1+ $\sigma$)=g_{i}+g_{i}$\sigma$^{-}=g_{j}+$\gamma$_{ij}g_{j}.

Modular Symbols to Modular Forms

Theorem (Eichler Shimura)\mathrm{S}\mathrm{m}\mathrm{b}_{ $\Gamma$}(\mathrm{S}\mathrm{y}\mathrm{m}^{k-2}(\mathbb{C}) \cong M_{k}( $\Gamma$)\oplus S_{k}( $\Gamma$)
So to compute M_{k}( $\Gamma$),

as Hecke‐modules.

weO Using Manin relations, write down a basis for\mathrm{S}\mathrm{m}\mathrm{b}_{ $\Gamma$}(\mathrm{S}\mathrm{y}\mathrm{m}^{k-2}(\mathb {C}).

$\Phi$ Compute matrices for action of U_{q} and T_{\ell} for small \ell.
O Diagonalize to get systems of Hecke eigenvalues \{a_{\ell}\}.
$\Phi$ These systems provide the Fourier coefficients for a basis ofeigenforms in M_{k}( $\Gamma$).

6p ‐adic Numbers

Fix p prime. Recall:v_{p}- For a, b prime to p,

setv_{p}(p^{v}\displaystyle \cdot\frac{a}{b})=v and |p^{v}\displaystyle \cdot\frac{a}{b}|_{p}=p^{-v}.

\mathbb{Q}_{p}- Completion of \mathb {Q} with norm |\cdot|_{p} Then \mathbb{Z}_{p}=\mathrm{t}z\in \mathbb{Q}_{p} : |z|\leq 1 }.
.

\ovalbox{\t smal REJ CT}_{p} −Alternately, \displaystyle \mathbb{Z}_{p}=\lim_{\leftar ow m}\mathbb{Z}/p^{m}\mathbb{Z} and \displaystyle \mathb {Q}_{p}=\mathb {Z}_{p}[\frac{1}{p}].
0Concretely, of the form \displaystyle \sum_{m=v}^{\infty}a_{m}p^{m} with a_{m}\in\{0, \cdots , p-1\}.
\mathrm{Q}

Computationally, represent as p^{v}\cdot u where u\in(\mathbb{Z}/p^{m}\mathbb{Z})^{\times},

p ‐adic Distributions\displaystyle \mathrm{A}-\{f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\in \mathbb{Q}_{p}[ z] :|a_{n}|\rightarrow 0\};\Vert f| =\sup_{z\in \mathbb{Z}_{p} |f(z)|.

\displaystyle \mathrm{D}-\mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{A}, \mathb {Q}_{p});\Vert $\mu$\Vert=\sup_{0\neq f\in \mathrm{A}\frac{| $\mu$(f)|}{|f|} .
\mathrm{A}_{k} −A with($\gamma$_{k}f)(z)=(a+cz)^{k}\displaystyle \cdot f(\frac{b+dz}{a+cz}) for $\gamma$=\left(\begin{ar ay}{l} ba\ cd \end{ar ay}\right)\in S_{0}(p)\mathrm{D}_{k}-\mathrm{D} with ( $\mu$|_{k} $\gamma$)(f)= $\mu$($\gamma$_{k}f).

V_{k}-\mathrm{S}\mathrm{y}\mathrm{m}^{k}(\mathbb{Q}_{p}^{2})=\mathbb{Q}_{p}[X, Y]_{k} with(P| $\gamma$)(X, \mathrm{Y})=P(dX-c\mathrm{Y}, -bX+a\mathrm{Y})

.

.

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Moments
The map

M:\displayst le\mathrm{D}\rightarow\prod_{j=0}^{\infty}\mathb {Q}_{p}$\mu$\mapsto( $\mu$(z^{j}) j=0^{\infty}

is injective, with image the bounded sequences.
The map$\rho$_{k}:\mathrm{D}_{k}\rightarrow V_{k}$\mu$\displaystyle\mapsto\int(\mathrm{Y}-zX)^{k}d$\mu$(z)=\sum_{j=0}^{k}(-1)^{j}\left(\begin{ar ay}{l k\ j \end{ar ay}\right)$\mu$(z^{j})X^{j}\mathrm{Y}^{k-j}is S_{0}(p) ‐equivariant.

Computing with Distributions\mathrm{D}^{0}- { $\mu$\in \mathrm{D} : $\mu$(z^{j})\in \mathbb{Z}_{p} for all j}.

\mathrm{F}\mathrm{i}1^{m}-\{ $\mu$\in \mathrm{D}^{0}:v_{p}( $\mu$(z^{j}))\geq m-j\}.
\mathcal{F}^{\prime m}-\mathrm{D}^{0}/\mathrm{F}\mathrm{i}1^{m} a finite \mathb {Z}_{p} ‐module.
,We will define Hecke operators via the action of S_{0}(p) and Fi \mathrm{l}^{} ischosen to be stable under this action.

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Overconvergent Modular Symbols$\Phi$Let N be prime to p and $\Gamma$=$\Gamma$_{0}(Np)\subset S_{0}(p)0An overconvergent modular symbol is an element of \mathrm{S}\mathrm{m}\mathrm{b}_{ $\Gamma$} (Dk).

.

Have Hecke operators.
0

Approximate by elements of \mathrm{S}\mathrm{m}\mathrm{b}_{ $\Gam a$}(\mathcal{F}_{k}^{\prime m}) Hecke operators,

descend.
0

The slope of an eigensymbol $\varphi$ is the valuation of theU_{p} ‐eigenvalue.
\mathrm{Q}Specialization map $\rho$^{*}:\mathrm{S}\mathrm{m}\mathrm{b}_{ $\Gamma$}(\mathrm{D}_{k})\rightar ow \mathrm{S}\mathrm{m}\mathrm{b}_{ $\Gamma$}(V_{k}) is surjective,
isomorphism on the slope <(k+1) piece.

Overconvergent Modular Symbols in Sage

Sage
Break for Sage demo: https: / \mathrm{c}\mathrm{l}\mathrm{o}\mathrm{u}\mathrm{d} sagemath. com.

9Appllcallon: p ‐adlc L ‐tunctlons
Classically, $\zeta$(1-k) p ‐adically interpolates for positive integers k.
Kummer congruences:if h\equiv k(\mathrm{m}\mathrm{o}\mathrm{d} $\phi$(p^{m}) then\displaystyle \frac{B_{h} {h}\equiv\frac{B_{k} {k}(\mathrm{m}\mathrm{o}\mathrm{d} p^{m}).

Can do the same for other L ‐functions. For example, if

f\in S_{k+2}( $\Gamma$,\overline{\mathbb{Q} ) is a slope h<k+1 eigenform, define the p ‐adic
L‐function of f to be the unique distribution $\mu$_{f} oncharacter of\mathb {Z}_{p}^{\times} so that ifx is a\mathb {Z}_{p}^{\times} with conductor p^{n} and 0\leq j\leq k then,$\mu$_{f}(z^{j}\displaystyle\cdotX)=\frac{1}{$\alpha$^{n}\cdot\frac{p^{n(j+1)}{(-2$\pi$i)^{j}\cdot\frac{j!}{$\tau$(\mathcal{X}^{-1})\cdot\frac{L(f,x'-1j+1)}{$\Omega$_{f}^{\pm}.
Here a is the U_{p} ‐eigenvalue of f,complex periods.

$\tau$($\chi$^{-1}) is a Gauss sum and $\Omega$_{f}^{\pm} are

Computation of p ‐adic L ‐functions

The classical construction of $\mu$_{f} involves an integral, the computationof which requires a Riemann sum. The resulting algorithm forcomputing $\mu$_{f} is exponential in the desired precision.
Pollack and Stevens show that there is an overconvergent

eigensymbol $\Phi$_{f} lifting the symbol $\varphi$_{f} so that,,$\mu$_{f}=$\Phi$_{f}(\{\infty\}-\{0\})|_{\mathbb{Z}_{p}^{\times}}.
The resulting algorithm for computing $\mu$_{f} is polynomial in the desired

precision.

10p ‐adic L ‐functions in Sage

Sage
Break for Sage demo: https: / \mathrm{c}\mathrm{l}\mathrm{o}\mathrm{u}\mathrm{d}.

sagemath. com

Hida Families

Le \displaystyle \mathrm{t}^{r}W=\lim_{\leftarrow m}(\mathbb{Z}/ $\phi$(p^{m})\mathbb{Z})\cong \mathbb{Z}/(p-1)\mathbb{Z}\times \mathbb{Z}_{p}.
Hida constructed families of overconvergent modular forms withvarying weight. These families
0

consisted of ordinary forms: slope 0,0extended over all of weight spac \mathrm{e}^{r}W,0have constant rank over weight space.

These families form a part of the eigencurve, a rigid analytic object

parameterizing overconvergent modular forms.

11Ongoing work: positive slope families

The remainder of the, eigencurve corresponds to families ofoverconvergent forms with positive slope. Want to compute power
series that give the Hecke eigenvalues as a function of varying weight.
These power series will be valid only in subsets of weight space (discsand annuli).
Idea

Use overconvergent modular symbols to find eigenvalues at specificweights and interpolate.

Overconvergent modular symbols are crucial since the weights will belarge.

More details: positive slope families

Solved Problem

Need to match corresponding eigenvalues between different weights.

Solution: since eigenvalues vary p ‐adically, their reduction modulo p
is constant over small discs. Can use reductions for varying Tp as asignature to match between different weights.
Unsolved Problem
In higher slope, finding eigenvalues at a fixed weight involves\displaytle\frac{U_p}{^h}For positive h we have been unable to avoid devastating
iteratingprecision loss.
.

,