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Author: William A. Stein
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% periods.tex
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% AUTHOR: WILLIAM STEIN and HELENA VERRILL
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% SUBJECT: NUMERICAL COMPUTATION OF PERIOD LATTICES ASSOCIATED TO NEWFORMS
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% DATE: APRIL 1999.
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\documentclass[11pt]{article}
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\include{macros}
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\title{Computing Period Lattices of Newforms\\
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(Preliminary)}
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\author{William Stein and Helena Verrill}
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\date{April, 1999}
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\begin{document}
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\maketitle
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\begin{abstract}
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We give an algorithm to numerically approximate
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the period lattice and real volume of any
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modular newforms $f\in S_k(\Gamma,\C)$
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of positive even weight. The computation
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of special values of the associated $L$-functions
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is also discussed. Numerical examples are provided.
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\end{abstract}
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%\section{Introduction}
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\section{Modular Symbols}
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Let $\Gamma=\Gamma_0(N)$ or $\Gamma_1(N)$, and let
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$k$ be an even positive integer. Let
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$\cM$ be the abelian group generated by symbols
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$\{\alpha,\beta\}$ with $\alpha, \beta\in\P^1(\Q)$
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modulo the relations
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$$\{\alpha,\beta\}+\{\beta,\gamma\}+\{\gamma,\alpha\} = 0$$
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and modulo any torsion.
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\begin{center}
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\begin{picture}(100,100)
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\put(0,10){\circle*{3}}
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\put(0,16){$\alpha$}
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\qbezier(0,10)(50,40)(100,10)
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\put(50,25){\vector(1,0){1}}
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\put(100,10){\circle*{3}}
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\put(100,16){$\beta$}
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\qbezier(100,10)(60,30)(50,100)
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\put(65,47){\vector(-1,2){1}}
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\put(50,100){\circle*{3}}
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\put(50,106){$\gamma$}
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\qbezier(0,10)(40,30)(50,100)
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\put(35,47){\vector(-1,-2){1}}
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\end{picture}
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\end{center}
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51
Let
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$$V_{k-2}=\Sym^{k-2}(\Z X + \Z Y)$$
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be the $\Z$-module of homogeneous polynomials in $X$ and $Y$
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of degree $k-2$.
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Define a left action of $g=\abcd{a}{b}{c}{d}\in \Gamma$ on the tensor product
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$V_{k-2}\tensor \cM$ by
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$$g(P(X,Y)\tensor\{\alpha,\beta\})
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= P(dX-bY,-cX+aY)\tensor\{g(\alpha),g(\beta)\}.$$
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The space of modular symbols of weight $k$ for $\Gamma$ is
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the largest quotient of $V_{k-2}\tensor\cM$ on which $\Gamma$
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acts trivially. If we let $I = \{1-g : g\in\Gamma\}\subset\Z[\Gamma]$
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then
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$$\M_k(\Gamma) = \frac{V_{k-2}\tensor\cM}{I(V_{k-2}\tensor\cM)}
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= H_0(\Gamma,V_{k-2}\tensor\cM).$$
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The cuspidal modular symbols are $\cS_k(\Gamma) = \ker(\delta)$,
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where
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$$\delta:\M_k(\Gamma)\ra \bigoplus_{[\alpha]\in \Gamma\backslash\P^1(\Q)}
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\Z[\alpha]$$
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is given by
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$$\delta(P(X,Y)\tensor\{\alpha,\beta\}) = ?.$$
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The Eichler-Shimura theory gives an isomorphism
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$$S_k(\Gamma,\C)\oplus\overline{S}_k(\Gamma,\C)\oplus\Eis(\Gamma,\C)
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\isom H^1(\Gamma,V_{k-2}^*\tensor\C)$$
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For example, when $k=2$ the map
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$$S_2(\Gamma,\C)\ra H^1(\Gamma,\C)=\Hom(\Gamma,\C)$$
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send $f(z)$ to the map $\gamma\mapsto \int_{0}^{\gamma(0)} f(z)dz$.
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The cup product pairing
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$$H_0(\Gamma,V_{k-2}\tensor\cM)\cross H^1(\Gamma,V_{k-2}^*\tensor\C)
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\ra H^0(\Gamma,\C)=\C$$
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gives rise to the period map we will study.
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\section{Computing Periods}
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Let $\Gamma=\Gamma_0(N)$ or $\Gamma_1(N)$. In this section
87
we describe how to numerically approximate the period lattice
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associated to a cuspidal newform
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$$f=\sum a_n q^n \in S_k(\Gamma,\C), \qquad k\geq 2.$$
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We do {\em not} require that the
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Fourier coefficients $a_n$ of $f$ lie in $\Q$, nor that the
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weight equals $2$. In the special case $k=2$ and all $a_n\in\Q$,
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see \cite{cremona}. The algorithm given only applies when
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$k$ is even, though we expect that there is a
95
generalization to odd $k$.
96
97
Let $\T=\Z[T_1,T_2,T_3,\ldots]$ denote the Hecke algebra, equipped
98
with its action on modular forms and modular symbols for $\Gamma$.
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Set $$\p_f = \Ann_\T(f).$$
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Let $f^{(1)},\ldots,f^{(d)}$ be the $\Gal(\Qbar/\Q)$ conjugates of $f$.
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Let $\cM_k(\Gamma,\Q)$ be the space of modular symbols of weight
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$k$ for $\Gamma$ (tensor $\Q$) and $\cS_k(\Gamma,\Z)\subset\cM_k(\Gamma,\Q)$
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the subspace of integral cuspidal symbols.
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There is a pairing
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$$\langle\quad,\quad\rangle : \cM_k(\Gamma,\Q)\cross S_k(\Gamma,\C) \ra \C$$
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$$\langle P(X,Y)\{\alp,\beta\}, \, f \rangle
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= 2\pi i \int_{\alpha}^{\beta} f(z)P(z,1) dz.$$
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The period map associated to $f$ is
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$$\Phi_f : \cM_k(\Gamma,\Q) \ra \C^d$$
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$$\Phi_f(x) = (\langle x,f^{(1)}\rangle, \ldots, \langle x,f^{(d)}\rangle).$$
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Let
112
$$\Lambda_f =\Phi_f(\cS_k(\Gamma,\Z))\subset\C^d.$$
113
114
The following is an algorithm to numerically
115
approximate $\Phi_f$ and $\Lambda_f$.
116
117
\begin{enumerate}
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\item {\bf [Compute left eigenspace.]}
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Compute a $\Q$-basis $\vphi_1,\ldots,\vphi_{2d}$ for
120
$$\Hom(\M_k(\Gamma,\Q),\Q)[\p_f].$$
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\item {\bf [Dual basis.]} Let $w_1,\ldots,w_{2d}$ be the $\Q$-basis for
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$\M_k(\Gamma,\Q)[\p_f]$ dual to $\vphi_1,\ldots,\vphi_{2d}$,
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so $\vphi_i(w_j)=\delta_{ij}$. It is not
124
necessary to compute the $w_i$ though this could
125
be done by inverting the matrix whose rows are the $\vphi_i$.
126
Observe that if $v\in\M_k(\Gamma,\Q)$ then
127
$$\langle v, f\rangle = \sum_{i=1}^{2d} \vphi_i(v) \langle w_i, f\rangle.$$
128
\item {\bf [Compute $\Z$-basis for $\Lambda_f$.]}
129
Compute a $\Z$-basis $z_1,\ldots,z_{2d}$
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for the lattice
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$$\{(\vphi_1(x),\ldots,\vphi_{2d}(x)) : x \in \cS_k(\Gamma,\Z) \}
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\subset \Q^{2d}.$$
133
\item {\bf [Find good periods.]}
134
Locate $v_1,\ldots v_{2d}\in \M_k(\Gamma,\Q)$ so that
135
\begin{itemize}
136
\item Each $v_i$ is of the form
137
$$P_g(X,Y)\{0,g(0)\}$$
138
with $g=\abcd{a}{b}{c}{d}\in \Gamma$, $c>0$, and
139
$$P_g(X,Y) = ((1-ad)X^2 - b(d-a)XY+b^2Y^2)^{\frac{k-2}{2}}.$$
140
Note that $P$ is an eigenvector for the action of $g$
141
on $V_{k-2}$ with eigenvalue $1$.
142
143
{\bf Remarks:}
144
\begin{enumerate}
145
\item This is the step which only makes sense for even weight.
146
We have not yet found a replacement for odd weight.
147
\item It is still necessary to prove the existence of enough $g$'s.
148
This we have not done. In practice a few randomly choosen $g$'s
149
seem to do the job.
150
\end{enumerate}
151
152
\item The matrix $(\vphi_i(v_j))$ is invertible.
153
\end{itemize}
154
\item {\bf [Compute approximate periods.]}
155
The form of each $v_i$ insures that we can obtain good numerical
156
approximations to $\langle v_i, f\rangle.$
157
Write $v_i = P_g(X,Y)\{0,g(0)\}$ with $g=\abcd{a}{b}{c}{d}$
158
and $c>0$.
159
We have
160
$\langle v_i, f\rangle = \sum_{n\geq 1} a_n c_n$, where
161
$$c_n=2\pi i \int_{0}^{g(0)} e^{2\pi i n z} P_g(z,1)dz .$$
162
To compute $c_n$, let $y_0=1/c$, $x_1=-d/c$, $x_2=a/c$.
163
Then
164
$$c_n = 2\pi i\int_{x_1+iy_0}^{x_2+iy_0} e^{2\pi i n z} P_g(z,1) dz.$$
165
\begin{proof}
166
Let $\alp=x_1+iy_0$ and $\beta=x_2+iy_0$.
167
Observe that $g(\alp)=\beta$ and (key fact) $P_g(aX+bY,cX+dY)=P_g(X,Y)$,
168
so $$g^{-1}(P_g(X,Y)\{x,y\}) = P_g(X,Y)\{g^{-1}(x),g^{-1}(y)\}$$
169
for any $x$ and $y$.
170
We thus have
171
\begin{eqnarray*}
172
0 &=& P_g(X,Y)(\{0,g(0)\} + \{g(0),g(\alp)\} + \{g(\alp),0\})\\
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&=& P_g(X,Y)\{0,g(0)\} + P_g(X,Y)\{0,\alp\} + P_g(X,Y)\{g(\alp),0\}\\
174
&=& P_g(X,Y)(\{0,g(0)\} - \{\alp,g(\alp)\})
175
\end{eqnarray*}
176
so $P_g(X,Y)\{0,g(0)\} = P_g(X,Y)\{\alp,g(\alp)\}$.
177
178
\end{proof}
179
180
The following formula is useful in
181
evaluating $c_n$.
182
$$
183
\int_{z_0}^{\infty} e^{2\pi i n z} z^m dz
184
= e^{2\pi i n z_0}
185
\sum_{s=0}^m \left\{
186
\frac{(-1)^s z_0^{m-s}}
187
{(2\pi i n)^{s+1}}
188
\cdot \prod_{j=(m+1)-s}^m j\right\}.$$
189
Since $y_0>0$, the $c_n\ra 0$ quickly.
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\begin{proof}
191
Repeated integration by parts.
192
\end{proof}
193
194
\item {\bf [Solve for periods of dual basis.]}
195
We have a matrix equation
196
$$
197
\left(\begin{matrix}\vphi_1(v_1)&\cdots &\vphi_{2d}(v_{1})\\
198
\cdots &\cdots&\cdots\\
199
\vphi_{1}(v_{2d})&\cdots &\vphi_{2d}(v_{2d})
200
\end{matrix}
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\right)
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\cdot
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\left(
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\begin{matrix}
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\langle w_1, f\rangle \\
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\ldots\\
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\langle w_{2d}, f\rangle
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\end{matrix}
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\right)
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=
211
\left(
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\begin{matrix}
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\langle v_1, f\rangle \\
214
\ldots\\
215
\langle v_{2d}, f\rangle
216
\end{matrix}
217
\right).$$
218
Inverting we obtain numerical approximations
219
$\alpha_i \sim \langle w_i, f\rangle$.
220
Compute the approximations $\langle v_i, f^{(j)}\rangle$ for
221
each of the Galois conjugates $f^{(j)}$ of $f$ in order
222
to obtain approximations
223
$$\alpha_i^{(j)}\sim \langle w_i, f^{(j)}\rangle\in\C,$$
224
for $1\leq i \leq 2d$ and $1\leq j\leq d$.
225
\item {\bf [Compute period map.]}
226
The period map
227
is approximated as
228
$$\Phi_f(v) \sim
229
\left(
230
\begin{matrix}\alp_1^{(1)}&\cdots &\alp_{2d}^{(1)}\\\cdots &\cdots&\cdots\\
231
\alp_1^{(d)}&\cdots &\alp_{2d}^{(d)}\\\end{matrix}\right)
232
\cdot
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\left(\begin{matrix}\vphi_1(v)\\
234
|\\
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\vphi_{2d}(v)
236
\end{matrix}
237
\right)$$
238
\item {\bf [Compute period lattice.]}
239
The (approximate) period lattice $\Lambda_f$ has as
240
basis the columns of the following product:
241
$$
242
\left(
243
\begin{matrix}\alp_1^{(1)}&\cdots &\alp_{2d}^{(1)}\\\cdots &\cdots&\cdots\\
244
\alp_1^{(d)}&\cdots &\alp_{2d}^{(d)}\\\end{matrix}\right)
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\cdot
246
\left(
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\begin{matrix}|&\cdots &|\\z_1&\cdots &z_{2d}\\|&\cdots &|\end{matrix}\right)
248
$$
249
\end{enumerate}
250
251
252
\subsection{Special Values}
253
Let
254
$$L(f,s) = \int_{0}^{\infty} f(z)z^{s-1}dz = \sum a_n n^{-s}$$
255
be the Mellin transform of $f$.
256
We can approximate the special
257
value $L(A_f,1)=\prod_i L(f^{(i)}, 1)$ by observering that
258
$$\Phi_f(Y^{k-2}\{0,\infty\}) = (L(f^{(1)},1),\ldots,L(f^{(d)},1)).$$
259
260
\subsection{The Real Period}
261
The complex torus $A_f = \C^d/\Lambda_f$ is of interest.
262
When $k=2$ it has the structure of abelian variety with a
263
good integral model, it is naturally a quotient of $\Jac(X_\Gamma)$.
264
265
The following construction is perfectly general and makes
266
sense for any integer weight $k\geq 2$. When we refer
267
to the BSD conjecture below, we are referring to the particular
268
case $k=2$.
269
270
Assume for the rest of this section
271
that the Fourier coefficients of $f$ are totally real.
272
Then the lattice $\Lambda_f$ is invariant under complex conjugation.
273
Thus we may consider the real points
274
$A_f(\R)=(\C^d/\Lambda_f)^+$ of the complex torus $A_f$.
275
Let
276
$$\tilde{\Omega}_f := \Vol(A_f(\R))$$
277
denote the volume of the identity component of
278
$A_f(\R)$ with respect to the measure on $A_f$ induced
279
by the standard Lebesgue measure on $\C^d$. Then
280
$$\tilde{\Omega}_f = \Vol (\R^d/\Lambda_f^+ )\cdot c_\infty$$
281
where $$c_\infty=\#(A_f(\R)/A_f(\R)^0)$$
282
is the number of connected components of $A_f(\R)$.
283
284
Define
285
$$\Omega_f = \frac{\tilde{\Omega}_f}{\Delta_f}$$
286
where $\Delta_f$ is the volume of the image of
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the Hecke algebra in $\C^d$ by the map
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$$T_p \ra (a_p^{(1)},\ldots,a_p^{(d)}).$$
289
{\bf Remarks:}
290
1) Note that $\Delta_f^2\in\Z$ though $\Delta_f$ need
291
not be rational. [I think...] \\
292
2) $\Delta_f$ is the determinant of a change of basis matrix relating
293
the basis $f^{(1)},\ldots,f^{(d)}$ to an integral basis. This
294
statement probably uses the duality between Hecke operators
295
and integral cusp forms.\\
296
3) Note that this $\Omega_f$ differs from the quantity
297
appearing in the BSD conjecture (which arises from
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a Neron differential) by the ``Manin constant''
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(conjecturally 1).
300
301
Here is how to compute $\tilde{\Omega}_f$.
302
303
\begin{enumerate}
304
\item {\bf [Compute conjugation on $Z = \Z z_1 + \cdots + \Z z_{2d}$]}
305
The action of complex conjugation on $\Lambda_f$ can be
306
computed as follows.
307
Note that conjugation commutes with the period map $\Phi_f$.
308
If $z_i$ lifts to $x\in \cS_k(\Gamma,\Z)$, then
309
$$z_i^{*} = (\vphi_1(x^*),\ldots,\vphi_{2d}(x^*)).$$
310
Compute the matrix $C$ representing conjugation with
311
respect to the basis $z_1,\ldots,z_{2d}$.
312
313
Here is how to compute $C$.
314
\begin{enumerate}
315
\item Let $P$ be the matrix whose rows are the $\vphi_i$.
316
Using Gauss elmination choose $2d$ linearly independent
317
columns $i_1,\ldots,i_{2d}$ of $P$. Let $B$ the matrix
318
consisting of these $2d$ columns.
319
\item Let $A$ be the matrix of complex conjugation on
320
$M_k(\Gamma)$ and let $A'$ be the matrix obtained by
321
deleting all but columns $i_1,\ldots,i_{2d}$ of $A$.
322
The product $PA'$ is a matrix whose columns are the
323
images of the columns $i_1,\ldots,i_{2d}$ under
324
complex conjugation.
325
\item The matrix of complex conjugation (on the column
326
space of $P$) with respect to the standard basis is
327
$$C'=PA'B^{-1}.$$
328
\item Let $Z$ be the matrix whose columns are
329
$z_1,\ldots, z_{2d}$. Then the matrix of $C$
330
with respect to the basis $z_i$ is:
331
$$C = Z^{-1}C'Z.$$
332
\end{enumerate}
333
334
335
Compute a $\Z$-basis $b_1,\ldots, b_d$ for $Z^+=\Ker(C-I)$.
336
337
\item {\bf [Compute $\Vol(A_f(\R)^0)=\Vol(\R^d/\Lambda_f^+)$]}
338
$$\Vol(\R^d/\Lambda_f^+) \sim
339
\det \left\{\left(
340
\begin{matrix}\alp_1^{(1)}&\cdots &\alp_{2d}^{(1)}\\\cdots &\cdots&\cdots\\
341
\alp_1^{(d)}&\cdots &\alp_{2d}^{(d)}\\\end{matrix}\right)
342
\cdot
343
\left(
344
\begin{matrix}|&\cdots &|\\b_1&\cdots &b_d\\|&\cdots &|\end{matrix}\right)
345
\right\}$$
346
Order the $b_i$ so that $\det>0$.
347
348
\item {\bf [Compute $c_{\infty}$]}
349
Let $\overline{C}$ be the the map induced by complex
350
conjugation on $\Lambda_f/2\Lambda_f = \Lambda_f\tensor\F_2$.
351
Then
352
$$c_\infty = 2^{\dim(\ker(\overline{C} - 1)) - d}.$$
353
\begin{proof}
354
We must compute the order of the component group
355
$$\Psi=\frac{A_f(\R)}{A_f(\R)^0} =
356
\frac{(\C^d/\Lambda_f)^+}{\R^d/\Lambda_f^+}.$$
357
For $v\in\C^d$ denote by $\overline{v}$ its complex conjugate and
358
by $[v]$ its image in $\C^d/\Lambda_f$.
359
If $[v]\in(\C^d/\Lambda_f)^+$ then $[v]=[\overline{v}]$ so
360
since $v+\overline{v}\in\R^d$ we have
361
$$2[v] = [v]+[\overline{v}]\in\R^d/\Lambda_f^+,$$
362
so $\Psi$ is annihilated by $2$.
363
Thus there is $\lambda\in\Lambda_f$ so that
364
$2v+\lambda\in\R^d$, and so
365
$v+\frac{1}{2}\lambda\in\R^d$, i.e., $v$ can be written
366
as something in $\frac{1}{2}\Lambda_f$ plus something in $\R^d$.
367
This means that $\Psi$ is generated by the image of
368
$(\frac{1}{2}\Lambda_f/\Lambda_f)^+$.
369
Thus
370
$$\Psi \isom \frac{(\frac{1}{2}\Lambda_f/\Lambda_f)^+}
371
{(\frac{1}{2}\Lambda_f\intersect\R^d)/\Lambda_f^+}
372
\isom \frac{(\Lambda_f/2\Lambda_f)^+}
373
{\Lambda_f^+/2\Lambda_f^+} $$
374
375
Consequently
376
$$\dim_{\F_2} \Psi = \dim (\Lambda_f/2\Lambda_f)^+
377
- \dim \Lambda_f^+ / 2\Lambda_f^+
378
= \dim(\ker(\overline{C} - 1)) - d.$$
379
\end{proof}
380
\end{enumerate}
381
382
\subsection{The Ratio $L(A_f,1)/\Omega_f$}
383
\begin{theorem}
384
Let $F(x) = (\vphi_1(x),\ldots,\vphi_{2d}(x)).$
385
Let $C$ be the map on the image of $F$ induced by the $*$ involution on
386
$\cS_k(\Gamma,\Z)$ and $\overline{C}$ its mod $2$ reduction.
387
Then
388
$$\left|\frac{L(A_f,1)}{\Omega_f}\right|
389
= [F(\cS_k(\Gamma,\Z))^+:F(\T e)]
390
\cdot 2^{\dim(\ker(\overline{C} - 1)) - d}$$
391
where $[A:B]$ is the absolute value of the determinant of a change
392
of basis matrix sending $A$ to $B$.
393
In particular the ratio is a rational number.
394
\end{theorem}
395
\begin{proof}
396
Let $\O$ be the image of
397
the Hecke algebra in $\C^d$ by the map
398
$$T_p \ra (a_p^{(1)},\ldots,a_p^{(d)}).$$
399
Thus $\Delta_f = \Vol(\O)=[\Z:\O]$.
400
We have
401
\begin{eqnarray*}
402
\frac{L(A_f,1)}{\Omega_f}
403
&=& \frac{\langle e,f^{(1)}\rangle\cdots \langle e,f^{(d)}\rangle}
404
{\Vol(\Phi_f(H_1(X_0(N),\Z))^+) / \Delta_f}\\
405
&=&\frac{[\Z^d : \Phi_f(e)\Z^d]}
406
{[\Z^d : \Phi_f(H_1)^+][\O:\Z^d]} \\
407
&=&\frac{[\O : \Phi_f(e)\O]}
408
{[\Z^d : \Phi_f(H_1)^+][\O:\Z^d]} \\
409
&=&\frac{[\Z^d : \Phi_f(e)\O]}
410
{[\Z^d : \Phi_f(H_1)^+]} \\
411
&=&\frac{[\Z^d : \Phi_f(\T e)]}
412
{[\Z^d : \Phi_f(H_1)^+]} \\
413
&=&[\Phi_f(H_1)^+ : \Phi_f(\T e)]\\
414
&=&[F(H_1)^+ : F(\T e)].
415
\end{eqnarray*}
416
417
\end{proof}
418
419
{\bf Weight 2 remark:} It is known (Mazur) that the Manin constant, which relates
420
our $\Omega$ to the one for the Neron model, is a unit
421
in $\Z[\frac{1}{2m}]$ where $m$ is the largest square
422
dividing the level $N$.
423
\begin{corollary}
424
The quotient of $L(A_f,1)$ by the real volume,
425
which appears in the BSD conjecture, a priori
426
a real number is in fact a rational number.
427
\end{corollary}
428
429
430
\section{Examples}
431
Let $f\in S_2(\Gamma_0(65),\C)$ be the unique newform
432
for which $a_2$ satisfies $a_2^2+2a_2-1=0$. Letting $\alp=a_2$ we have
433
$$f = q + \alpha q^2 + (\alpha+1)q^3
434
+(-2\alp-1)q^4 + q^5 + (-\alp+1)q^6
435
+(-2\alp)q^7 + (\alp-2)q^8+\cdots$$
436
As we will see, the associated quotient $A=A_f$ of $J_0(65)$
437
has analytic rank $0$.
438
439
We first compute $\cS_2(\Gamma,\Z)\subset \cM_2(\Gamma,\Q)$.
440
With respect to our chosen basis, the matrix of $T_2$ on
441
$\cM_2(\Gamma,\Q)$ is
442
$$T_2 =
443
\left(\begin{array}{ccccccccccccc}
444
{3}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\
445
{0}&{1}&{2}&{1}&{1}&{1}&{1}&{1}&{1}&{1}&{0}&{0}&{0}\\
446
{1}&{0}&{-1}&{0}&{0}&{0}&{1}&{1}&{1}&{1}&{0}&{0}&{1}\cr
447
{-1}&{0}&{0}&{-2}&{-1}&{-1}&{0}&{0}&{0}&{0}&{0}&{-1}&{-1}\cr
448
{0}&{-1}&{-1}&{0}&{1}&{1}&{-1}&{-2}&{0}&{0}&{1}&{1}&{0}\cr
449
{0}&{2}&{1}&{0}&{0}&{0}&{1}&{1}&{0}&{0}&{0}&{1}&{1}\cr
450
{-1}&{0}&{0}&{0}&{0}&{0}&{-1}&{0}&{0}&{0}&{1}&{0}&{0}\cr
451
{0}&{-1}&{-1}&{-1}&{-2}&{0}&{0}&{-2}&{-2}&{-1}&{0}&{0}&{0}\cr
452
{0}&{1}&{1}&{0}&{2}&{-1}&{1}&{2}&{3}&{0}&{-1}&{-1}&{0}\cr
453
{0}&{1}&{2}&{1}&{0}&{1}&{0}&{-1}&{-1}&{1}&{1}&{1}&{0}\cr
454
{-1}&{-1}&{-1}&{-1}&{-1}&{0}&{0}&{-1}&{-1}&{0}&{0}&{0}&{-1}\cr
455
{0}&{-1}&{-1}&{-1}&{0}&{0}&{0}&{-1}&{0}&{0}&{1}&{-1}&{1}\cr
456
{1}&{1}&{1}&{0}&{0}&{1}&{1}&{0}&{0}&{0}&{0}&{2}&{1}\cr
457
\end{array}\right).$$
458
The left eigenspace corresponding to $f$ is the left kernel
459
of $T_2^2+2 T_2-1$ which is
460
$$P=\left(
461
\begin{array}{ccccccccccccc}
462
{0}&{0}&{1}&{1}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{1}&{0}\\
463
{0}&{-1}&{0}&{-1}&{-1}&{-1}&{1}&{0}&{1}&{1}&{1}&{0}&{1}\\
464
{1}&{1}&{0}&{1}&{1}&{1}&{1}&{2}&{1}&{0}&{1}&{0}&{0}\\
465
{-1}&{-1}&{-1}&{2}&{1}&{-1}&{-2}&{1}&{0}&{1}&{-2}&{-1}&{1}\\
466
\end{array}
467
\right)$$
468
The rows of $P$ are $\vphi_1,\ldots,\vphi_4$.
469
470
\comment{
471
The lattice
472
$$\{(\vphi_1(x),\ldots,\vphi_{4}(x)) : x \in \cS_k(\Gamma,\Z) \}
473
\subset \Q^{4}$$
474
is simply the lattice spanned by the columns of $P$, which
475
is the full $\Z^4$ (observe that columns 1,2,3,5 have determinant
476
$2$ so span a sublattice of index $2$ in $\Z^4$, and column
477
7 does not lie in it).
478
}
479
480
We try several $g\in\Gamma_0(65)$:
481
\begin{eqnarray*}
482
v_1&=&\{0,\abcd{22}{1}{65}{3}(0)\} = [0,0,0,1,0,0,0,0,0,0,1,0,-1]^t\\
483
v_2&=&\{0,\abcd{-32}{-1}{65}{2}(0)\}= [0,0,-1,1,0,0,1,0,0,0,1,0,-1]^t\\
484
v_3&=&\{0,\abcd{-16}{-1}{65}{4}(0)\}= [0,0,0,0,1,-1,1,0,-1,0,1,0,-1]^t\\
485
v_4&=&\{0,\abcd{11}{1}{65}{6}(0)\}=[0,1,0,0,-1,1,0,-1,1,0,-1,-1,0]^t\\
486
\end{eqnarray*}
487
Then
488
$$\vphi_i(v_j) =
489
\left(\begin{matrix}
490
{1}&{-1}&{2}&{-1}\cr
491
{0}&{0}&{3}&{-2}\cr
492
{0}&{0}&{1}&{-3}\cr
493
{-1}&{-1}&{-1}&{-1}\cr
494
\end{matrix}\right)$$
495
which has rank $4$.
496
497
The period integrals, to 15 decimals are as follows.
498
\begin{eqnarray*}
499
\langle v_1, f\rangle&=& -1.37443706596536 + 0.768794716989823i\\
500
\langle v_2, f\rangle&=& -2.74887413193073 \\
501
\langle v_3, f\rangle&=& -1.94374753931650 \\
502
\langle v_4, f\rangle&=& -0.569310473351139 + 1.85603463243761i\\
503
\langle v_1, f^{(2)}\rangle&=& -1.15118942200858 + 3.35725985432803i\\
504
\langle v_2, f^{(2)}\rangle&=& -2.30237884401716\\
505
\langle v_3, f^{(2)}\rangle&=& 1.62802769346498\\
506
\langle v_4, f^{(2)}\rangle&=& 2.77921711547356 - 1.39062256407339i
507
\end{eqnarray*}
508
Inverting the matrix $\vphi_i(v_j)$ we obtain
509
$$(\alpha_i^{(j)})^t =
510
\left(\begin{matrix}
511
{{0.531535414512712} - {0.543619957723896}i}&{{0.212623571040667} + {2.37394120920071}i}\cr
512
{{0.220169177572770} - {1.31241467471371}i}&{{-0.513318708886579} - {0.983318645127320}i}\cr
513
{{-0.622732473879884}}&{{-1.45188455985449}}\cr
514
{{0.440338355145540}}&{{-1.02663741777315}}\cr
515
\end{matrix}\right)$$
516
517
518
\begin{verbatim}
519
P=[0,0,1,1,0,0,0,0,0,0,0,1,0;
520
0,-1,0,-1,-1,-1,1,0,1,1,1,0,1;
521
1,1,0,1,1,1,1,2,1,0,1,0,0;
522
-1,-1,-1,2,1,-1,-2,1,0,1,-2,-1,1];
523
524
A=[1, -1, 2, -1; 0, 0, 3, -2; 0, 0, 1, -3; -1, -1, -1, -1]
525
526
{H1Z=
527
[0,0,0,0,0,0,0,0,0,0;
528
-1,0,-1,-1,0,0,-1,-1,-1,-1;
529
1,0,0,0,0,0,0,0,0,0;
530
0,1,0,0,0,0,0,0,0,0;
531
0,0,0,0,0,-1,0,0,0,0;
532
0,0,1,0,0,0,0,0,0,0;
533
0,0,0,1,0,0,0,0,0,0;
534
0,0,0,0,1,0,0,0,0,0;
535
0,0,0,0,0,1,0,0,0,0;
536
0,0,0,0,0,0,1,0,0,0;
537
0,0,0,0,0,0,0,1,0,0;
538
0,0,0,0,0,0,0,0,1,0;
539
0,0,0,0,0,0,0,0,0,1];}
540
541
\\ integral basis for image of S_2 in columns space of P
542
{Z=[1, 0, 0, 0;
543
1, 2, 0, 0;
544
-1,-2, 2,-1;
545
0,-2, 1, 3];}
546
547
548
\\ conjugation on M_2
549
CC = [1,0,0,0,0,0,0,0,0,0,0,0,0;
550
0,0,0,0,0,0,0,-1,0,0,1,1,1;
551
0,0,0,-1,0,0,0,0,0,0,0,0,0;
552
0,0,-1,0,0,0,0,0,0,0,0,0,0;
553
0,0,1,0,0,0,0,1,1,1,0,-1,-1;
554
0,0,0,0,0,0,1,0,0,0,0,1,1;
555
0,0,0,0,0,1,0,0,0,0,0,0,-1;
556
0,0,0,0,0,0,0,1,0,0,0,-1,0;
557
0,0,-1,0,1,0,0,-1,0,-1,0,1,1;
558
0,0,1,1,0,0,0,0,0,1,0,0,0;
559
0,1,0,0,0,0,0,1,0,0,0,-1,-1;
560
0,0,0,0,0,0,0,0,0,0,0,-1,0;
561
0,0,0,0,0,0,0,0,0,0,0,1,1];
562
563
\\ The matrix of C with respect to the basis Z=[z_1,...,z_4]
564
C=[-1, 0, 0, 0;
565
0, -1, 0, 0;
566
-1, -2, 1, 0;
567
1, 0, 0, 1];
568
569
The +1 eigenspace is:
570
span z_3, z_4.
571
572
Using the below alp matrix we obtain the real period:
573
574
alp*[z_3;z_4] = alp*[0, 0; 0, 0; 2,-1; 1, 3]
575
=[-0.8051265926142280223898673966 1.943747539316507071177084460]
576
[-3.930406537482142877473965080 -1.628027693464980131583752011]
577
578
which has determinant
579
8.950486425265488291492806649
580
581
We also have
582
matrank(C*Mod(1,2)-1) = 1
583
so
584
c_infinity = 2.
585
586
Thus \tilde{Omega} = 17.90097285053097658298561329.
587
588
Delta = |1 1 | = 2*sqrt(2)
589
|a2 a2bar|
590
So,
591
Omega = 6.328949646223367581007308099
592
593
The image of {0,oo} is simply [0,0,1,-1]~, so
594
alp*[0,0,1,-1]~
595
=[-1.063070829025425322615855453,
596
-0.4252471420813350514566677322];
597
598
Taking the product of the two entries gives
599
600
L(A_f,1)=L(f,1)*L(fbar,1)
601
= 0.4520678318730976843576858890
602
603
We also get, for free by writing [0,0,1,-1]~ in terms
604
of Z that the image of (0)-(oo) in A_f is rational 7 torsion.
605
606
The characteristic polynomial of T_3 on A_f is:
607
x^2-2
608
so 14 is an upper bound on the torsion.
609
610
Thus
611
L(A_f,1) 1
612
---------- = 0.07142857142857142857143189343 ~ ----
613
Omega 14
614
615
616
Using the method of graphs and my component
617
group formula David Kohel and I computed the tamagawa numbers and
618
got
619
c5 = 7 c13 = 1
620
621
The BSD conjecture asserts that (assume it hence):
622
623
1 [Sha] * c5 *c13 [Sha] * 7
624
-------- = ------------------------- = -------------------------
625
14 [Torsion]*[Torsiondual] [torsion]*[torsion dual]
626
627
Thus
628
[Sha] = [torsion]*[torsion dual] / 2*7^2.
629
630
Thus assuming BSD, the torsion on one of A_f or A_fdual must
631
be 14.
632
633
It could easily be the case that [torsion]=14, [torsion dual]=7
634
and [Sha]=1.
635
636
\end{verbatim}
637
638
639
\section{Tables}
640
641
\section{Source Code}
642
The following pari code can be used to compute the period
643
integrals $\langle P_g\{0,g(0)\}, f\rangle$.
644
\begin{verbatim}
645
\\ file: period.gp
646
\\ DESCRIPTION:
647
\\ Compute period integrals by integrating along a cleverly
648
\\ chosen arc in the upper half plane.
649
\\
650
\\ AUTHOR: William Stein and Helena Verrill
651
\\ LAST MODIFIED: April, 1999.
652
653
ncols(v)=matsize(v)[2];
654
655
\\ Compute the line integral
656
\\ / x2+iy0
657
\\ |
658
\\ 2PiI*| f(z) * z^m
659
\\ |
660
\\ / x1+iy0
661
\\ where
662
\\ a=[a1,a2,a3,a4,...] defines the series
663
\\ f(z) = a1*z + a2*z^2 + a3*z^3 + ...
664
665
{lineintegral(y0,x1,x2,m,a,
666
prec,n,c,ans)=
667
c=vector(prec,n, lineval(x2+I*y0,m,n) - lineval(x1+I*y0,m,n));
668
ans=sum(n=1, prec, a[n] * c[n]);
669
print("Int(2*Pi*I*f(z)*", z^m,",",x1+I*y0,",",x2+I*y0,") = ",ans);
670
return(ans);
671
}
672
673
{lineval(z,m,n,
674
s,c,i)=
675
c=2*Pi*I*n;
676
2*Pi*I
677
* exp(c*z)
678
* sum(s=0,m,(-1)^s* (z^(m-s)) / (c^(s+1)) * prod(i=m-s+1,m,i));
679
}
680
681
\\ Let g be in SL_2(Z) and k be an even positive integer.
682
\\ This function computes an eigenvector element of
683
\\ Sym^(k-2)(Z) = homogeneous poly's of degree k-2 in X and Y
684
\\ Which has eigenvalue +1 for g.
685
\\ Note: the output is a vector of coefficients.
686
{syminvariant(g,k)=
687
((1-g[1,1]*g[2,2]) *X^2
688
- g[1,2]*(g[2,2]-g[1,1])*X
689
+ g[1,2]^2
690
)^((k-2)/2)
691
}
692
693
\\ Let f be a weight k newform for Gamma_0(N) and fix
694
\\ g=[a,b;c,d] in Gamma_0(N) such that c>0.
695
\\ Let P(X,Y) be the eigenvector computed by syminvariant(g,k).
696
\\ This function computes the period
697
\\ < P(X,Y)*{0,g(0)}, f(z) >
698
{gf_period(g,a,k,prec=0,
699
P,x1,x2,y0,i,j,m,v,ans)=
700
if(prec==0,prec=ncols(a));
701
if(prec>ncols(a),
702
print("gf_period: WARNING desired precision exceeds that of f.")
703
);
704
705
P=syminvariant(g,k);
706
print("gf_period: precision = ",prec);
707
print("< (",P,")*{0,",if(g[2,2]==0,"oo",g[1,2]/g[2,2]),
708
"}, ",sum(j=1,13,a[j]*q^j)+O(q^14)," >");
709
710
y0=1/g[2,1];
711
x1=-g[2,2]*y0;
712
x2=g[1,1]*y0;
713
714
v=vector(k-2+1,i,0);
715
ans=sum(m=0,k-2,
716
if(polcoeff(P,m)!=0,
717
v[m+1]=polcoeff(P,m)*lineintegral(y0,x1,x2,m,a,prec),
718
0
719
);
720
);
721
ans;
722
}
723
724
\\ quick computation of eta
725
{myeta(q,prec,
726
n)=
727
sum(n=-prec,prec,
728
(-1)^n * q^((3*n-1)*n/2))
729
+O(q^((3*prec-1)*prec/2)) ;
730
}
731
732
\\ our favorite.
733
{f1(prec, \\ eta(q^3)^8, weight 4, level 9.
734
f) =
735
f = myeta(q,prec);
736
Vec(subst(f,q,q^3)^8)
737
}
738
739
{f2(prec, \\ eta(q^6)^4, weight 2, level 36.
740
f) =
741
f = myeta(q,prec);
742
Vec(subst(f,q,q^2)^4*subst(f,q,q^4)^4)
743
}
744
745
746
\\ list some elements of Gamma_0(N)
747
{mlift(c,d,N,
748
v)=
749
v=bezout(c,d);
750
return([v[2],-v[1];c,d]);
751
}
752
\end{verbatim}
753
754
\begin{thebibliography}{HHHHHHH}
755
\bibitem[CR]{cremona} J.E. Cremona,
756
{\em Algorithms for modular elliptic curves, 2nd edition},
757
Cambridge University Press, (1997).
758
\end{thebibliography}
759
760
\end{document}
761
762
763
764
The matrix alpha_i^j which computes period map, when right compl with phi.
765
766
alp=[0.5315354145127126613078931891330698990 - 0.5436199577238963826250859980838456612*I, 0.2201691775727704371403309739132151733 - 1.312414674713719213072658569868641118*I, -0.6227324738798844483352409500442398090 + 3.665476401263631858808075880749397435 E-23*I, 0.4403383551455408742806145034199040494 + 3.510503045937607708297890680758835419 E-23*I; 0.2126235710406675257282129991395067099 + 2.373941209200713376138704039517591586*I, -0.5133187088865787957602987003974718253 - 0.9833186451273203717704335635086521968*I, -1.451884559854492642976877604131230096 - 2.130221455047120662019817425995001094 E-23*I, -1.026637417773157591520209871918494907 - 8.778132686392049792197488630777259782 E-24*I]
767