 CoCalc Public Fileswww / tables / Notes / manin-distrib.tex
Author: William A. Stein
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10\title{\bf\Huge \mbox{The Manin index of optimal new quotients}\vspace{5ex}}
11\author{\LARGE William A. Stein}
12\date{\Large October 1, 1999\vspace{2ex}\\}
14\newcounter{Pagecount}
15\markboth{}{W.\thinspace{}A. Stein,  The Manin index of optimal new quotients}
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97\begin{document}
98
99%\maketitle
100\pagenumbering{Roman}
101%\setcounter{page}{0}
102\Page{The Birch and Swinnerton-Dyer conjecture}
103Consider the optimal quotient $A=A_f$ associated to a newform $f$:
104$$\xymatrix{ 105 & *+++{IJ_0(N)}\[email protected]{^(->}[d]\\ 106 & J_0(N)\[email protected]{->>}[d]^{\pi}\\ 107f = \sum a_n q^n \[email protected]{|~>}[r] & *+++{A}\[email protected]{^(->}[r]^{\iota}&{\cA}}$$
108Here $I := \ker (\T \onto \Z[\ldots a_n \ldots])$ is a prime ideal.
109
110\begin{thm}[Agash\'e,---]
111$$\frac{L(A,1)}{\Omega_{A}}\cdot c_A = 112 [\pi_*(H_1(X_0(N),\Z)^+) : \pi_*(\T \e)]/c_{\infty}$$
113\end{thm}
114Here
115\begin{bulletlist}
116\item $\cA :=$ N\'{e}ron model of $A$ over $\Z$.
117\item $\Omega_{A} :=$ volume of $A(\R)$ wrt basis for
118      $H^0(\cA,\Omega_{A})$.
119\item $c_\infty :=$ number of connected components of $A(\R)$, a power of $2$.
120\item {\bf Manin index}
121$$c_A := [S_2(N;\Z)[I] : \qexp(\pi^* \iota^* H^0(\cA,\Omega_{\cA}))] 122 \in\Z_{>0}$$
123\end{bulletlist}
124
125\begin{conj}[Birch, Swinnerton-Dyer]
126$$\displaystyle \frac{L(A,1)}{\Omega_{A}} 127 = \frac{\#\Sha(A/\Q)\cdot \prod c_p} 128 {\#A(\Q) \cdot \#\Adual(\Q)}.$$
129\end{conj}
130
131\Page{Motivation}
132{\bf Problem.}
133Verify the BSD conjecture, up to a power of~$2$,
134for specific~$A_f$ at square-free level.
135
136\begin{ex}
137$p=2333$
138$$J_0(2333) \isog A \cross B \cross C$$
139$\dim A = 4$, $\dim B = 89$, $\dim C=101$.
140$$L(A,1)=L(B,1)=0.$$
141$$\frac{L(C,1)}{\Omega_C} \cdot c_C 142 = \frac{83341^2}{11\cdot 53}\cdot 2^?$$
143$$c_p = 11\cdot 53 = \#C(\Q) = \#C^{\vee}(\Q)= \numer((p-1)/12).$$
144If we know that $c_C$ is a power of two then,\\
145to verify BSD up to $2$-powers, must show
146that $\#\Sha=83341^2\cdot 2^?$.\\
147In this case we are lucky:  $83341\mid \#(A\intersect C)$,
148which can (probably) be used to show that
149 $83341\mid \#\Sha$.
150\end{ex}
151
152\begin{ex}
153$p=2111$ is prime.
154$$J_0(2111) \isog A \cross B$$
155with $\dim A = 64$, $\dim B = 112$.\\
156$L(A,1)=0$
157$$\frac{L(B,1)} 158 {\Omega_B}\cdot c_B = \frac{211}{5}\cdot 2^?.$$
159$$c_p = 5\cdot 211 = \#B(\Q) = \#B^{\vee}(\Q) = \numer((p-1)/12).$$
160If we know that $c_B$ is a power of $2$, then
161to verify BSD, must show that $\#\Sha=211^2\cdot 2^?/c_B$.\\
162\end{ex}
163
164
165
166\Page{The Manin index}
167
168\begin{conj}[Agashe, ---, Manin when $\dim A=1$]\label{myconj}
169$c_A = 1.$
170\end{conj}
171\vspace{-2ex}
172
173{\bf Caution:} Without the condition that~$A$ is new, the conjecture
174is false. The Manin index of $J_0(33)$ is~$3$, because
175of a mod~$3$ congruence.
176
177A.~Logan is skeptical of Conjecture~\ref{myconj}.
178
179{\bf Evidence:} See Amod's talk from Wednesday.
180
181{\em Empirical evidence for the Birch and
182Swinnerton-Dyer conjectures for
183modular Jacobians of genus~2 curves},\\
184Flynn, Lepr\'evost, Schaeffer, ---, Stoll, Wetherell.\\
185We considered $28$ optimal quotients of dimension~$2$ that
186happened to be Jacobians.
187\begin{bulletlist}
188\item We computed $\Omega_A$ directly from a minimal
189proper regular models.
190\item We then computed the volume of $A(\R)$ with respect
191to $S_2(N;\Z)[I]$, and got the same answer, to several
192decimal places.
193\end{bulletlist}
194The non square-free levels treated:
195 $$N=3^2\cdot 7,\quad 3^2\cdot 13,\quad 5^3,\quad 3^3\cdot 5,\quad 3\cdot 7^2, 196 \quad 5^2\cdot 7,\quad 2^2\cdot 47,\quad 3^3\cdot 7.$$
197In every case, $c_A = 1$.
198
199
200\Page{Bounding the Manin index}
201
202We present a slight generalization of a theorem of Mazur,
203from his {\em Rational of prime degree} paper.
204\begin{thm}[Mazur when $\dim A=1$, ---]
205Let~$m$ be the largest square dividing~$N$.
206Let $A$ be an optimal quotient of $J_0(N)^{\new}$.
207Then $c_A$ is a unit in $\Z[\frac{1}{2m}]$.
208\end{thm}
209\begin{proof}
210We give the proof assuming that $A=A_f$.
211\begin{bulletlist}
212\item $R:=\Z[\frac{1}{2m}]$.
213\item $\cJ :=$ Neron model of $J_0(N)$ over $R$.
214\item $\cA :=$ ditto for $A$.
215\item $X :=$ smooth locus of the minimal proper regular model of $X_0(N)$.
216\end{bulletlist}
217
218Consider
219$$\xymatrix{ 220 *+++{H^0(\cA,\Omega_{\cA/R})}\[email protected]{^(->}[rr]^{\pi^*}_{\text{pullback}}&& 221 {H^0(\cJ,\Omega_{\cJ/R})}\ar[r]^{\isom}& 222 {H^0(X,\Omega_X)}\ar[rr]^{\qexp}&&R[[q]].}$$
223\begin{bulletlist}
224\item $\pi^*$ is an inclusion because $H^0(\cA,\Omega_{\cA/R})$
225is free ($\cA/R$ is smooth) and $\pi^*_\C$ is an inclusion.
226\item The middle isomorphism uses Mich\'{e}le Raynaud's result:
227  $$\cPic^0(X)\isom \cJ^0,$$
228induces $H^1(X,\O_X)\xrightarrow{\,\isom\,} H^1(\cJ,\O_{\cJ})$
229on tangent spaces.  Now apply Grothendieck duality.
230\item $\qexp$ comes from Tate curve:
231$$\tau:\Spec R[[q]] \ra X.$$
232Formal completion of $X$ at $\bifR$ is $(\Spec R)[[q]]$.
233\end{bulletlist}
234
235
236\Page{Proof (continued)}
237{\bf To show:} Image of $H^0(\cA,\Omega_{\cA})$ saturated.
238Then  $c_A\in R^*$ since $S_2(N,R)[I]\subset R[[q]]$ is saturated.
239
240Consider
241$$\xymatrix{ 242 0\ar[r]& H^0(\cA,\Omega_{\cA})\ar[rr]^{\quad\qexp} 243 &&R[[q]]\ar[r]&D\ar[r]&0.}$$
244Is $D$ torsion free?  Let $\ell\nmid 2m$ be a prime; tensor with $\Fl$:
245$$\xymatrix{ 246 &{\Tor^1(D,\Fl)}\[email protected]{=}[d]\\ 2470\ar[r]&D[\ell]\ar[r]&H^0(\cA,\Omega_{\cA})\tensor\Fl\ar[r]& 248 {\Fl[[q]]}\ar[r]&D\tensor\Fl\ar[r]&0}$$
249Thus
250$$D[\ell]=0 \iff 251H^0(\cA,\Omega_{\cA})\tensor\Fl\ra\Fl[[q]]\text{ injective}$$
252Since $\ell\nmid 2m$, we have $e(\Ql)=1<\ell-1$, so [Mazur, Cor.~1.1]
253gives an exact sequence
254$$255 0\ra H^0(\cA/\Zl,\Omega_{\cA/\Zl})\ra 256 H^0(\cJ/\Zl,\Omega_{\cJ/\Zl})\ra 257 H^0(\cB/\Zl,\Omega_{\cB/\Zl})\ra 0.$$
258Here $B=\ker(\pi:J\ra A)$.
259Since $H^0(\cB/\Zl,\Omega_{\cB/\Zl})$ is torsion free,
260$$\xymatrix{ 261 *+++{H^0(\cA/\Zl,\Omega_{\cA/\Zl})\tensor\Fl} 262 \[email protected]{^(->}[r]& 263 {H^0(\cJ/\Zl,\Omega_{\cJ/\Zl})\tensor\Fl}}$$
264Fact (see e.g., pg 183 of Eisenbud's book):
265 $\Zl/R$ is flat.\\
266Cohomology commutes with flat base change'':
267  $$H^0(\cA,\Omega_{\cA})\tensor\Zl \isom H^0(\cA/\Zl,\Omega_{\cA/\Zl})$$
268
269
270\Page{Proof (continued)}
271$$\xymatrix{ 272*+++{H^0(\cA,\Omega_{\cA})\tensor\Fl} 273 \[email protected]{^(->}[rrr]\ar[dddrrr]_{\txt{injective?}\quad}&&& 274 H^0(\cJ,\Omega_{\cJ})\tensor\Fl\ar[d]^{\isom}\\ 275&&& 276 H^0(X,\Omega_{X})\tensor\Fl\ar[d]^{\isom}\\ 277&&& 278 H^0(X_{\Fl},\Omega_{X_{\Fl}})\ar[d]^{\qexp}\\ 279&&& 280 {\Fl[[q]].} 281}$$
282The isomorphism
283 $H^0(X,\Omega_{X})\tensor\Fl\isom 284 H^0(X_{\Fl},\Omega_{X_{\Fl}})$
285uses cohomological trickery and that $k=2$, $N\geq 5$.
286
287 If $\ell\nmid N$, then $\qexp$ is injective
288    (see Deligne-Rapoport, Theorem 3.9).
289Thus $H^0(\cA,\Omega_{\cA})\tensor\Fl \hookrightarrow \Fl[[q]]$,
290so $D[\ell]=0$, as required.
291
292
293If $\ell\mid N$ then $\qexp$ is not necessarily injective.
294However, $\ell$ exactly divides $N$. Deligne-Rapoport model
295of $X_{\Fl}$:
296
297
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507	(2992,815)(3010,827)(3029,840)
508	(3048,853)(3069,866)(3090,878)
509	(3112,891)(3135,904)(3159,917)
510	(3183,930)(3208,942)(3234,953)
511	(3259,964)(3285,974)(3311,984)
512	(3336,993)(3362,1001)(3387,1008)
513	(3412,1014)(3437,1019)(3462,1023)
514	(3487,1027)(3510,1030)(3534,1031)
515	(3557,1032)(3582,1033)(3607,1033)
516	(3633,1032)(3659,1030)(3686,1027)
517	(3714,1024)(3742,1020)(3770,1015)
518	(3798,1010)(3826,1003)(3854,997)
519	(3881,989)(3908,981)(3934,973)
520	(3960,964)(3985,954)(4009,945)
521	(4033,935)(4056,924)(4078,913)
522	(4100,902)(4121,890)(4142,878)
523	(4163,865)(4184,852)(4205,838)
524	(4227,823)(4248,808)(4270,792)
525	(4291,775)(4313,758)(4334,741)
526	(4355,724)(4376,706)(4396,688)
527	(4417,671)(4436,653)(4455,636)
528	(4474,619)(4492,603)(4509,587)
529	(4526,571)(4543,556)(4559,541)
530	(4575,527)(4592,511)(4609,496)
531	(4626,481)(4644,466)(4662,450)
532	(4680,435)(4699,419)(4718,404)
533	(4737,389)(4757,374)(4776,359)
534	(4796,344)(4816,330)(4836,317)
535	(4855,304)(4874,291)(4894,279)
536	(4912,268)(4931,257)(4950,246)
537	(4968,237)(4987,227)(5006,218)
538	(5025,209)(5045,200)(5066,191)
539	(5087,183)(5109,175)(5132,167)
540	(5156,160)(5180,153)(5205,147)
541	(5230,141)(5255,136)(5280,131)
542	(5306,128)(5331,125)(5356,123)
543	(5381,121)(5405,121)(5429,121)
544	(5453,122)(5476,124)(5500,127)
545	(5521,130)(5543,134)(5565,139)
546	(5588,144)(5611,150)(5634,157)
547	(5658,165)(5682,174)(5707,183)
548	(5731,193)(5756,203)(5780,214)
549	(5804,226)(5828,238)(5851,250)
550	(5874,262)(5895,275)(5916,288)
551	(5937,300)(5956,313)(5974,326)
552	(5992,339)(6008,352)(6025,364)
553	(6043,380)(6061,396)(6078,412)
554	(6096,429)(6112,447)(6129,465)
555	(6145,483)(6161,502)(6177,522)
556	(6193,541)(6208,561)(6222,581)
557	(6236,600)(6250,619)(6263,638)
558	(6276,657)(6288,675)(6301,692)
559	(6313,710)(6325,727)(6337,744)
560	(6349,762)(6362,779)(6375,797)
561	(6389,815)(6404,834)(6419,852)
562	(6435,871)(6451,889)(6467,908)
563	(6484,925)(6501,943)(6518,959)
564	(6535,975)(6552,990)(6569,1004)
565	(6586,1018)(6603,1030)(6620,1041)
566	(6637,1052)(6653,1061)(6669,1070)
567	(6686,1078)(6703,1085)(6721,1092)
568	(6740,1099)(6760,1104)(6780,1109)
569	(6801,1113)(6822,1116)(6844,1119)
570	(6866,1120)(6888,1120)(6911,1120)
571	(6933,1118)(6956,1115)(6978,1111)
572	(7000,1106)(7022,1100)(7044,1094)
573	(7065,1086)(7087,1077)(7106,1069)
574	(7125,1060)(7144,1049)(7164,1039)
575	(7184,1027)(7205,1014)(7227,1000)
576	(7249,986)(7272,971)(7295,955)
577	(7319,938)(7343,921)(7367,903)
578	(7392,885)(7416,866)(7441,848)
579	(7465,829)(7490,810)(7514,792)
580	(7537,773)(7561,755)(7584,737)
581	(7607,720)(7630,703)(7652,685)
582	(7675,669)(7697,652)(7720,635)
583	(7743,619)(7767,602)(7791,585)
584	(7815,568)(7840,551)(7866,534)
585	(7892,517)(7918,500)(7944,483)
586	(7971,466)(7998,449)(8024,433)
587	(8051,417)(8077,402)(8103,387)
588	(8128,373)(8153,359)(8178,346)
589	(8201,334)(8225,322)(8247,311)
590	(8269,301)(8291,291)(8312,281)
591	(8337,271)(8362,260)(8387,251)
592	(8412,242)(8439,233)(8466,224)
593	(8495,215)(8526,207)(8559,198)
594	(8593,189)(8629,180)(8666,171)
595	(8702,162)(8737,154)(8770,147)
596	(8799,141)(8823,135)(8840,132)
597	(8852,129)(8859,128)(8862,127)
598\end{picture}
599}
600\vspace{2ex}
601
602The intersections are supersingular points; the Atkin-Lehner
603involution~$w_{\ell}$ swaps~$\pmb{0}$ and$\pmb{\infty}$.
604
605Consider $g\in H^0(\cA,\Omega_{\cA})\tensor\Fl$ such that
606$\qexp(g)=0$.
607Then~$g$ vanishes on the component of~$X_{\Fl}$ containing~$\infty$.
608Since $\cA$ is new and corresponds to a single eigenform,
609$w_\ell(g) = \pm g$, so $\qexp(w_\ell(g))=0$; hence~$g$
610vanishes on the other component of $X_{\Fl}$.  This forces $g=0$,
611and hence $D[\ell]=0$.
612\end{proof}
613
614
615
616\end{document}
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