Author: William A. Stein
1
\documentclass{slides}
2
\usepackage{amsmath}
3
\usepackage{amssymb}
4
\usepackage[all]{xy}
5
6
\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
7
\newcommand{\textcyr}{%
8
{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}%
9
\fontshape{n}\selectfont #1}}
10
\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
11
12
\newcommand{\hra}{\hookrightarrow}
13
\newcommand{\C}{\mathbf{C}}
14
\newcommand{\F}{\mathbf{F}}
15
\newcommand{\Q}{\mathbf{Q}}
16
\newcommand{\ra}{\rightarrow}
17
\newcommand{\T}{\mathbf{T}}
18
\newcommand{\Z}{\mathbf{Z}}
19
\newcommand{\tor}{\mbox{\tiny\rm tor}}
20
\newcommand{\Ind}{\mbox{\rm Ind}}
21
\newcommand{\vis}{\mbox{\rm vis}}
22
\newcommand{\m}{\mathfrak{m}}
23
\newcommand{\ncisom}{\approx} % noncanonical isomorphism
24
25
26
\title{Visible Shafarevich-Tate groups
27
of modular abelian varieties\footnote{Thanks
28
for inviting me!}\vspace{1em}\\
29
{\small (Utrecht Arithmetic Geometry Workshop)}}
30
\author{William A. Stein}
31
\date{Tuesday, 27 June 2000}
32
33
\begin{document}
34
\maketitle
35
36
\begin{slide}
37
38
Let $f=\sum a_n q^n\in S_2(\Gamma_0(N);\C)$ be a newform.\\
39
\mbox{}$\qquad f \leadsto$ optimal quotient $A_f$ of $J_0(N)$
40
$$\xymatrix{ 41 A_f^{\vee}{^(->}[r] & J_0(N){->>}[d]\\ 42 & A_f 43 }$$
44
Shafarevich-Tate group of $A$:
45
$$\Sha(A) := \ker\left(H^1(\Q,A) \ra \prod_v H^1(\Q_v,A)\right)$$
46
{\bf Conjecture:} (Birch, Swinnerton-Dyer, Tate)\vspace{-3ex}
47
\begin{enumerate}
48
\item {\em Formula:} $\Sha(A_f)$ is finite and
49
$$L(A_f,1) = 50 \frac{\Omega_{A_f}\cdot 51 \prod_{p\mid N} c_p}{\#A_f(\Q)\cdot \#A_f^{\vee}(\Q)}\cdot 52 \#\Sha(A_f).$$
53
\vspace{-5ex}
54
\item {\em Rank:} Order of vanishing of $L(A_f,s)$ at $1$ equals
55
rank of $A_f(\Q)$. (Regulators, etc.)
56
57
\end{enumerate}
58
59
60
\end{slide}
61
62
\begin{slide}
63
\head{Mazur: Visualize $\Sha$!}
64
65
Given $A \hra J$, let $B=J/A$. We have:
66
$$\xymatrix{ 67 0\ar[r] 68 &{A(\Q)} 69 \ar[r] 70 & {J(\Q)} 71 \ar[r] 72 & {B(\Q)} 73 \arr[d]d[][dlll]d[][dll]\\ 74 &H^1(\Q,A)\ar[r] 75 &H^1(\Q,J)\ar[r] 76 &{\cdots}\\ 77 } 78$$
79
%$$0 \ra A(\Q) \ra J(\Q) \ra B(\Q) \ra H^1(\Q,A) \ra H^1(\Q,J) \ra \cdots$$
80
The {\em visible or effaceable part} of $H^1(\Q,A)$ is the image
81
of $B(\Q)$; equivalently, the kernel of
82
$H^1(\Q,A) \ra H^1(\Q,J)$.
83
Visible, in the sense that torsor is fiber over point in $J(\Q)$.
84
85
{\bf Observation:} (Greenberg, ---) Any $c\in H^1(\Q,A)$,
86
is visible in $J=\mbox{\rm Res}_{K/\Q}(A_K)$, where~$K$ splits~$c$.\\
87
{P{\tiny ROOF}:}
88
Natural map $\iota:A\hookrightarrow J$ induces
89
$$H^1(\Q,A) \xrightarrow{\iota_*} H^1(\Q,J) 90 = H^1(K,A)$$
91
(by Shapiro's Lemma), so $\iota_*(c) = 0.$
92
93
Conrad: Abelian varieties over infinite fields embed in Jacobians.
94
95
{\bf Mazur's query:} Is $\Sha(A_f^{\vee})$ visible in $J_0(N)$?
96
97
\end{slide}
98
99
\begin{slide}
100
\head{The data ($N$ prime)}
101
Assume BSD. There are $38$ rank zero $A_f^{\vee}$
102
of prime level $\leq 2161$ with nontrivial odd part of $\Sha$.
103
Of these, $22$ have all this $\Sha$ visible. E.g.,
104
\begin{center}
105
\begin{tabular}{lcll}
106
\hspace{1em}$A_f$ & dim & $\sqrt{\#\Sha/ 2^?}$ & notes\\\hline
107
{\bf 389} & 20 & 5 &\\
108
{\bf 433} & 16 & 7 &\\
109
{\bf 563} & 31& 13 &\\
110
{\bf 571} & 2 & 3 &\\
111
{\bf 709} & 30 & 11 &\\
112
{\bf 997} & 42 & 9 & \\
113
{\bf 1061} & 46 & 151 & \\
114
{\bf 1091} & 62 & 7 & inv\\
115
{\bf 1171} & 53 & 11 & \\
116
{\bf 1283} & 62 & 5 & inv\\
117
{\bf 1429} & 53 & 5 & inv\\
118
{\bf 1481} & 62 & 14 & inv\\
119
$\cdots$ & $\cdots$ & $\cdots$ \\
120
{\bf 2111} & 112 & 211 & inv, not eisen\\
121
{\bf 2333} & 101 & 83341 & \\
122
\end{tabular}
123
\end{center}
124
125
\end{slide}
126
127
\begin{slide}
128
129
{\bf Theorem 1. (---)}
130
{\em Let $A, B \subset J$ s.t. $(A \cap B)(\C)$ and $A(\Q)$ finite.
131
Assume $B$ has toric reduction at each $\ell \mid N_J$.
132
Let $p$ be an odd prime (or principle prime ideal) not dividing $N_J$,
133
the orders of comp.\ grps.\ of $A$ and $B$,
134
$(J/B)(\Q)_{\tor}$, $B(\Q)_{\tor}$, and $B[p]\subset A$. Then
135
$$B(\Q)/p B(\Q) \subset \Sha(A)^{\vis}.\vspace{-5ex}$$}
136
137
{P{\tiny ROOF SKETCH}:} Since $B[p]\subset A\cap B$,
138
$B\ra C$ factors through $p$:
139
$$\xymatrix{ 140 & & B\ar[d] \ar[r]^{p}& B\ar[d]\\ 141 0\ar[r]& A\ar[r]&J\ar[r]&C\ar[r] &0.}$$
142
{\bf Key diagram:}
143
$$\xymatrix{ 144 0 \ar[r] & B(\Q) \ar[r]^{p}\ar[d] & B(\Q)\ar[dr]^{\pi} \ar[r]\ar[d] 145 & B(\Q)/pB(\Q)\ar[r]\ar[d] & 0\\ 146 0 \ar[r] & J(\Q)/A(\Q)\ar[r] & C(\Q) \ar[r] & H^1(\Q,A)^{\text{vis}} \ar[r] & 0\\ 147 }$$
148
Snake lemma implies $B(\Q)/p B(\Q)$ injects into $H^1(\Q,A)^{\text{vis}}$.
149
Use subtle exactness properties of N\'eron models
150
to get into $\Sha(A)^{\text{vis}}$.
151
\end{slide}
152
153
\begin{slide}
154
\head{Visibility at higher level}
155
Theorem 1 can be refined: condition that $p\nmid$ geometric component
156
group of $B$ replaced by
157
$$p \nmid \prod_{\ell} 158 \#\Phi_{B,\ell}(\F_\ell)\cdot \#H^1(\F_\ell,\Phi_{B,\ell}).$$
159
160
A while ago, Cremona and Mazur discovered three elliptic curves, {\bf
161
2849A}, {\bf 4343B}, {\bf 5389A}, in which BSD predicts odd invisible
162
$\Sha$.\vspace{-3ex}
163
164
\begin{center}
165
\begin{tabular}{lcl}
166
\hspace{1em}$A_f$ & $\sqrt{\#\Sha/ 2^?}$ & becomes visible\\\hline
167
{\bf 2849A} & $3$ & $8547 = 3\cdot 2849 = 3\cdot 7\cdot 11\cdot 37$\\
168
{\bf 5389A} & $3$ & $37723 = 7 \cdot 5389 = 7\cdot 17\cdot 317$\\\hline
169
{\bf 1429B} & $5$ & $2858 = 2\cdot 1429$ (dim $B=2$) \\
170
\end{tabular}\hspace{1em}
171
\end{center}
172
Now we know {\em unconditionally} that there is odd visible $\Sha$!
173
174
{\bf Conjecture:} Let $c\in \Sha(A_f^{\vee})[p]$. Then there exists an integer
175
$M$ such that $\iota_{*}(c)=0$ for $\iota:A_f^{\vee}\ra J_0(NM)$ one of the
176
natural maps. (Require $p\nmid \deg(\iota)$.)
177
178
\end{slide}
179
180
181
\begin{slide}
182
\head{Evidence for the BSD conjecture}
183
In the eventually
184
visible examples above, we have {\em unconditionally proved} that
185
up to a $2$-power $\Sha(A_f)$ is as big as BSD predicts.
186
187
Vanishing of $L(B,1)$ usually forces vanishing of $L(A_f,1)$ modulo $p$,
188
so we are very unlikely to observe that $\Sha(A_f)$ is bigger than
189
BSD predicts.
190
\end{slide}
191
192
193
\begin{slide}
194
195
196
{\bf Hard open problem:} {\em Suppose $L(E,1)$ and $L'(E,1)$ vanish.
197
Show $E(\Q)$ is infinite.}
198
199
Formulate visible analogue of BSD conjecture.
200
Deduce the existence of nontrivial visible
201
elements of $\Sha(A_f)$ in new way, and
202
conclude that the rank of $E$ is $>0$ without computing
203
points on~$E$.
204
205
{\em Test question:} Prove that the first elliptic curve of rank $>1$
206
has rank $>1$ {\em without} explicitely finding any points.
207
This is the curve $E$ labeled {\bf 389A}.
208
209
Strategy: Let~$A$ be {\bf 389E}, so $E \subset A\cap E$.
210
Use Euler system of Heegner points'' to deduce that
211
$5$ divides cardinality of {\em visible part} of $\Sha(A)$,
212
then use the key diagram
213
to deduce that $E(\Q)/5 E(\Q)$ is nontrivial
214
(assume $E(\Q)=0$ and use the snake lemma to derive a
215