Open in CoCalc
1% periods.tex
2
3% AUTHOR:  WILLIAM STEIN and HELENA VERRILL
4% SUBJECT: NUMERICAL COMPUTATION OF PERIOD LATTICES ASSOCIATED TO NEWFORMS
5% DATE:    APRIL 1999.
6
7\documentclass[11pt]{article}
8\include{macros}
9\title{Computing Period Lattices of Newforms\\
10(Preliminary)}
11\author{William Stein and Helena Verrill}
12\date{April, 1999}
13\begin{document}
14\maketitle
15\begin{abstract}
16We give an algorithm to numerically approximate
17the period lattice and real volume of any
18modular newforms $f\in S_k(\Gamma,\C)$
19of positive even weight.  The computation
20of special values of the associated $L$-functions
21is also discussed. Numerical examples are provided.
22\end{abstract}
23
24%\section{Introduction}
25
26\section{Modular Symbols}
27Let $\Gamma=\Gamma_0(N)$ or $\Gamma_1(N)$, and let
28$k$ be an even positive integer.  Let
29$\cM$ be the abelian group generated by symbols
30$\{\alpha,\beta\}$ with $\alpha, \beta\in\P^1(\Q)$
31modulo the relations
32$$\{\alpha,\beta\}+\{\beta,\gamma\}+\{\gamma,\alpha\} = 0$$
33and modulo any torsion.
34\begin{center}
35\begin{picture}(100,100)
36\put(0,10){\circle*{3}}
37\put(0,16){$\alpha$}
38\qbezier(0,10)(50,40)(100,10)
39\put(50,25){\vector(1,0){1}}
40\put(100,10){\circle*{3}}
41\put(100,16){$\beta$}
42\qbezier(100,10)(60,30)(50,100)
43\put(65,47){\vector(-1,2){1}}
44\put(50,100){\circle*{3}}
45\put(50,106){$\gamma$}
46\qbezier(0,10)(40,30)(50,100)
47\put(35,47){\vector(-1,-2){1}}
48\end{picture}
49\end{center}
50
51Let
52  $$V_{k-2}=\Sym^{k-2}(\Z X + \Z Y)$$
53be the $\Z$-module of homogeneous polynomials in $X$ and $Y$
54of degree $k-2$.
55Define a left action of $g=\abcd{a}{b}{c}{d}\in \Gamma$ on the tensor product
56$V_{k-2}\tensor \cM$ by
57$$g(P(X,Y)\tensor\{\alpha,\beta\}) 58 = P(dX-bY,-cX+aY)\tensor\{g(\alpha),g(\beta)\}.$$
59The space of modular symbols of weight $k$ for $\Gamma$ is
60the largest quotient of $V_{k-2}\tensor\cM$ on which $\Gamma$
61acts trivially.  If we let $I = \{1-g : g\in\Gamma\}\subset\Z[\Gamma]$
62then
63$$\M_k(\Gamma) = \frac{V_{k-2}\tensor\cM}{I(V_{k-2}\tensor\cM)} 64 = H_0(\Gamma,V_{k-2}\tensor\cM).$$
65The cuspidal modular symbols are $\cS_k(\Gamma) = \ker(\delta)$,
66where
67 $$\delta:\M_k(\Gamma)\ra \bigoplus_{[\alpha]\in \Gamma\backslash\P^1(\Q)} 68 \Z[\alpha]$$
69is given by
70 $$\delta(P(X,Y)\tensor\{\alpha,\beta\}) = ?.$$
71
72
73The Eichler-Shimura theory gives an isomorphism
74$$S_k(\Gamma,\C)\oplus\overline{S}_k(\Gamma,\C)\oplus\Eis(\Gamma,\C) 75 \isom H^1(\Gamma,V_{k-2}^*\tensor\C)$$
76For example, when $k=2$ the map
77$$S_2(\Gamma,\C)\ra H^1(\Gamma,\C)=\Hom(\Gamma,\C)$$
78send $f(z)$ to the map $\gamma\mapsto \int_{0}^{\gamma(0)} f(z)dz$.
79
80The cup product pairing
81$$H_0(\Gamma,V_{k-2}\tensor\cM)\cross H^1(\Gamma,V_{k-2}^*\tensor\C) 82 \ra H^0(\Gamma,\C)=\C$$
83gives rise to the period map we will study.
84
85\section{Computing Periods}
86Let $\Gamma=\Gamma_0(N)$ or $\Gamma_1(N)$.  In this section
87we describe how to numerically approximate the period lattice
88associated to a cuspidal newform
89  $$f=\sum a_n q^n \in S_k(\Gamma,\C), \qquad k\geq 2.$$
90We do {\em not} require that the
91Fourier coefficients $a_n$ of $f$ lie in $\Q$, nor that the
92weight equals $2$.  In the special case $k=2$ and all $a_n\in\Q$,
93see \cite{cremona}.  The algorithm given only applies when
94$k$ is even, though we expect that there is a
95generalization to odd $k$.
96
97Let $\T=\Z[T_1,T_2,T_3,\ldots]$ denote the Hecke algebra, equipped
98with its action on modular forms and modular symbols for $\Gamma$.
99Set $$\p_f = \Ann_\T(f).$$
100Let $f^{(1)},\ldots,f^{(d)}$ be the $\Gal(\Qbar/\Q)$ conjugates of $f$.
101Let $\cM_k(\Gamma,\Q)$ be the space of modular symbols of weight
102$k$ for $\Gamma$ (tensor $\Q$) and $\cS_k(\Gamma,\Z)\subset\cM_k(\Gamma,\Q)$
103the subspace of integral cuspidal symbols.
104There is a pairing
105$$\langle\quad,\quad\rangle : \cM_k(\Gamma,\Q)\cross S_k(\Gamma,\C) \ra \C$$
106$$\langle P(X,Y)\{\alp,\beta\}, \, f \rangle 107 = 2\pi i \int_{\alpha}^{\beta} f(z)P(z,1) dz.$$
108The period map associated to $f$ is
109$$\Phi_f : \cM_k(\Gamma,\Q) \ra \C^d$$
110$$\Phi_f(x) = (\langle x,f^{(1)}\rangle, \ldots, \langle x,f^{(d)}\rangle).$$
111Let
112   $$\Lambda_f =\Phi_f(\cS_k(\Gamma,\Z))\subset\C^d.$$
113
114The following is an algorithm to numerically
115approximate $\Phi_f$ and $\Lambda_f$.
116
117\begin{enumerate}
118\item {\bf [Compute left eigenspace.]}
119Compute a $\Q$-basis $\vphi_1,\ldots,\vphi_{2d}$ for
120$$\Hom(\M_k(\Gamma,\Q),\Q)[\p_f].$$
121\item {\bf [Dual basis.]} Let $w_1,\ldots,w_{2d}$ be the $\Q$-basis for
122$\M_k(\Gamma,\Q)[\p_f]$ dual to $\vphi_1,\ldots,\vphi_{2d}$,
123so $\vphi_i(w_j)=\delta_{ij}$.  It is not
124necessary to compute the $w_i$ though this could
125be done by inverting the matrix whose rows are the $\vphi_i$.
126Observe 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 $\Lambda_f$.]}
129Compute a $\Z$-basis $z_1,\ldots,z_{2d}$
130for the lattice
131    $$\{(\vphi_1(x),\ldots,\vphi_{2d}(x)) : x \in \cS_k(\Gamma,\Z) \} 132 \subset \Q^{2d}.$$
133\item {\bf [Find good periods.]}
134Locate $v_1,\ldots v_{2d}\in \M_k(\Gamma,\Q)$ so that
135\begin{itemize}
136\item Each $v_i$ is of the form
137         $$P_g(X,Y)\{0,g(0)\}$$
138with $g=\abcd{a}{b}{c}{d}\in \Gamma$, $c>0$, and
139$$P_g(X,Y) = ((1-ad)X^2 - b(d-a)XY+b^2Y^2)^{\frac{k-2}{2}}.$$
140Note that $P$ is an eigenvector for the action of $g$
141on $V_{k-2}$ with eigenvalue $1$.
142
143{\bf Remarks:}
144\begin{enumerate}
145\item This is the step which only makes sense for even weight.
146We have not yet found a replacement for odd weight.
147\item It is still necessary to prove the existence of enough $g$'s.
148This we have not done.  In practice a few randomly choosen $g$'s
149seem 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.]}
155The form of each $v_i$ insures that we can obtain good numerical
156approximations to $\langle v_i, f\rangle.$
157Write $v_i = P_g(X,Y)\{0,g(0)\}$ with $g=\abcd{a}{b}{c}{d}$
158and $c>0$.
159We have
160$\langle v_i, f\rangle = \sum_{n\geq 1} a_n c_n$, where
161$$c_n=2\pi i \int_{0}^{g(0)} e^{2\pi i n z} P_g(z,1)dz .$$
162To compute $c_n$, let $y_0=1/c$, $x_1=-d/c$, $x_2=a/c$.
163Then
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}
166Let $\alp=x_1+iy_0$ and $\beta=x_2+iy_0$.
167Observe that $g(\alp)=\beta$ and (key fact) $P_g(aX+bY,cX+dY)=P_g(X,Y)$,
168so $$g^{-1}(P_g(X,Y)\{x,y\}) = P_g(X,Y)\{g^{-1}(x),g^{-1}(y)\}$$
169for any $x$ and $y$.
170We thus have
171\begin{eqnarray*}
1720 &=& P_g(X,Y)(\{0,g(0)\} + \{g(0),g(\alp)\} + \{g(\alp),0\})\\
173  &=& 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*}
176so $P_g(X,Y)\{0,g(0)\} = P_g(X,Y)\{\alp,g(\alp)\}$.
177
178\end{proof}
179
180The following formula is useful in
181evaluating $c_n$.
182$$183 \int_{z_0}^{\infty} e^{2\pi i n z} z^m dz 184 = e^{2\pi i n z_0} 185 \sum_{s=0}^m \left\{ 186 \frac{(-1)^s z_0^{m-s}} 187 {(2\pi i n)^{s+1}} 188 \cdot \prod_{j=(m+1)-s}^m j\right\}.$$
189Since $y_0>0$, the $c_n\ra 0$ quickly.
190\begin{proof}
191Repeated integration by parts.
192\end{proof}
193
194\item {\bf [Solve for periods of dual basis.]}
195We have a matrix equation
196$$197\left(\begin{matrix}\vphi_1(v_1)&\cdots &\vphi_{2d}(v_{1})\\ 198 \cdots &\cdots&\cdots\\ 199 \vphi_{1}(v_{2d})&\cdots &\vphi_{2d}(v_{2d}) 200 \end{matrix} 201\right) 202\cdot 203\left( 204\begin{matrix} 205 \langle w_1, f\rangle \\ 206 \ldots\\ 207 \langle w_{2d}, f\rangle 208\end{matrix} 209\right) 210= 211\left( 212\begin{matrix} 213 \langle v_1, f\rangle \\ 214 \ldots\\ 215 \langle v_{2d}, f\rangle 216\end{matrix} 217\right).$$
218Inverting we obtain numerical approximations
219$\alpha_i \sim \langle w_i, f\rangle$.
220Compute the approximations $\langle v_i, f^{(j)}\rangle$ for
221each of the Galois conjugates $f^{(j)}$ of $f$ in order
222to obtain approximations
223$$\alpha_i^{(j)}\sim \langle w_i, f^{(j)}\rangle\in\C,$$
224for $1\leq i \leq 2d$ and $1\leq j\leq d$.
225\item {\bf [Compute period map.]}
226The period map
227is approximated as
228$$\Phi_f(v) \sim 229 \left( 230\begin{matrix}\alp_1^{(1)}&\cdots &\alp_{2d}^{(1)}\\\cdots &\cdots&\cdots\\ 231 \alp_1^{(d)}&\cdots &\alp_{2d}^{(d)}\\\end{matrix}\right) 232\cdot 233\left(\begin{matrix}\vphi_1(v)\\ 234 |\\ 235 \vphi_{2d}(v) 236 \end{matrix} 237\right)$$
238\item {\bf [Compute period lattice.]}
239The (approximate) period lattice $\Lambda_f$ has as
240basis the columns of the following product:
241$$242 \left( 243\begin{matrix}\alp_1^{(1)}&\cdots &\alp_{2d}^{(1)}\\\cdots &\cdots&\cdots\\ 244 \alp_1^{(d)}&\cdots &\alp_{2d}^{(d)}\\\end{matrix}\right) 245\cdot 246\left( 247\begin{matrix}|&\cdots &|\\z_1&\cdots &z_{2d}\\|&\cdots &|\end{matrix}\right) 248$$
249\end{enumerate}
250
251
252\subsection{Special Values}
253Let
254$$L(f,s) = \int_{0}^{\infty} f(z)z^{s-1}dz = \sum a_n n^{-s}$$
255be the Mellin transform of $f$.
256We can approximate the special
257value $L(A_f,1)=\prod_i L(f^{(i)}, 1)$ 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}
261The complex torus $A_f = \C^d/\Lambda_f$ is of interest.
262When $k=2$ it has the structure of abelian variety with a
263good integral model, it is naturally a quotient of $\Jac(X_\Gamma)$.
264
265The following construction is perfectly general and makes
266sense for any integer weight $k\geq 2$.   When we refer
267to the BSD conjecture below, we are referring to the particular
268case $k=2$.
269
270Assume for the rest of this section
271that the Fourier coefficients of $f$ are totally real.
272Then the lattice $\Lambda_f$ is invariant under complex conjugation.
273Thus we may consider the real points
274$A_f(\R)=(\C^d/\Lambda_f)^+$ of the complex torus $A_f$.
275Let
276   $$\tilde{\Omega}_f := \Vol(A_f(\R))$$
277denote the volume of the identity component of
278$A_f(\R)$ with respect to the measure on $A_f$ induced
279by the standard Lebesgue measure on $\C^d$.  Then
280$$\tilde{\Omega}_f = \Vol (\R^d/\Lambda_f^+ )\cdot c_\infty$$
281where $$c_\infty=\#(A_f(\R)/A_f(\R)^0)$$
282is the number of connected components of $A_f(\R)$.
283
284Define
285$$\Omega_f = \frac{\tilde{\Omega}_f}{\Delta_f}$$
286where $\Delta_f$ is the volume of the image of
287the Hecke algebra in $\C^d$ by the map
288$$T_p \ra (a_p^{(1)},\ldots,a_p^{(d)}).$$
289{\bf Remarks:}
2901) Note that $\Delta_f^2\in\Z$ though $\Delta_f$ need
291not be rational. [I think...]  \\
2922) $\Delta_f$ is the determinant of a change of basis matrix relating
293the basis $f^{(1)},\ldots,f^{(d)}$ to an integral basis.  This
294statement probably uses the duality between Hecke operators
295and integral cusp forms.\\
2963) Note that this $\Omega_f$ differs from the quantity
297appearing in the BSD conjecture (which arises from
298a Neron differential) by the Manin constant''
299(conjecturally 1).
300
301Here is how to compute $\tilde{\Omega}_f$.
302
303\begin{enumerate}
304\item {\bf [Compute conjugation on $Z = \Z z_1 + \cdots + \Z z_{2d}$]}
305The action of complex conjugation on $\Lambda_f$ can be
306computed as follows.
307Note that conjugation commutes with the period map $\Phi_f$.
308If $z_i$ lifts to $x\in \cS_k(\Gamma,\Z)$, then
309$$z_i^{*} = (\vphi_1(x^*),\ldots,\vphi_{2d}(x^*)).$$
310Compute the matrix $C$ representing conjugation with
311respect to the basis $z_1,\ldots,z_{2d}$.
312
313Here is how to compute $C$.
314\begin{enumerate}
315\item Let $P$ be the matrix whose rows are the $\vphi_i$.
316Using Gauss elmination choose $2d$ linearly independent
317columns $i_1,\ldots,i_{2d}$ of $P$.  Let $B$ the matrix
318consisting 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
321deleting all but columns $i_1,\ldots,i_{2d}$ of $A$.
322The product $PA'$ is a matrix whose columns are the
323images of the columns $i_1,\ldots,i_{2d}$ under
324complex conjugation.
325\item The matrix of complex conjugation (on the column
326space 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 $C$
330with respect to the basis $z_i$ is:
331    $$C = Z^{-1}C'Z.$$
332\end{enumerate}
333
334
335Compute 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$$\Vol(\R^d/\Lambda_f^+) \sim 339\det \left\{\left( 340\begin{matrix}\alp_1^{(1)}&\cdots &\alp_{2d}^{(1)}\\\cdots &\cdots&\cdots\\ 341 \alp_1^{(d)}&\cdots &\alp_{2d}^{(d)}\\\end{matrix}\right) 342\cdot 343\left( 344\begin{matrix}|&\cdots &|\\b_1&\cdots &b_d\\|&\cdots &|\end{matrix}\right) 345\right\}$$
346Order the $b_i$ so that $\det>0$.
347
348\item {\bf [Compute $c_{\infty}$]}
349Let $\overline{C}$ be the the map induced by complex
350conjugation on $\Lambda_f/2\Lambda_f = \Lambda_f\tensor\F_2$.
351Then
352$$c_\infty = 2^{\dim(\ker(\overline{C} - 1)) - d}.$$
353\begin{proof}
354We must compute the order of the component group
355$$\Psi=\frac{A_f(\R)}{A_f(\R)^0} = 356 \frac{(\C^d/\Lambda_f)^+}{\R^d/\Lambda_f^+}.$$
357For $v\in\C^d$ denote by $\overline{v}$ its complex conjugate and
358by $[v]$ its image in $\C^d/\Lambda_f$.
359If $[v]\in(\C^d/\Lambda_f)^+$ then $[v]=[\overline{v}]$ so
360since $v+\overline{v}\in\R^d$ we have
361$$2[v] = [v]+[\overline{v}]\in\R^d/\Lambda_f^+,$$
362so $\Psi$ is annihilated by $2$.
363Thus 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
366as something in $\frac{1}{2}\Lambda_f$ plus something in $\R^d$.
367This means that $\Psi$ is generated by the image of
368$(\frac{1}{2}\Lambda_f/\Lambda_f)^+$.
369Thus
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
375Consequently
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}
384Let $F(x) = (\vphi_1(x),\ldots,\vphi_{2d}(x)).$
385Let $C$ 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.
387Then
388$$\left|\frac{L(A_f,1)}{\Omega_f}\right| 389 = [F(\cS_k(\Gamma,\Z))^+:F(\T e)] 390 \cdot 2^{\dim(\ker(\overline{C} - 1)) - d}$$
391where $[A:B]$ is the absolute value of the determinant of a change
392of basis matrix sending $A$ to $B$.
393In particular the ratio is a rational number.
394\end{theorem}
395\begin{proof}
396Let $\O$ be the image of
397the Hecke algebra in $\C^d$ by the map
398$$T_p \ra (a_p^{(1)},\ldots,a_p^{(d)}).$$
399Thus $\Delta_f = \Vol(\O)=[\Z:\O]$.
400We 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
420our $\Omega$ to the one for the Neron model, is a unit
421in $\Z[\frac{1}{2m}]$ where $m$ is the largest square
422dividing the level $N$.
423\begin{corollary}
424The quotient of $L(A_f,1)$ by the real volume,
425which appears in the BSD conjecture, a priori
426a real number is in fact a rational number.
427\end{corollary}
428
429
430\section{Examples}
431Let $f\in S_2(\Gamma_0(65),\C)$ be the unique newform
432for which $a_2$ satisfies $a_2^2+2a_2-1=0$.  Letting $\alp=a_2$ we have
433  $$f = q + \alpha q^2 + (\alpha+1)q^3 434 +(-2\alp-1)q^4 + q^5 + (-\alp+1)q^6 435 +(-2\alp)q^7 + (\alp-2)q^8+\cdots$$
436As we will see, the associated quotient $A=A_f$ of $J_0(65)$
437has analytic rank $0$.
438
439We first compute $\cS_2(\Gamma,\Z)\subset \cM_2(\Gamma,\Q)$.
440With respect to our chosen basis, the matrix of $T_2$ on
441$\cM_2(\Gamma,\Q)$ is
442$$T_2 = 443\left(\begin{array}{ccccccccccccc} 444 {3}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ 445 {0}&{1}&{2}&{1}&{1}&{1}&{1}&{1}&{1}&{1}&{0}&{0}&{0}\\ 446 {1}&{0}&{-1}&{0}&{0}&{0}&{1}&{1}&{1}&{1}&{0}&{0}&{1}\cr 447 {-1}&{0}&{0}&{-2}&{-1}&{-1}&{0}&{0}&{0}&{0}&{0}&{-1}&{-1}\cr 448 {0}&{-1}&{-1}&{0}&{1}&{1}&{-1}&{-2}&{0}&{0}&{1}&{1}&{0}\cr 449 {0}&{2}&{1}&{0}&{0}&{0}&{1}&{1}&{0}&{0}&{0}&{1}&{1}\cr 450 {-1}&{0}&{0}&{0}&{0}&{0}&{-1}&{0}&{0}&{0}&{1}&{0}&{0}\cr 451 {0}&{-1}&{-1}&{-1}&{-2}&{0}&{0}&{-2}&{-2}&{-1}&{0}&{0}&{0}\cr 452 {0}&{1}&{1}&{0}&{2}&{-1}&{1}&{2}&{3}&{0}&{-1}&{-1}&{0}\cr 453 {0}&{1}&{2}&{1}&{0}&{1}&{0}&{-1}&{-1}&{1}&{1}&{1}&{0}\cr 454 {-1}&{-1}&{-1}&{-1}&{-1}&{0}&{0}&{-1}&{-1}&{0}&{0}&{0}&{-1}\cr 455 {0}&{-1}&{-1}&{-1}&{0}&{0}&{0}&{-1}&{0}&{0}&{1}&{-1}&{1}\cr 456 {1}&{1}&{1}&{0}&{0}&{1}&{1}&{0}&{0}&{0}&{0}&{2}&{1}\cr 457 \end{array}\right).$$
458The left eigenspace corresponding to $f$ is the left kernel
459of $T_2^2+2 T_2-1$ which is
460$$P=\left( 461\begin{array}{ccccccccccccc} 462 {0}&{0}&{1}&{1}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{1}&{0}\\ 463 {0}&{-1}&{0}&{-1}&{-1}&{-1}&{1}&{0}&{1}&{1}&{1}&{0}&{1}\\ 464 {1}&{1}&{0}&{1}&{1}&{1}&{1}&{2}&{1}&{0}&{1}&{0}&{0}\\ 465 {-1}&{-1}&{-1}&{2}&{1}&{-1}&{-2}&{1}&{0}&{1}&{-2}&{-1}&{1}\\ 466\end{array} 467\right)$$
468The rows of $P$ are $\vphi_1,\ldots,\vphi_4$.
469
470\comment{
471The lattice
472    $$\{(\vphi_1(x),\ldots,\vphi_{4}(x)) : x \in \cS_k(\Gamma,\Z) \} 473 \subset \Q^{4}$$
474is simply the lattice spanned by the columns of $P$, which
475is 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
4777 does not lie in it).
478}
479
480We try several $g\in\Gamma_0(65)$:
481\begin{eqnarray*}
482v_1&=&\{0,\abcd{22}{1}{65}{3}(0)\} =  [0,0,0,1,0,0,0,0,0,0,1,0,-1]^t\\
483v_2&=&\{0,\abcd{-32}{-1}{65}{2}(0)\}=   [0,0,-1,1,0,0,1,0,0,0,1,0,-1]^t\\
484v_3&=&\{0,\abcd{-16}{-1}{65}{4}(0)\}=  [0,0,0,0,1,-1,1,0,-1,0,1,0,-1]^t\\
485v_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*}
487Then
488$$\vphi_i(v_j) = 489\left(\begin{matrix} 490 {1}&{-1}&{2}&{-1}\cr 491 {0}&{0}&{3}&{-2}\cr 492 {0}&{0}&{1}&{-3}\cr 493 {-1}&{-1}&{-1}&{-1}\cr 494\end{matrix}\right)$$
495which has rank $4$.
496
497The 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*}
508Inverting 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}
519P=[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
524A=[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;
5291,0,0,0,0,0,0,0,0,0;
5300,1,0,0,0,0,0,0,0,0;
5310,0,0,0,0,-1,0,0,0,0;
5320,0,1,0,0,0,0,0,0,0;
5330,0,0,1,0,0,0,0,0,0;
5340,0,0,0,1,0,0,0,0,0;
5350,0,0,0,0,1,0,0,0,0;
5360,0,0,0,0,0,1,0,0,0;
5370,0,0,0,0,0,0,1,0,0;
5380,0,0,0,0,0,0,0,1,0;
5390,0,0,0,0,0,0,0,0,1];}
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
549CC = [1,0,0,0,0,0,0,0,0,0,0,0,0;
5500,0,0,0,0,0,0,-1,0,0,1,1,1;
5510,0,0,-1,0,0,0,0,0,0,0,0,0;
5520,0,-1,0,0,0,0,0,0,0,0,0,0;
5530,0,1,0,0,0,0,1,1,1,0,-1,-1;
5540,0,0,0,0,0,1,0,0,0,0,1,1;
5550,0,0,0,0,1,0,0,0,0,0,0,-1;
5560,0,0,0,0,0,0,1,0,0,0,-1,0;
5570,0,-1,0,1,0,0,-1,0,-1,0,1,1;
5580,0,1,1,0,0,0,0,0,1,0,0,0;
5590,1,0,0,0,0,0,1,0,0,0,-1,-1;
5600,0,0,0,0,0,0,0,0,0,0,-1,0;
5610,0,0,0,0,0,0,0,0,0,0,1,1];
562
563\\ The matrix of C with respect to the basis Z=[z_1,...,z_4]
564C=[-1, 0, 0, 0;
565   0, -1, 0, 0;
566  -1, -2, 1, 0;
567   1, 0, 0, 1];
568
569The +1 eigenspace is:
570   span z_3, z_4.
571
572Using the below alp matrix we obtain the real period:
573
574alp*[z_3;z_4] = alp*[0, 0; 0, 0; 2,-1; 1, 3]
575=[-0.8051265926142280223898673966  1.943747539316507071177084460]
576 [-3.930406537482142877473965080  -1.628027693464980131583752011]
577
578which has determinant
579   8.950486425265488291492806649
580
581We also have
582   matrank(C*Mod(1,2)-1) = 1
583so
584  c_infinity = 2.
585
586Thus   \tilde{Omega} = 17.90097285053097658298561329.
587
588Delta = |1    1   |  = 2*sqrt(2)
589        |a2  a2bar|
590So,
591  Omega = 6.328949646223367581007308099
592
593The image of {0,oo} is simply [0,0,1,-1]~, so
594alp*[0,0,1,-1]~
595=[-1.063070829025425322615855453,
596  -0.4252471420813350514566677322];
597
598Taking the product of the two entries gives
599
600L(A_f,1)=L(f,1)*L(fbar,1)
601 = 0.4520678318730976843576858890
602
603We also get, for free by writing [0,0,1,-1]~ in terms
604of Z that the image of (0)-(oo) in A_f is rational 7 torsion.
605
606The characteristic polynomial of T_3  on A_f is:
607      x^2-2
608so 14 is an upper bound on the torsion.
609
610Thus
611   L(A_f,1)                                           1
612  ----------   =  0.07142857142857142857143189343  ~ ----
613   Omega                                              14
614
615
616Using the method of graphs and my component
617group formula David Kohel and I computed the tamagawa numbers and
618got
619     c5 = 7       c13 = 1
620
621The BSD conjecture asserts that (assume it hence):
622
623    1         [Sha] * c5 *c13                   [Sha] * 7
624--------   = -------------------------   =  -------------------------
625   14         [Torsion]*[Torsiondual]        [torsion]*[torsion dual]
626
627Thus
628    [Sha] = [torsion]*[torsion dual] / 2*7^2.
629
630Thus assuming BSD, the torsion on one of A_f or A_fdual must
631be 14.
632
633It could easily be the case that [torsion]=14, [torsion dual]=7
634and [Sha]=1.
635
636\end{verbatim}
637
638
639\section{Tables}
640
641\section{Source Code}
642The following pari code can be used to compute the period
643integrals $\langle P_g\{0,g(0)\}, f\rangle$.
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
652
653ncols(v)=matsize(v);
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,-v;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},
757Cambridge University Press, (1997).
758\end{thebibliography}
759
760\end{document}
761
762
763
764The matrix alpha_i^j which computes period map, when right compl with phi.
765
766alp=[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