Author: William A. Stein
1\documentclass{article}
2\usepackage[british]{babel}
3\usepackage{amsmath,amssymb,amsthm,mathrsfs}
4\usepackage[mathcal]{euscript}
5\usepackage[a4paper,body={14cm,23cm}, includefoot]{geometry}
6\usepackage[dvips]{color}
8
9\newcommand{\william}{\footnote{[[William: #1]]}\marginpar{\hfill {\sf[[\thefootnote]]}}}
10
11\newcommand{\christian}{\footnote{[[Christian: #1]]}\marginpar{\hfill {\sf[[\thefootnote]]}}}
12
13
14%--   Cyrillic
15\input cyracc.def \font\tencyr=wncyr10 \def\russe{\tencyr\cyracc}
16\def\Sha{\text{\russe{Sh}}}
17
18%--   nicer <=
19\renewcommand{\geq}{\geqslant}
20\renewcommand{\leq}{\leqslant}
21
22%--   Colours
23\definecolor{dgreen}{rgb}{0,0.4,0.4}
24\definecolor{dblue}{rgb}{0,0,0.7}
25\definecolor{red}{rgb}{1,0,0}
26
27%--   Theorems
28\newtheoremstyle{mythm}{11pt}{11pt}{\it\color{dblue}}{}{\bf\color{dblue}}{.}{\newline}{}
29\theoremstyle{mythm}
30\newtheorem{thm}{Theorem}
31\newtheorem{thmkato}[thm]{Kato's Theorem}
32
33\newtheoremstyle{mypl}{11pt}{11pt}{\it\color{dblue}}{}{\bf\color{dblue}}{.}{ }{}
34\theoremstyle{mypl}
35\newtheorem{prop}[thm]{Proposition}
36\newtheorem{lem}[thm]{Lemma}
37\newtheorem{cor}{Corollary}
38
39\newtheoremstyle{mycon}{11pt}{11pt}{\it\color{dgreen}}{}{\bf\color{dgreen}}{.}{ }{}
40\theoremstyle{mycon}
41\newtheorem{con}{Conjecture}
42
43%named versions
44\newenvironment{theorem}{\begin{thm} {\bf (#1)}}{\end{thm}}
45
46\newenvironment{conjecture}{\begin{con} {\bf (#1)}}{\end{con}}
47
48
49
50%-- Operators
51\DeclareMathOperator{\Aut}{Aut}
52\DeclareMathOperator{\Hom}{Hom}
53\DeclareMathOperator{\HH}{H}
54\DeclareMathOperator{\Gl}{GL}
55\DeclareMathOperator{\Gal}{Gal}
56\DeclareMathOperator{\rk}{rank}
57\DeclareMathOperator{\Reg}{Reg}
58\DeclareMathOperator{\tors}{tors}
59\DeclareMathOperator{\ord}{ord}
60\DeclareMathOperator{\Col}{Col}
61
62%-- Abbrevations
63\newcommand{\sss}{\scriptscriptstyle}
64\newcommand{\liminj}{\varinjlim}
65\newcommand{\limproj}{\varprojlim}
66\newcommand{\vu}{\upsilon}
67\newcommand{\mf}{\mathfrak}
68
69%-- Fields
70\newcommand{\QQ}{\mathbb{Q}}
71\newcommand{\RR}{\mathbb{R}}
72\newcommand{\CC}{\mathbb{C}}
73\newcommand{\ZZ}{\mathbb{Z}}
74\newcommand{\FF}{\mathbb{F}}
75%\newcommand{\QZ}{{{}^{\QQ_p}\!/\!{}_{\ZZ_p}}}
76\newcommand{\QZ}{{\QQ_p/\ZZ_p}}
77
78
79%-- Iwasawa theory
80\newcommand{\QQinf}{{}_\infty\QQ}
81\newcommand{\Kinf}{{}_\infty K}
82\newcommand{\QQn}{{}_n\QQ}
83\newcommand{\QQnplusone}{{}_{n+1}\QQ}
84\newcommand{\Hinf}{{}_{\infty\!} \HH}
85\newcommand{\Hinfloc}{{}_{\infty\!} \HH_{\text{loc}}^1}
86\newcommand{\Hinfglob}{{}_{\infty\!} \HH_{\text{glob}}^1}
87\newcommand{\Ginf}{{}_{\infty\!} G}
88\newcommand{\cinf}{{}_{\infty} c}
89
90\DeclareMathOperator{\an}{an}
91\DeclareMathOperator{\tor}{tor}
92
93
94%-- Elliptic Curves
95\newcommand{\Lstar}{L^{\ast}(E,1)}
96\newcommand{\Sel}{\mathcal{Sel}}
97\newcommand{\Rel}{\mathcal{R}}
98\newcommand{\Tp}{T_{\! p}}
99\newcommand{\Xinf}{X(E/\QQinf)}
100
101\newcommand{\LL}{\mathcal{L}}
102\newcommand{\Linv}{\mathscr{L}}
103
104%-- Arrows
105
106\newcommand{\IncTo}{\hookrightarrow}
107\newcommand{\rTo}{\longrightarrow}
108
109%-- small-er indices
110
111\newcommand{\OmegaE}{\Omega_{\sss E}}
112\newcommand{\qE}{q_{\sss E}}
113\newcommand{\ZeroE}{O_{\sss E}}
114\newcommand{\fE}{f_{\sss E}}
115\newcommand{\etaE}{\eta_{\sss E}}
116\newcommand{\omegaE}{\omega_{\sss E}}
117
118
119%--
120
121\newcommand{\manque}{%
122 \begin{center}
123 {\color{red}\bfseries\large [\dots #1 \dots ] }
124 \end{center}
125}
126
127
128\newcommand{\note}{{\sf\small[[#1]]}}
129%-- Title.
130
131\begin{document}
132
133\author{William Stein and Christian Wuthrich}
134\title{Computions About Tate-Shafarevich Groups Using Iwasawa Theory}
135
136\maketitle
137
138\abstract{We explain how to combine deep results from Iwasawa theory
139with explicit computation to obtain information about
140$p$-parts of Shafarevich-Tate groups of elliptic curves over $\QQ$.
141This method provides a practical way to compute
142$\Sha(E/\QQ)[p]$ in many cases when traditional $p$-descent methods
143are completely impractical.}
144
145
146\section{Introduction}\label{ranksha_sec}
147
148\william{Be sure to cite \cite{colmez}, perin-riou, etc.}
149\william{In sections 3--5, it would be good to have an actual
150short (!) illustrative example in each section.}
151
152
153 Let $E$ be an elliptic curve defined over $\QQ$ and let
154 \begin{equation}\label{w_eq}
155   y^2 \, + \, a_1\, x\,y\, + \,a_3\,y \,=\, x^3 \, + \, a_2\, x^2\, + \,a_4\,x\, + \,a_6
156 \end{equation}
157be a choice of global minimal Weierstrass equation for $E$.
158 The Mordell proved that the set of rational points
159 $E(\QQ)$ is an abelian group of finite rank $r=\rk(E(\QQ))$.
160 Birch and Swinnerton-Dyer then conjectured that
161 $162 r = \ord_{s=1} L(E,s), 163$
164 where $L(E,s)$ is the Hasse-Weil $L$-function of $E$
165 (see Conjecture~\ref{bsd_con} below).
166 We call $r_{\an} = \ord_{s=1} L(E,s)$ the analytic
167 rank of $E$.
168
169There is no known provably correct general
170algorithm to compute $r$, but one can computationally
171obtain upper and lower bounds in any particular case.
172One way to give a lower bound on $r$ is to  search for linearly independent points
173of small height via the method of descent, which involves searching for points of
174even smaller  height on a collection of auxiliary curves.
175Complex and $p$-adic Heegner points constructions can also be used in some
176cases to bound the rank from below.
177To give a computable upper bound on the rank $r$,
178 apart from the case of analytic ranks $0$ and $1$ when Kolyvagin's work on Euler systems
179  can be applied, the only general way of obtaining an upper bound is by doing an $n$-descent
180  for some integer $n>1$. The 2-descents implemented by J. Cremona~\cite{cremona}
181  and Denis Simon \cite{simon}, and the $3$ and $4$ descents in Magma, are particularly
182  powerful. But they may fail in practice to compute the exact  rank
183  due to the presence of $2$ or $3$-torsion elements in the Tate-Shafarevich group.
184
185 The Tate-Shafarevich group, denoted by $\Sha(E/\QQ)$, is a torsion abelian group associated to $E/\QQ$. It is the kernel of the localisation map
186 \begin{equation*}
187  0\rTo \Sha(E/\QQ) \rTo \HH^1(\QQ,E)\rTo \prod_\vu\HH^1(\QQ_\vu,E)
188 \end{equation*}
189 where the product runs over all places $\vu$ in $\QQ$. The arithmetic importance of this group lies in its geometric interpretation. There is a bijection from $\Sha(E/\QQ)$ to the $\QQ$-isomorphism classes of principal homogeneous spaces $C/\QQ$ of $E$ which have points everywhere locally. In particular, $C$ is a curve of genus 1 defined over $\QQ$ whose Jacobian is isomorphic to $E$. nontrivial elements in $\Sha(E/\QQ)$ correspond to curves $C$ which defy the Hasse principle.
190
191\begin{conjecture}{Shafarevich and Tate}\label{consha_con}
192	The group $\Sha(E/\QQ)$ is finite.
193\end{conjecture}
194
195
196 These two invariants, the rank $r$ and the Tate-Shafarevich group $\Sha(E/\QQ)$ are encoded in the Selmer group.
197Let $E(p)$ denote the $\Gal(\bar\QQ/\QQ)$-module of all torsion points of $E$ whose orders are powers of $p$. The Selmer group $\Sel_p(E/\QQ)$  is defined by
198the following exact sequence:
199 \begin{equation*}
200  0\rTo \Sel_p(E/\QQ)\rTo \HH^1(\QQ,E(p))\rTo \prod_\vu \HH^1(\QQ_\vu,E)\, .
201 \end{equation*}
202 Likewise, for any positive integer $n$, the $n$-Selmer group is defined by
203 the exact sequence
204 $$0 \to \Sel^{(n)}(E/\QQ) \to \HH^1(\QQ,E[n])\rTo \prod_\vu \HH^1(\QQ_\vu,E)$$
205 where $E[n]$ is the subgroup of elements of order dividing $n$ in $E$.
206
207 It follows from the Kummer sequence that
208 there are short exact sequences
209 $$210 0\rTo E(\QQ)/n E(\QQ) \rTo \Sel^{(n)}(E/\QQ)\rTo \Sha(E/\QQ)[n]\rTo 0\,. 211$$
212 and
213 \begin{equation*}
214  0\rTo E(\QQ)\otimes \QZ \rTo \Sel_p(E/\QQ)\rTo \Sha(E/\QQ)(p)\rTo 0\,.
215 \end{equation*}
216 If the Tate-Shafarevich group is finite, then the $\ZZ_p$-corank
217 of $\Sel_p(E/\QQ)$ is equal to  the rank $r$ of $E(\QQ)$.
218
219 The finiteness of $\Sha(E/\QQ)$ is only known for curves of analytic rank $0$ and $1$
220 in which case computation of Heegner points and Kolyvagin's work on Euler systems
221 gives an explicit computable multiple of its order.
222The group $\Sha(E/\QQ)$ is not known to be finite for even a single elliptic curve
223with $r_{\an}\geq 2$.  In such cases, the best one can do using current techniques
224is hope to bound the $p$-part $\Sha(E/\QQ)(p)$ of $\Sha(E/\QQ)$, for specific
225primes $p$.  Even this might not a priori be possible, since it is not known that
226$\Sha(E/\QQ)(p)$ is finite.  However, if it were the case that $\Sha(E/\QQ)(p)$
227is finite (as Conjecture~\ref{consha_con} asserts), then this could be verified
228by computing Selmer groups $\Sel^{(p^n)}(E/\QQ)$ for
229sufficiently many $n$ (see, e.g., \cite{stoll}).  Note that practical
230computation of $\Sel^{(p^n)}(E/\QQ)$
231is prohibitively difficult for all but a few very small $p^n$.
232
233The algorithm in this paper gives another method for computing an upper
234bound on the order of $\Sha(E/\QQ)(p)$,  for most primes $p$.
235We will  exclude $p=2$, since traditional descent methods work well
236at $p=2$, and Iwasawa theory is not as well developed for $p=2$.
237We also exclude primes $p$ such that $E$ has additive reduction
238at $p$  (see Section~\ref{sec:additive}).
239The algorithm requires that the full Mordell-Weil group $E(\QQ)$ is known.
240
241
242%% -------------------------------------------------------------------------
243\section{The Birch and Swinnerton-Dyer conjecture}\label{bsd_sec}
244
245If Conjecture~\ref{bsd_con} below were true, it would yield
246an algorithm to compute both the rank $r$ and the order
247of $\Sha(E/\QQ)$.
248
249Let $E$ be an elliptic curve over $\QQ$, and
250 let $L(E,s)$ be the Hasse-Weil $L$-function associated to the $\QQ$-isogeny class of $E$.
251According to \cite{bcdt} (which completes work initiated in \cite{}), the function
252$L(E,s)$  is holomorphic on the whole complex plane.
253 Let $\omegaE$ be the invariant differential $dx/(2y+a_1 x+a_3)$ of
254 a minimal Weierstrass equation~\eqref{w_eq} of $E$. We write
255 $\OmegaE=\int_{E(\RR)} \omegaE \in \RR_{>0}$ for the N\'eron period of $E$.
256 \begin{conjecture}{Birch and Swinnerton-Dyer}\label{bsd_con}
257  \begin{enumerate}
258  \item The order of vanishing of the Hasse-Weil function $L(E,s)$ at $s=1$ is equal to the rank $r=\rk(E(\QQ))$.
259  \item The leading term $\Lstar$ of the Taylor expansion of $L(E,s)$ at $s=1$ satisfies
260  \begin{equation}\label{bsd_eq}
261   \frac{\Lstar}{\OmegaE} = \frac{\prod_\vu c_\vu\cdot \#\Sha(E/\QQ)}{(\#E(\QQ)_{\tors})^2}\cdot\Reg(E/\QQ)
262  \end{equation}
263   where the Tamagawa numbers are denoted by $c_\vu$ and $\Reg(E/\QQ)$
264   is the regulator of $E$, i.e., the discriminant of the  N\'eron-Tate canonical
265   height pairing on $E(\QQ)$.
266  \end{enumerate}
267 \end{conjecture}
268 \begin{prop}
269 If Conjecture~\ref{bsd_con} is true, then there is an algorithm to compute~$r$
270 and $\#\Sha(E/\QQ)$.
271 \end{prop}
272 \begin{proof}
273 The proof is well known, but we repeat it here since it illustrates several key ideas.
274 By naively searching for points in $E(\QQ)$ we obtain a lower bound on $r$,
275 which is closer and closer to the true rank $r$, the longer we run the search.
276 At some point this lower bound will equal $r$, but without using further information
277 we do not know when that will occur.  As explained, e.g., in \cite{cremona:algs},
278 we can for any $k$ compute $L^{(k)}(E,1)$ to any desired precision.
279 Such computations yield upper bounds on $r_{\an}$.  In particular, if
280 we compute $L^{(k)}(E,1)$ and it is nonzero (to the precision of our computation),
281 then $r_{\an} < k$.  Eventually this method will also converge to give an upper
282 bound on $r_{\an}$, though again without further information we do not know
283 when our computed upper bound on $r_{\an}$ equals to the true value
284 of $r_{\an}$.   However, if we know Conjecture~\ref{bsd_con}, we know that
285 $r = r_{\an}$, hence at some point the lower bound on $r$ computed using
286 point searches, will equal the upper bound on $r_{\an}$ computed using
287 the $L$-series.  At this point, by Conjecture~\ref{bsd_con} we know the
288 true value of $r$.
289
290 Once $r$ is known, one can compute $E(\QQ)$ via a point search (and
291 saturation \cite{cremona??}), hence we can approximate $\Reg(E/\QQ)$
292 to any desired precision.  All other quantities in \ref{bsd_eq} can also be
293 approximated to any desired precision.  Solving for  $\#\Sha(E/\QQ)$
294 in \ref{bsd_eq} and computed all other quantities to large enough precision
295 to determine $\#\Sha(E/\QQ)$ then determines $\#\Sha(E/\QQ)$, as claimed.
296   \end{proof}
297
298 Note that the conjecture~\eqref{bsd_eq} is also invariant under isogenies
299 defined over~$\QQ$ (see Cassels~\cite{cassels}).
300
301
302%% -------------------------------------------------------------------------
303\section{The $p$-adic $L$-function}\label{lp_sec}
304
305We will assume for the rest of this article that $E$ does not admit complex multiplication (CM),
306though CM curves are an area of active research for these methods (\cite{rubin, etc}).
307
308 In order to  formulate a $p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer, one needs first a $p$-adic version of the analytic function $L(E,s)$. Mazur and Swinnerton-Dyer~\cite{mazurswd} have found such a function. We refer to~\cite{mtt} for details on the construction and the historic references.
309
310 Let $\pi\colon X_0(N)\rTo E$ be the modular parametrisation of $E$ and let $c_{\pi}$ be the Manin constant, i.e., the positive integer satisfying $c_\pi\cdot\pi^{*}\omegaE = 2\pi i f(\tau) d\tau$ with
311 $f$ the newform associated to $E$. Manin conjectured that $c_{\pi}=1$, and much
312 work has been done toward this conjecture (\cite{edixhoven, me}).
313
314 Given a rational number $r$, consider
315 the image $\pi_{*}(\{r\})$ in $H_1(E,\RR)$ of the path joining $r$ to $i\,\infty$ in the upper half plane.
316 Define
317 \begin{equation*}
318  \lambda^{+}(r) =\frac{c_{\pi}}{2}\cdot \left( \int_{\pi_{*}(\{r\})} \omegaE + \int_{\pi_{*}(\{-r\})} \omegaE \right) = \pi i \cdot \left ( \int_r^{i\infty} f(\tau)\, d\tau + \int_{-r}^{i\infty} f(\tau)\, d\tau \right )
319 \end{equation*}
320 There is a basis $\{\gamma_{+},\gamma_{-}\}$ of $H_1(E,\ZZ)$ such that $\int_{\gamma_{+}} \omegaE$ is equal to $\OmegaE$ if $E(\RR)$ is connected and to $\tfrac{1}{2}\,\OmegaE$ otherwise.
321  By a theorem of Manin~\cite{manin}, we know that $\lambda^{+}(r)$ belongs to
322  $\QQ\cdot \OmegaE$. We define the modular symbol $[r]^{+}\in\QQ$ to be
323    \begin{equation*}
324    [r]^{+} \cdot \OmegaE  = \lambda^{+}(r)
325  \end{equation*}
326  for all $r\in\QQ$.
327 In particular we have $^{+}=L(E,1)\cdot \OmegaE^{-1}$.
328 The quantity $[r]^{+}$ can be computed either
329 algebraically using modular symbols and linear algebra
330 (\cite{cremona:algs}) or numerically, by approximating both
331 $\OmegaE$ using the Gauss arithmetic-geometry mean
332  and $\lambda^{+}(r)$ by summing a
333  rapidly convergent series, and
334  bounding the denominator of $\lambda^{+}(r)/\OmegaE$
335  using results about modular symbols.\william{This is probably way too vague -- I'm being lazy.}
336
337 Let $p$ be a prime of semistable reduction. We write\footnote{%
338 	The context should make it clear if we speak about $a_p$ or $a_2$ and $a_3$ as in~\eqref{w_eq}.
339 } $a_p$ for the trace of Frobenius.
340 Suppose first that $E$ has good reduction at $p$. Then $N_p=p+1-a_p$ is the number of points on $\tilde{E}(\FF_p)$. Let $X^2 -a_p\cdot X +p$ be the characteristic polynomial of Frobenius and let $\alpha\in\bar\QQ_p$ be a root of this polynomial such that $\ord_p(\alpha) <1$. There are two different possible choices if $E$ has supersingular reduction and there is a single possibility for primes where $E$ has good ordinary reduction.
341  Now if $E$ has multiplicative reduction at $p$, then $a_p$ is $1$ if it is split multiplicative and $a_p$ is $-1$ if it is nonsplit multiplicative reduction.
342  In either multiplicative case, we have to take $\alpha=a_p$.
343
344 Define a measure on $\ZZ_p^\times$ with values in $\QQ(\alpha)$ by
345 \begin{equation*}
346  \mu_{\alpha} ( a + p^k \ZZ_p ) = \frac{1}{\alpha^k}\cdot \left[\frac{a}{p^k}\right]^{+} -\frac{1}{\alpha^{k+1}}\cdot \left[\frac{a}{p^{k-1}}\right]^{+}
347 \end{equation*}
348 for any $k\geq 1$ and $a\in\ZZ_p^\times$. Given a continuous character $\chi$ on $\ZZ_p^\times$ with values in the completion $\CC_p$ of the algebraic closure of  $\QQ_p$, we may integrate $\chi$ against $\mu_{\alpha}$.
349 Any invertible element $x$ of $\ZZ_p^{\times}$ can be written as $\omega(x)\cdot \langle x\rangle$ where $\omega(x)$ is
350 a $(p-1)$st
351 root of unity and $\langle x\rangle$ belongs to $1+2p\ZZ_p$. We define the analytic $p$-adic $L$-function by
352 \begin{equation*}
353   L_\alpha (E,s) = \int_{\ZZ_p^\times} \langle x\rangle^{s-1} \, d\mu_{\alpha}(x)
354   \quad\text{ for all $s\in\ZZ_p$.}
355 \end{equation*}
356 where by $\langle x\rangle^{s-1}$ we mean $\exp_p((s-1)\cdot \log_p(\langle x\rangle ))$. The function $L_\alpha(E,s)$ extends to a locally analytic function in $s$ on the disc defined by $\vert s- 1\vert < 1$ (see \S~13 in~\cite{mtt}).
357
358 Let $\Ginf$ be the Galois group of the cyclotomic extension $\QQ(\mu_{p^\infty})$
359 obtained by adjoining to $\QQ$ all $p$-power roots of unity. By $\kappa$ we denote the cyclotomic character $\Ginf\rTo \ZZ_p^\times$.
360 Because the cyclotomic character is an isomorphism,
361 choosing a topological generator $\gamma$ in $\Gamma = \Ginf^{4(p-1)}$ amounts to picking
362 an element $\kappa(\gamma)$ in $1+2p\ZZ_p^\times$.
363 With this choice, we may convert the function $L_{\alpha}(E,s)$ into a $p$-adic power series in $T = \kappa(\gamma)^{s-1}-1$. We write $\LL_{\alpha}(E,T)$ for this series in $\QQ_p(\alpha)[\![T]\!]$. We have
364 \begin{equation}\label{eqn:Lpser}
365  \LL_{\alpha}(E,T) = \int_{\ZZ_p^\times} (1+T)^{\frac{\log(x)}{\log(\kappa(\gamma))}} d\mu_\alpha(x)\,.
366 \end{equation}
367 As in~\cite{pollack}, we define
368 the polynomial\william{$\log(a)$ below doesn't make
369 sense without further explanation,
370 since $a\in (\ZZ/p^k\ZZ)^*$.  Likewise for the substitution
371 $a=\omega(b)\cdot \kappa(\gamma)^j$ below.  Presumably we are making a fixed choice of lifts to $\ZZ_p^*$?}
372 \begin{align*}
373  P_n &= \sum_{a\in(\ZZ/p^k\ZZ)^{\times}}
374  \left[\frac{a}{p^k}\right]^{+} \cdot (1+T)^{ \frac{\log(a)}{\log(\kappa(\gamma))}} \\
375   & = \sum_{j=0}^{p^{k-1}-1}\, \sum_{b=1}^{p-1}\, \left[\frac{\omega(b)\cdot \kappa(\gamma)^j}{p^k}\right]^{+} \cdot (1+T)^j,
376 \end{align*}
377 where we changed the summation by putting $a = \omega(b) \cdot \kappa(\gamma)^j$.
378 Then the approximation as a Riemann sum of the above integral for $\LL_{\alpha}(E,T)$ can be written as
379 \begin{equation*}
380  \LL_{\alpha}(E,T)  = \lim_{k\to\infty} \left( \frac{1}{\alpha^k} \cdot P_{k} - \frac{1}{\alpha^{k+1}}\cdot P_{k-1} \right)\,.
381 \end{equation*}
382
383 \subsection{The $p$-adic multiplier}
384 For a prime of good reduction, we define the $p$-adic multiplier by
385 \begin{equation}\label{epsp1}
386  \epsilon_p = \left(1-\tfrac{1}{\alpha}\right)^2 \,.
387 \end{equation}
388 For a prime of bad multiplicative reduction, we put
389 \begin{equation*}
390  \epsilon_p = \left(1-\tfrac{1}{\alpha}\right) =\begin{cases} 0\quad &\text{if $p$ is split multiplicative and } \\
391  2 &\text{ if $p$ is nonsplit.}
392   \end{cases}
393 \end{equation*}
394
395 \subsection{Interpolation property}
396 The $p$-adic $L$-function constructed above satisfies a desired interpolation property with respect to the complex $L$-function. For instance, we have that
397 \begin{equation*}
398  \LL_{\alpha}(E,0) = L_{\alpha}(E,1) = \int_{\ZZ_p^\times} d\mu_{\alpha} = \epsilon_p \cdot\frac{L(E,1)}{\OmegaE}\,.
399 \end{equation*}
400 A similar formula holds when integrating nontrivial characters of $\ZZ_p^\times$ against $\mu_\alpha$. If $\chi$ is the character on $\Ginf$ sending $\gamma$ to a root of unity $\zeta$ of exact order $p^n$, then
401 \begin{equation*}
402  \LL_{\alpha}(E,\zeta) = \frac{1}{\alpha^{n+1}}\cdot \frac{p^{n+1}}{G(\chi^{-1})}\cdot \frac{L(E,\chi^{-1},1)}{\OmegaE}\,.
403 \end{equation*}
404 Here $G(\chi^{-1})$ is the Gauss sum and $L(E,\chi^{-1},1)$ is the Hasse-Weil $L$-function of $E$ twisted by $\chi^{-1}$.
405
406 \subsection{The good ordinary case}
407 Suppose now that the reduction of the elliptic curve at the prime $p$ is good and ordinary, so
408 $a_p$ is not divisible by $p$.
409 As mentioned before, in this case there is
410 a unique choice of root
411 $\alpha$ of the characteristic polynomial
412 $x^2 - a_p x + p$ that satisfies $\ord_p(\alpha) < 1$.
413 Since $\alpha$ is an algebraic integer, this implies
414 that $\ord_p(\alpha)=0$, so $\alpha$ is a unit
415 in $\ZZ_p$. We get therefore a unique $p$-adic $L$-function that we will denote simply by $\LL_p(E,T) = \LL_{\alpha}(E,T)$. It is proved in~\cite{wuthkato} that
416 \begin{prop}
417  Let $E$ be an elliptic curve with good ordinary reduction
418  at a prime $p > 2$. Then the series $\LL_p(E,T)$ belongs to $\ZZ_p[\![T]\!]$.
419 \end{prop}
420 Note that $\ord_p(\epsilon_p)$ is equal to $-2\,\ord_p(N_p)$ where $N_p=p+1-a_p$ is the number of points in the reduction $\tilde E(\FF_p)$ at $p$. % even when p=2 !!
421
422\subsection{Multiplicative case}
423We have to seperate the case of split from the case of nonsplit multiplicative reduction. In fact if the reduction is nonsplit, then the description of the good ordinary case applies just the same. But if the reduction is split multiplicative (the exceptional case'' in~\cite{mtt}), then the $p$-adic $L$-series must have a trivial zero, i.e., $\LL_p(E,0) = 0$ because $\epsilon_p =0$. By a result of Greenberg and Stevens~\cite{grste} (see also~\cite{koblp} for a simple proof), we know that
424\begin{equation*}
425  \left.\frac{d\, \LL_p(E,T)}{d\,T}\right\vert_{T=0}  = \frac{1}{\log_p\kappa(\gamma)}\cdot \frac{\log_p(\qE)}{\ord_p(\qE)} \cdot \frac{L(E,1)}{\OmegaE}
426\end{equation*}
427where $\qE$ denotes the Tate period of $E$ over $\QQ_p$.
428This will replace the interpolation formula.
429Note that it is now known thanks to~\cite{steph} that $\log_p(\qE)$ is nonzero. Hence we define the $p$-adic $\Linv$-invariant as
430\begin{equation}\label{epsp2}
431  \Linv_p = \frac{\log_p(\qE)}{\ord_p(\qE)} \neq 0\,.
432\end{equation}
433We refer to~\cite{colmezlinvariant} for a detailed discussion of the different $\Linv$-invariants and their connections.
434
435 \subsection{The supersingular case}
436 In the supersingular case, that is when $a_p\equiv 0\pmod{p}$, we have two roots $\alpha$ and $\beta$ both of valuation $\tfrac{1}{2}$. A careful analysis of the functions $\LL_{\alpha}$ and $\LL_{\beta}$ can be found in~\cite{pollack}. The series $\LL_{\alpha}(E,T)$ will not have integral coefficients in $\QQ_p(\alpha)$. Nevertheless one can still extract two integral series $\LL_p^{\pm}(E,T)$. We will not need this description.
437
438 There is a way of rewriting the $p$-adic $L$-series which relates more easily to the $p$-adic height defined in the next section. We follow Perrin-Riou's description in~\cite{pr00}.
439
440 As before $\omegaE$ denotes the chosen invariant differential on $E$. Let $\etaE=x\cdot \omegaE$. The pair  $\{\omegaE,\etaE\}$ forms a basis of the Dieudonn\'e module $D_p(E) = \QQ_p\otimes\HH^1_{\text{dR}}(E/\QQ)$. This $\QQ_p$-vector space comes equipped with a (geometric) Frobenius $\varphi$ acting on it linearly. Its characteristic polynomial is equal to $X^2 - p^{-1}\,a_p \, X + p^{-1}$.
441
442 Write $\LL_{\alpha}(E,T)$ as $G(T) + \alpha \cdot H(T)$ with $G(T)$ and $H(T)$ in $\QQ_p[\![T]\!]$. Then we define
443 \begin{equation*}
444  \LL_p(T) = G(T)\cdot \omegaE + a_p \cdot H(T)\cdot \omegaE - p\cdot H(T)\cdot \varphi(\omegaE)\,.
445 \end{equation*}
446 This is a formal power series with coefficients in $D_p(E)\otimes \QQ_p[\![T]\!]$ which contains exactly the same information as $\LL_{\alpha}(E,T)$. See~\cite{pr00} for a direct definition. The $D_p$-valued $L$-series satisfies again certain interpolation properties,\footnote{%
447 Perrin-Riou writes in~\cite{pr00} the multiplier as $(1-\varphi)^{-1}\cdot (1-p^{-1}\varphi^{-1})$ and she multiplies the right hand side with $L(E/\QQ_p,1)^{-1}=N_p\cdot p^{-1}$. It is easy to see that $(1-\varphi)\cdot (1-p^{-1}\varphi^{-1}) = 1 -\varphi - (\varphi - a_p \cdot p^{-1}) + p^{-1} = N_p\cdot p^{-1}$.
448  } e.g.
449 \begin{equation*}
450  (1-\varphi)^{-2} \, \LL_p(0) = \frac{L(E,1)}{\OmegaE}\cdot \omegaE \quad\in D_p(E)\,.
451 \end{equation*}
452
453
455 The case of additive reduction is much harder to treat. We have not tried to include the possibility of additive reduction in our algorithm. Note that there are two interesting paper of Delbourgo~\cite{delbourgo98} and~\cite{delbourgo02} on this subject. We will not refer to this case anymore throughout the paper.
456
457%% -------------------------------------------------------------------------
458\section{$p$-adic heights}\label{hp_sec}
459
460 The second term to be generalised in the Birch-Swinnerton-Dyer formula is the real valued regulator. In $p$-adic analogues of the conjecture it is replaced
461 by a $p$-adic regulator, which is defined using a $p$-adic analogue of the
462 height pairing. We follow here the generalised version~\cite{prbe}, \cite{pr00},
463 and \cite{mst}.
464
465 Let $\nu$ be an element of the Dieudonn\'e module $D_p(E)$. We will define a $p$-adic height function $h_\nu\colon E(\QQ)\rTo \QQ_p$ which depends linearly on the vector $\nu$. Hence it is sufficient to define it on the basis $\omega=\omegaE$ and $\eta=\etaE$.
466
467 If $\nu=\omega$, then we define
468 \begin{equation*}
469  h_\omega(P)=-\log(P)^2
470 \end{equation*}
471  where $\log$ is the linear extension of the $p$-adic elliptic logarithm $\log_{\hat E}\colon \hat E(p\ZZ_p)\rTo p\ZZ_p$ defined on the formal group $\hat E$.
472
473 For $\nu=\eta$, we define first the $p$-adic sigma function of Bernardi $\sigma(z)$ as in~\cite{bernardi}. Denote by $t=-\tfrac{x}{y}$ the uniformizer at $\ZeroE$ and write $z(t) = \log_{\hat E}(t)$. Define the Weierstrass $\wp$-function as usual by
474 \begin{equation*}
475  \wp(t) = x(t)+\frac{a_1^2+4\,a_2}{12} \in\QQ((t))
476 \end{equation*}
477 Here $a_1$ and $a_2$ are the coefficients of the minimal Weierstrass equation~\eqref{w_eq} of $E$. The function $\wp(t)$ is a solution to the usual differential equation. We define the sigma-function of Bernardi to be a solution of the equation
478 \begin{equation*}
479   - \wp(t)  = \frac{d}{\omegaE}\left(\frac{1}{\sigma}\cdot\frac{d\sigma}{\omegaE}\right)
480 \end{equation*}
481 such that $\sigma(0)=0$ and $\sigma(t(-P))=-\sigma(t(P))$.
482 This provides us with a series
483 \begin{equation*}
484  \sigma(t) = t + \frac{a_1}{2}\,t^2 + \frac{a_1^2+a_2}{3}\,t^3+\frac{a_1^3+2a_1a_2+3a_3}{4}\,t^4+\cdots  \in \QQ(\!(t)\!)\,.
485 \end{equation*}
486 As a function on the formal group $\hat E(p\ZZ_p)$ it converges for $\ord_p(t) > \tfrac{1}{p-1}$.
487
488  Given a point $P$ in $E(\QQ)$ there exists a multiple $m\cdot P$ such that $\sigma(t(P))$ converges and such that $m\cdot P$ has good reduction at all primes. Denote by $e(m\cdot P)\in\ZZ$ the square root of the denominator of the $x$-coordinate of $m\cdot P$.  Now define
489  \begin{equation*}
490   h_{\eta}(P) = \frac{2}{m^2} \cdot \log_p\left (\frac{\sigma(t(m\cdot P))}{e(m\cdot P)}\right )
491  \end{equation*}
492%\manque{factors correct } YES.
493 It is proved in~\cite{bernardi} that this function is quadratic and satisfies the parallelogram law.
494
495  Finally, if $\nu= a\, \omega+b\,\eta$ then put
496  \begin{equation*}
497   h_\nu(P) = a \, h_{\omega}(P) + b\, h_{\eta}(P)\,.
498  \end{equation*}
499  This quadratic function induces a bilinear symmetric pairing $\langle\cdot,\cdot\rangle_{\nu}$ with values in $\QQ_p$.
500
501  \subsection{The good ordinary case}
502  Since we have only a single $p$-adic $L$-function in the case that the reduction is good ordinary, we have also to pin down a canonical choice of a $p$-adic height function. This was first done by Schneider~\cite{schneider1} and Perrin-Riou~\cite{pr82}. We refer to~\cite{mt} and~\cite{mst} for more details.
503
504  Let $\nu_{\alpha}= a \, \omega + b\,\eta$ be an eigenvector of $\varphi$ on $D_p(E)$ associated to the eigenvalue $\tfrac{1}{\alpha}$. The value $e_2 =\mathbf{E}_2(E,\omegaE) = -12\cdot \tfrac{a}{b}$ is the value of the Katz $p$-adic Eisenstein series of weight $2$ at $(E,\omegaE)$.
505  Then, if $P$ has good reduction at all primes and lies in the range of convergence of $\sigma(t)$, we define the canonical $p$-adic height of $P$ to be
506  \begin{align}
507   \hat h_p (P) &= \frac{1}{b}\cdot h_{\nu_{\alpha}}(P) \notag\\
508   				&= -\frac{a}{b} \cdot z(P)^2 +2\, \log\left (\frac{\sigma(t(P))}{e( P)}\right ) \notag\\
509				&= 2\,\log_p \left ( \frac{\exp(\frac{e_2}{24} \log(P)^2)\cdot \sigma(t(P))}{e(P)} \right) = 2\, \log_p \left ( \frac{\sigma_p(t(P))}{e(P)} \right) \label{hpeq}
510  \end{align}
511  The function $\sigma_p(t)$, defined by the last line, is called the canonical sigma-function, see~\cite{mt}, it is known to lie in $\ZZ_p[\![t]\!]$.
512  The $p$-adic height defined here is up to the factor $2$ the same as in~\cite{mst}.\footnote{This factor is needed if one does not want to modify the $p$-adic version of the Birch and Swinnerton-Dyer conjecture~\ref{pbsd_ord_con}.}
513
514  We write $\langle \cdot,\cdot\rangle_p$ for the canonical $p$-adic height pairing on $E(\QQ)$ associated to $\hat h_p$ and $\Reg_p(E/\QQ)$ for its determinant.
515
516  \begin{conjecture}{Schneider~\cite{schneider1}}\label{conreg_con}
517    The canonical $p$-adic height is nondegenerate on the free part of $E(\QQ)$. In other words, the canonical $p$-adic regulator $\Reg_p(E/\QQ)$ is nonzero.
518  \end{conjecture}
519
520  Apart from the special case treated in~\cite{bertrand} of curves with complex multiplication of rank $1$, there are hardly any results on this conjecture. See also~\cite{wuth04}.
521
522  \subsection{The multiplicative case}
523  In the case of multiplicative reduction, one may use Tate's $p$-adic uniformization (see~\cite{sil2}). We have an explicit description of the height pairing in~\cite{schneider1}. If one wants to have the same closed formula in the $p$-adic version of the Birch and Swinnerton-Dyer conjecture for multiplicative primes as for other ordinary primes, the $p$-adic height has to be changed slightly. We use here the description of the $p$-adic regulator given in section II.6 of~\cite{mtt}. Alas, their formula is not correct as explained by Werner in~\cite{werner}.
524
525  Let $\qE$ be the Tate parameter of the elliptic curve over $\QQ_p$, i.e., we
526  have a homomorphism $\psi\colon \bar\QQ_p^\times \rTo E(\bar\QQ_p)$ whose kernel is
527  precisely $\qE^\ZZ$. The image of $\ZZ_p^\times$ under $\psi$ is equal to the subgroup of points of $E(\QQ_p)$ lying on the connected component of the N\'eron model of $E$. Now let $C$ be the constant such that $\psi^*(\omegaE) = C \cdot \frac{du}{u}$ where $u$ is a uniformiszr of $\QQ_p^\times$ at $1$. The value of the weight~2 $p$-adic Eisenstein series can then be computed as
528  \begin{equation*}
529   e_2 =\mathbf{E}_2(E,\omegaE) = C^2 \cdot\left ( 1- 24 \cdot \sum_{n\geq 1 } \sum_{d\mid n} d. \cdot q^n \right )
530  \end{equation*}
531  Then we use the formula of the good ordinary case to define the canonical $\sigma$
532  function $\sigma_p(t(P)) = \exp(\frac{e_2}{24} z(P)^2)\cdot \sigma(t(P))$.
533  If the reduction is nonsplit multiplicative, then we
534  use the formula~\eqref{hpeq} for the good ordinary case.
535
536  Suppose now that the reduction is split multiplicative.
537 Let $P$ be a point in $E(\QQ)$ having good reduction at all finite places and with trivial reduction at $p$. Then
538 \begin{equation*}
539  \hat h_p(P) = 2 \log_p\left ( \frac{\sigma_p(t(P))}{e(P)} \right) + \frac{\log_p(u(P))^2}{\log(\qE)}
540 \end{equation*}
541where $u(P)$ is the unique element of $\ZZ_p^\times$ mapping to $P$ under the
542Tate parametrisation~$\psi$.
543 The $p$-adic regulator is formed as before but with this modified $p$-adic height~$\hat h_p$.
544
545  \subsection{The supersingular case}
546  In the supersingular case, we cannot find a canonical $p$-adic height with values in $\QQ_p$. Instead, the height will have values  in the Dieudonn\'e module $D_p(E)$.
547  The main references for this height are~\cite{prbe} and~\cite{pr00}.
548
549  Suppose that $\nu = a\,\omega + b\,\eta$ is any element of $D_p(E)$ not lying in $\QQ_p\,\omegaE$.
550 It can be easily checked that the value of
551 \begin{equation*}
552  H_p(P) = \frac{1}{b} \cdot ( h_{\nu}(P) \cdot\omega - h_{\omega}(P)\cdot \nu )\quad\in D_p
553 \end{equation*}
554  is independent of the choice of $\nu$. We will call this
555  the $D_p$-valued height on $E(\QQ)$.
556
557  On $D_p(E)$ there is a alternating bilinear form $[\cdot,\cdot]$ characterised by the property that $[\omegaE,\etaE]=1$. Write $\Reg_{\nu}\in\QQ_p$ for the regulator
558  of $h_{\nu}$ on a $\ZZ$-basis of the free part of $E(\QQ)$ with respect
559  to some decomposition $E(\QQ) = F \oplus E(\QQ)_{\tor}$ (since the height
560  is $0$ on torsion, the choice of decomposition does not matter). Then
561  \begin{equation*}
562   \Reg_p(E/\QQ) = \frac{\Reg_{\nu}\cdot \nu'-\Reg_{\nu'}\cdot \nu}{[\nu',\nu]}\quad\in D_p(E)
563  \end{equation*}
564  is independent of the choice of $\nu$ and $\nu'$ in $D_p(E)$, as long as they do not belong to $\QQ_p\,\omegaE$. We call this the $D_p$-valued regulator of $E/\QQ$.
565
566  It is not difficult to see that $\Reg_p(E/\QQ) = H_p(P)$ if the curve is of rank $1$ with generator $P$. If $E(\QQ)$ is finite, then $\Reg_p(E/\QQ)$ is simply $\omegaE$. In both these cases the $D_p$-valued regulator can not vanish.
567
568  If one restricts any $p$-adic height $h_{\nu}$ to the fine Mordell-Weil group defined  in~\cite{wuthfine} to be the kernel
569  \begin{equation*}
570   \mathfrak{M}(E/\QQ) = \ker\left(E(\QQ)\otimes \ZZ_p\rTo \widehat{E(\QQ_p)} \right),
571  \end{equation*}
572  where $\widehat{E(\QQ_p)}$ is the $p$-adic completion of $E(\QQ_p)$.
573The restricted height is then
574independent of the chosen element $\nu$ in $D_p(E)$. We call its regulator the fine
575regulator, which is an element of $\QQ_p$ defined up to multiplication by a
576unit in $\ZZ_p$.
577
578  In general, the $D_p$-valued regulator is 0
579  if and only if the fine regulator vanishes.
580
581  \begin{conjecture}{Perrin-Riou~\cite[Conjecture 3.3.7.i]{prfourier93}}\label{conreg_ss_con}
582   The fine regulator of $E/\QQ$ is nonzero for all primes $p$. In particular, $\Reg_p(E/\QQ)\neq 0$ for all primes where $E$ has supersingular reduction.
583  \end{conjecture}
584
585
586  \subsection{Normalisation}
587  In view of Iwasawa theory, it is actually natural to normalise the heights and the regulators depending on the choice of the generator $\gamma$. In this way the heights depend on the choice of an isomorphism $\Gamma\rTo\ZZ_p$ rather than on the $\ZZ_p$-extension only.
588This normalization
589can be achieved by simply dividing $\hat h_p(P)$ and $h_{\nu}(P)$ by $\kappa(\gamma)$. The regulators will be divided by $\kappa(\gamma)^r$ where $r$ is the rank of $E(\QQ)$.
590  Hence we write
591  \begin{equation*}
592   \Reg_{\gamma}(E/\QQ) = \frac{\Reg_p(E/\QQ)}{\kappa(\gamma)^r}
593  \end{equation*}
594
595%% -------------------------------------------------------------------------
596\section{The $p$-adic Birch and Swinnerton-Dyer conjecture}\label{pbsd_sec}
597
598 \subsection{The ordinary case}
599 The following conjecture is due to Mazur, Tate and Teitelbaum~\cite{mtt}. Rather than formulating it for the function $L_{\alpha}(E,s)$, we state it directly for the series $\LL_p(E,T)$. It is then a statement about the development of this function at $T=0$ rather than at $s=1$.
600
601 \begin{conjecture}{Mazur, Tate and Teitelbaum~\cite{mtt}}\label{pbsd_ord_con}
602  Let $E$ be an elliptic curve with good ordinary reduction or with  multiplicative reduction at a prime $p$.
603  \begin{itemize}
604  \item The order of vanishing of the $p$-adic $L$-function $\LL_p(E,T)$ at $T=0$ is equal to the rank $r$, unless $E$ has split multiplicative reduction at $p$ in which case the order of vanishing is equal to $r+1$.
605  \item The leading term $\LL_p^{\ast}(E,0)$ satisfies
606  \begin{equation}\label{pbsd_ord_eq}
607 \LL_p^{\ast}(E,0) =  \epsilon_p\cdot \frac{\prod_\vu c_\vu\cdot \#\Sha(E/\QQ)}{(\#E(\QQ)_{\text{tors}})^2}\cdot\Reg_{\gamma}(E/\QQ)
608  \end{equation}
609  unless the reduciton is split multiplicative in which case the leading term is
610  \begin{equation}
611 \LL_p^{\ast}(E,0) =  \Linv_p\cdot \frac{\prod_\vu c_\vu\cdot \#\Sha(E/\QQ)}{(\#E(\QQ)_{\text{tors}})^2}\cdot\Reg_{\gamma}(E/\QQ).
612  \end{equation}
613  \end{itemize}
614  \william{Are the conjectures only up to a $p$-adic unit or are the {\em conjectures}
615  really on the nose supposed to be true?}
616 \end{conjecture}
617
618\subsection{The supersingular case}
619 The conjecture in the case of supersingular reduction is due to Bernardi and Perrin-Riou in~\cite{prbe} and~\cite{pr00}. The conjecture relates here an algebraic and an analytic value
620 in the $\QQ_p$-vector space $D_p(E)$ of dimension 2. The fact of having two
621 coordinates was used cleverly by Kurihara and Pollack in~\cite{kuriharapollack} to construct global points via a $p$-adic analytic computation.
622
623 We say that an
624 element $a(T)\cdot\omegaE + b(T)\cdot\etaE$ in $D_p(E)\otimes \QQ_p[\![T]\!]$
625 has order $d$ at $T=0$ if $d$ is equal to the minimum of the orders of $a(T)$ and $b(T)$.
626
627 \begin{conjecture}{Bernardi and Perrin-Riou~\cite{prbe}}\label{pbsd_ss_con}
628  Let $E$ be an elliptic curve with good supersingular reduction at a prime $p$.
629  \begin{itemize}
630  \item The order of vanishing of the $D_p$-valued $L$-function $\LL_p(E,T)$ at $T=0$ is equal to the rank $r$.
631  \item The leading term $\LL_p^{\ast}(E,0)$ satisfies
632  \begin{equation}\label{pbsd_ss_eq}
633   \left (1-\varphi\right)^{-2}\cdot\LL_p^{\ast}(E,0)  =  \frac{\prod_\vu c_\vu\cdot\#\Sha(E/\QQ)}{(\#E(\QQ)_{\text{tors}})^2}\cdot \Reg_{\gamma}(E/\QQ)\quad \in D_p(E)
634  \end{equation}
635  \end{itemize}
636 \end{conjecture}
637
638%% -------------------------------------------------------------------------
639\section{Iwasawa theory of elliptic curves}\label{iwasawa_sec}
640  We suppose from now on that $p>2$.
641  Let $\QQinf$ be the Galois extension of $\QQ$ whose Galois group is $\Gamma$. It is the unique $\ZZ_p$-extension of $\QQ_p$. Let $\Lambda$ be the completed group algebra $\ZZ_p[\![\Gamma]\!]$.
642  We use the fixed topological generator $\gamma$ of $\Gamma$ to identify $\Lambda$ with $\ZZ_p[\![T]\!]$ by sending $\gamma$ to $1+T$.
643  It is well-known that any finitely generated $\Lambda$-module admits
644  a decomposition as a direct sum of
645  elementary $\Lambda$-modules. Denote by $\QQn$ the
646  $n$\textsuperscript{th} layer of the $\ZZ_p$-extension. As before, we may define the $p$-Selmer group over $\QQn$ by
647 the exact sequence
648 \begin{equation*}
649  0\rTo \Sel_p(E/\QQn)\rTo \HH^1(\QQn,E(p))\rTo \prod_\vu \HH^1(\QQn_\vu,E)
650 \end{equation*}
651 \william{Would you be opposed to using the notation $\Sel_p(\QQn, E)$?  It's clearer
652 and easier to read in this case.}
653 with the product running over all places $\vu$ of $\QQn$. Moreover, we define $\Sel_p(E/\QQinf)$ to be the limit $\liminj \Sel_p(E/\QQn)$ following the maps induced by the restriction maps $\HH^1(\QQn,E(p))\rTo \HH^1(\QQnplusone,E(p))$. The group $\Sel_p(E/\QQinf)$ contains essentially the information about the growth of the rank of $E(\QQn)$ and of the size of $\Sha(E/\QQn)(p)$ as $n$ tends to infinity. We will consider the Pontryagin dual
654 \begin{equation*}
655  X(E/\QQinf) = \Hom\left(\Sel_p(E/\QQinf),\QZ\right)
656 \end{equation*}
657 which is a finitely generated $\Lambda$-module (see~\cite{coatessujatha}).
658
659 \subsection{The ordinary case}
660 Assume now that the reduction at $p$ is good and ordinary or of multiplicative type. It was shown by Kato in~\cite{kato} that $X(E/\QQinf)$ is a torsion $\Lambda$-module. Hence by the decomposition theorem, we may associated to it a characteristic series $\fE(T)$ in $\Lambda$. The
661 series
662 \begin{equation}\label{eqn:fE}
663 \fE(T)\in\ZZ_p[\![T]\!]
664 \end{equation}
665 is well-defined up to multiplication by a unit in $\Lambda^{\!\times}$.
666
667 In analogy to the zeta-function of a variety over a finite field, one
668 should think of $\fE(T)$ as a generating function encoding the growth
669 of the rank and the Tate-Shafarevich group. For instance, the zeros of $\fE(T)$ at roots of unity whose orders are powers of $p$ describe the growth
670 of the rank. Since a nonzero power series with coefficients in
671 $\ZZ_p$ can only have finitely many zeros, one can
672 show that the rank of $E(\QQn)$ has to stabilize in
673 the tower $\QQ_n$. In other words, the Mordell-Weil
674 group  $E(\QQinf)$ is still of finite rank.
675 \william{Shouldn't we cite Lichtenbaum here too?}
676
677The following relatively old result is due to
678Schneider~\cite{schneider2} and
679Perrin-Riou~\cite{pr82}. The multiplicative case is due to Jones~\cite{jones89}.
680\begin{thm}[Schneider, Perrin-Riou, Jones]\label{perrinriouschneider_thm}
681 The order of vanishing of $\fE(T)$ at $T=0$ is at least equal to the rank $r$.
682 It is equal to $r$ if and only if the $p$-adic height pairing is nondegenerate (conjecture~\ref{conreg_con}) and the $p$-primary part of the Tate-Shafarevich group $\Sha(E/\QQ)(p)$ is finite (conjecture~\ref{consha_con}). In this case the leading term of the series $\fE(T)$ has the same valuation as
683 \begin{equation*}
684  \epsilon_p\cdot\frac{ \prod_\vu c_\vu\cdot \#\Sha(E/\QQ)(p)}
685  {(\#E(\QQ)(p))^2}\cdot\Reg_{\gamma}(E/\QQ)
686 \end{equation*}
687 unless the reduction is split multiplicative in which case the same formula holds with $\epsilon_p$ replaced by $\Linv_p$.
688\end{thm}
689
690
691 \subsection{The supersingular case}
692
693 The supersingular case is much more complicated, since the $\Lambda$-module $X(E/\QQinf)$ is not torsion. A very beautiful approach to the supersingular case has been found by Pollack~\cite{pollack} and Kobayashi~\cite{kobayashi}. As mentioned above there exists two $p$-adic series
694 $\LL_p^{\pm}(E,T)$ to which will correspond
695 two new Selmer groups $X^{\pm}(E/\QQinf)$ which now are $\Lambda$-torsion. Despite the advantages of this $\pm$-theory, we are using the approach of Perrin-Riou here. See section~3 in~\cite{pr00}.
696
697 Let $\Tp E$ be the Tate module and define $\Hinfloc$ to be the projective limit of the cohomology groups $\HH^1(\QQn_{\mf p},\Tp E)$ following the corestriction maps. Here $\QQn_{\mf p}$ is the localisation of $\QQn$ at the unique prime $\mf p$ above $p$. Perrin-Riou~\cite{prcol} has constructed a $\Lambda$-linear Coleman
698 map $\Col$ from $\Hinfloc$ to a sub-module of $\QQ_p[\![T]\!]\otimes D_p(E)$.
699
700 Define the fine Selmer group to be the kernel
701 \begin{equation*}
702  \Rel(E/\QQn) = \ker\left ( \Sel(E/\QQn) \rTo E(\QQn_{\mf p})\otimes\QZ\right)\,.
703 \end{equation*}
704 It is again a consequence of the work of Kato \william{Give a reference.} that
705 the Pontryagin dual $Y(E/\QQinf)$ of $\Rel(E/\QQinf)$ is a $\Lambda$-torsion module. Denote by $g_E(T)$ its characteristic series.
706
707 Let $\Sigma$ be any finite set of places in $\QQ$ containing the places of bad reduction for $E$ and the places $\infty$ and $p$. By $G_{\Sigma}(\QQn)$, we denote the Galois group of the maximal extension of $\QQn$ unramified at all places which do not lie above $\Sigma$. Next we define $\Hinfglob$ as the projective limit of $\HH^1(G_{\Sigma}(\QQn),\Tp E)$. It is a $\Lambda$-module of rank $1$ and it is actually independent of the choice of $\Sigma$.
708
709 Choose now any element $\cinf$ in $\Hinfglob$ such that $Z_c =\Hinfglob/(\Lambda\cdot \cinf)$ is $\Lambda$-torsion. Typically the zeta element'' of Kato could be such a choice.\william{Huh?}
710 Write $h_c(T)$ for the characteristic series of $Z_c$. Then we define an algebraic equivalent of the $D_p(E)$-valued $L$-series by
711 \begin{equation*}
712  \fE(T) = g_E(T)\cdot \Col(\cinf)\cdot h_c(T)^{-1} \in \QQ_p[\![T]\!]\otimes D_p(E)
713 \end{equation*}
714 where by $\Col(\cinf)$ we mean the image of the localisation of $\cinf$ to $\Hinfloc$ under the Coleman map $\Col$. The resulting series $\fE(T)$ is
715independent of the choice of $\cinf$. Of course, $\fE(T)$ is again only defined up to multiplication by a unit in $\Lambda^{\!\times}$.
716
717 Again we have an Euler-characteristic result due to Perrin-Riou~\cite{prfourier93}:
718
719\begin{thm}[Perrin-Riou]\label{perrinriou_thm}
720 The order of vanishing of $\fE(T)$ at $T=0$ is at least equal to the rank $r$.
721 It is equal to $r$ if and only if the $D_p(E)$-valued regulator $\Reg_p(E/\QQ)$ is nonzero (conjecture~\ref{conreg_ss_con}) and the $p$-primary part of the Tate-Shafarevich group $\Sha(E/\QQ)(p)$ is finite (conjecture~\ref{consha_con}). In this case the leading term of the series $(1-\varphi)^{-2}\,\fE(T)$ has the same valuation as
722 \begin{equation*}
723 \prod_\vu c_\vu\cdot \#\Sha(E/\QQ)(p)\cdot  \Reg_{\gamma}(E/\QQ)
724 \end{equation*}
725\end{thm}
726
727 Note that we simplified the right hand term in comparison to~\eqref{pbsd_ss_eq}, because $N_p\equiv 1 \pmod{p}$ and hence $\#E(\QQ)_{\text{tors}}$ must be $p$-adic unit
728 if the reduction at~$p$ is supersingular.
729
730%% -------------------------------------------------------------------------
731\section{The Main Conjecture}\label{mainconjecture_sec}
732
733The main conjecture links the two $p$-adic power series (\ref{eqn:Lpser}) and (\ref{eqn:fE})
734of the previous sections. We formulate everything now simultaneously for the
735ordinary and the supersingular case, even if they are of quite different nature.
736We still assume that $p\neq 2$.
737
738\begin{conjecture}{Main conjecture of Iwasawa theory for elliptic curves}\label{mainconjecture_con}
739 If $E$ has good or nonsplit multiplicative reduction at $p$, then
740 there exists an element $u(T)$ in $\Lambda^{\!\times}$ such that $\LL_p(E,T) = \fE(T)\cdot u(T)$. If the reduction of $E$ at $p$ is split multiplicative, then there exists such a $u(T)$ in $T\cdot \Lambda^{\!\times}$.
741\end{conjecture}
742
744To the elliptic curve $E$ we attach the mod-$p$ representation
745\begin{equation*}
746 \bar\rho_p\colon \Gal(\bar \QQ/\QQ)\rTo \Aut(E[p])\cong \Gl_2(\FF_p)
747\end{equation*}
748of the absolute Galois group of $\QQ$.
749Serre proved that $\bar\rho_p$ is almost always
750surjective (note that by hypothesis $E$ does not have complex multiplication)
751and that for semistable curves surjectivity can only fail when there
752is an isogeny of degree $p$ defined over $\QQ$. See~\cite{serregl2} and~\cite{serrewiles}.
753
754\begin{thmkato}\label{katodiv_thm}
755 Suppose that $E$ has semistable reduction at $p$ and that  $\bar\rho_p$ is either surjective or that its image is contained in a Borel subgroup. Then there exists a series $d(T)$ in $\Lambda$ such that $\LL_p(E,T) = \fE(T)\cdot d(T)$. If the reduction is split multiplicative then $T$ divides $d(T)$.
756\end{thmkato}
757
758The main ingredient for this theorem is in theorem 17.4 in~\cite{kato} for the good ordinary case when $\bar\rho_p$ is surjective, or in~\cite{wuthkato} when there is a $p$-isogeny. The exceptional case refers to~\cite{kkt} and~\cite{koblp}. The statement of the main conjecture for supersingular primes is known to be equivalent to Kato's formulation in Conjecture~12.10 in~\cite{kato} and to Kobayashi's version in~\cite{kobayashi}.
759
760In particular the theorem applies to all odd primes $p$ if $E$ is a semistable curve.
761For the remaining cases, e.g., if the image of $\bar\rho_p$ is contained in the normalizer of a Cartan subgroup, one obtains only a weaker statement:
762\begin{thmkato}\label{ncartan_thm}
763 Suppose the image of $\bar\rho_p$ is not contained in a Borel subgroup of $\Gl_2(\FF_p)$ and that $\bar\rho_p$ is not surjective, then there is an integer $m\geq 0$ such that $\fE(T)$ divides $p^m\cdot\LL_p(E,T)$.
764\end{thmkato}
765
766Greenberg and Vatsal~\cite{grvat} have shown that in certain cases the main conjecture holds. There is hope that the main conjecture will be proved soon for primes $p$ subject to certain conditions. We are awaiting the forthcoming paper of Skinner and Urban.
767
768%% -------------------------------------------------------------------------
769\section{If the $L$-series does not vanish}\label{rank0_sec}
770Suppose the Hasse-Weil $L$-function $L(E,s)$ does not vanish at $s=1$. In this case
771Kolyvagin proved that $E(\QQ)$ and $\Sha(E/\QQ)$ are finite. In particular
772Conjecture~\ref{consha_con} is valid; also,  Conjectures~\ref{conreg_con}
773and~\ref{conreg_ss_con} are trivially true in this case.
774
775 Let $p>2$ be a prime of semistable reduction such that the representations $\bar\rho_p$ is either surjective or has its image contained in a Borel subgroup of $\Gl_2(\FF_p)$. By the interpolation property, we know that $\LL_p(E,0)$ is nonzero, unless~$E$
776 has split multiplicative reduction.
777
778 \subsection{The good ordinary case}
779 In the ordinary case  we have
780 \begin{equation*}
781  \epsilon_p^{-1}\cdot \LL_p(E,0) = \frac{L(E,1)}{ \OmegaE} = ^{+},
782 \end{equation*}
783 which is a nonzero rational number by~\cite{manin}.
784 Using the theorem\footnote{In the case of analytic
785 rank 0, the theorem is actually relatively easy and well explained in~\cite{coatessujatha}.}
786 of Perrin-Riou and Schneider~\ref{perrinriouschneider_thm} in the first line
787 and Kato's theorem~\ref{katodiv_thm} on the main conjecture in the
788 second line,\william{What does it mean in the first line''?  In the second line'' ??}
789 we find that
790 \begin{align*}
791  \ord_p \left (\epsilon_p\cdot\frac{ \prod_\vu c_\vu\cdot \#\Sha(E/\QQ)(p)}{(\#E(\QQ)(p))^2}\right) =&
792  	\ord_p(\fE(0)) \\
793	\leq& \ord_p(\LL_p(E,0)) \\
794	&= \ord_p \left (\frac{L(E,1)}{ \OmegaE} \right)
795	 + \ord_p(\epsilon_p)\,.
796 \end{align*}
797 Hence, we have the following
798 upper bound on the $p$-primary part of the Tate-Shafarevich group which is sharp under the assumption of the main conjecture:
799 \begin{equation}\label{sha_bound_r0_eq}
800  \ord_p \left( \Sha(E/\QQ)(p) \right) \leq \ord_p\left(\frac{L(E,1)}{\OmegaE}\right)-\ord_p\left(\frac{\prod c_\vu}{(\#E(\QQ)_{\text{tors}})^2}\right)\,.
801 \end{equation}
802 This bound agrees with the Birch and Swinnerton-Dyer conjecture.
803 \william{This is stronger than what I stated in my previous bsd computation
804 paper.}
805
806 \subsection{The multiplicative case}
807  If the reduction is not split, then the above holds just the same.\william{Why?}
808  If instead the reduction is split multiplicative, we have
809  that $\LL_p(E,0) =0$ and \begin{equation*}
810   \LL_p'(E,0)=\Linv_p\cdot\frac{L(E,1)}{ \OmegaE} =\Linv_p\cdot ^{+} \neq 0\,.
811  \end{equation*}
812 Since the $p$-adic multiplier is the same on the algebraic as on the analytic side, we can once again compute it as above to obtain the same bound~\eqref{sha_bound_r0_eq} again.
813
814 \subsection{The supersingular case}
815  For the supersingular $D_p(E)$-valued series, we have
816  \begin{equation*}
817   (1-\varphi)^{-2}\cdot\LL_p(E,0) = \frac{L(E,1)}{ \OmegaE} \cdot \omegaE= ^{+} \cdot \omegaE
818  \end{equation*}
819  which is a nonzero element of $D_p(E)$.
820  The $D_p(E)$-valued regulator $\Reg_p(E/\QQ)$ is equal to $\omegaE$. We may therefore concentrate solely on the coordinate in $\omegaE$. Write $\ord_p(\fE(0))$ for the $p$-adic valuation of the leading coefficient of the $\omegaE$-coordinate of $\fE(T)$.
821  Again we obtain an inequality by using theorem~\ref{perrinriou_thm}
822 \begin{align*}
823  \ord_p \left( \prod_\vu c_\vu\cdot \#\Sha(E/\QQ)(p) \right) =&
824  	\ord_p((1-\varphi)^{-2}\,\fE(0)) \\
825	\leq& \ord_p((1-\varphi)^{-2}\,\LL_p(E,0)) \\
826	&= \ord_p \left (\frac{L(E,1)}{ \OmegaE} \right)\,.
827	 \end{align*}
828
829 \subsection{Conclusion}
830 Summarising the above computations, we have
831 \begin{thm}
832  Let $E$ be an elliptic curve such that $L(E,1)\neq 0$. Then $\Sha(E/\QQ)$ is finite and
833  \begin{equation*}
834   \# \Sha(E/\QQ) \leq C\cdot\frac{L(E,1)}{\OmegaE}\cdot\frac{(\#E(\QQ)_{\tors})^2}{\prod c_\vu}
835  \end{equation*}
836  where $C$ is a product of a power of $2$ and of power of
837  primes of additive reduction  and of powers of
838  primes for which the representation $\bar\rho_p$ is not
839  surjective and there is no isogeny of degree $p$ on $E$ defined over $\QQ$.
840
841  In particular if $E$ is semistable, then $C$ is a power of $2$.
842 \end{thm}
843
844 This improves Corollary~3.5.19 in~\cite{eulersystems}.
845
846%% -------------------------------------------------------------------------
847 \section{If the $L$-series vanishes to the first order}\label{rank1_sec}
848
849 We suppose for this section that $E$ has good and ordinary reduction at $p$ and that the complex $L$-series $L(E,s)$ has a zero of order $1$ at $s=1$. The method of Heegner
850 points and the theorem of Kolyvagin show again that $\Sha(E/\QQ)$ is finite and that
851 the rank of $E(\QQ)$ is equal to $1$. Let $P$ be a choice of generator of the free
852 part of the Mordell-Weil group (modulo torsion).
853 Suppose that the $p$-adic height $\hat h_p(P)$ is nonzero.
854  Thanks to a theorem of Perrin-Riou in~\cite{prheegner},
855  we must have the following equality of rational numbers
856  \begin{equation*}
857   \frac{1}{\Reg(E/\QQ)}\cdot \frac{L'(E,1)}{\OmegaE} =\frac{1}{\Reg_p(E/\QQ)}\cdot \frac{\LL_p'(E,0)}{(1-\tfrac{1}{\alpha})^2\cdot \log(\kappa(\gamma))}
858  \end{equation*}
859  where, on the left hand side, we have the canonical real-valued regulator $\Reg(E/\QQ)=\hat h(P)$ and the leading coefficient of $L(E,s)$, while, on the right hand side, we have the $p$-adic regulator $\Reg_p(E/\QQ)=\hat h_p(P)$ and the leading term of the $p$-adic $L$-series. By the conjecture of Birch and Swinnerton-Dyer (or its $p$-adic analogue), this rational number should be equal to $\prod c_\vu\cdot \#\Sha(E/\QQ)\cdot (\#E(\QQ)_{\tors})^{-2}$. By Kato's theroem, one knows that the characteristic series $\fE(T)$ of the Selmer group divides $\LL_p(E,T)$; at least up to a power of $p$. Hence the series $\fE(T)$ has a zero of order $1$ at $T=0$ and its leading term divides the above rational number in $\QQ_p$ (here we use that $E(\QQ)$ has rank
860  $1$ so $T\mid f_E(T)$). Hence we have
861  \begin{thm}
862   	Let $E/\QQ$ be an elliptic curve with good ordinary reduction at the odd prime $p$.
863	Suppose that the representation $\bar\rho_p$ is surjective onto $\Gl_2(\FF_p)$ or that the curve admits an isogeny of degree $p$ defined over $\QQ$.
864	 If $L(E,s)$ has a simple zero at $s=1$, then the $p$-primary part of $\Sha(E/\QQ)$ is finite and its valuation is bounded by
865	\begin{equation*}
866	 \ord_p(\# \Sha(E/\QQ)(p) )\leq \ord_p\left( \frac{(\#E(\QQ)_{\tors})^2}{\prod c_\vu}\cdot \frac{1}{\Reg(E/\QQ)}\cdot \frac{L'(E,1)}{\OmegaE} \right)
867	\end{equation*}
868  \end{thm}
869  In other words the Birch and Swinnerton-Dyer conjecture if true up to a factor involving only bad and  supersingular primes, and primes for which the representation is neither surjective nor has its image contained in a Borel subgroup.
870
871%% -------------------------------------------------------------------------
872\section{The algorithm}\label{algorithm_sec}
873
874\subsection{The rank}
875Let $E/\QQ$ be an elliptic curve.
876Suppose we are in the situation that we have found $n$ linearly independent points. We wish to prove that $n$ is equal to the rank $r=\rk(E(\QQ))$.
877
878For this purpose, we choose a prime $p$ satisfying the following conditions
879\begin{itemize}
880  \item $p > 2$,
881  \item $E$ has good reduction at $p$.
882\end{itemize}
883By computing the analytic $p$-adic $L$-function $\LL_p(E,T)$ to a certain precision, we find an upper bound, say $b$, on the order of vanishing of $\LL_p(E,T)$ at $T=0$. Then
884\begin{equation*}
885 b\,\geq \,\ord_{T=0} \LL_p(E,1) \,\geq\, \ord_{T=0} \fE(T)\, \geq\, r
886\end{equation*}
887by Kato's theorem and by the theorems~\ref{perrinriouschneider_thm} and~\ref{perrinriou_thm}. Hence we have an upper bound on the rank $r$. In case $b$ is different from $n$, we can either increase the precision or we can change the prime $p$. Note that the $p$-adic Birch and Swinnerton-Dyer conjecture tells us exactly what the needed precision should be for being able to distinguish the leading coefficient from zero.
888
889\william{The procedure described in this section is {\em NOT} an algorithm.   It
890depends on the $p$-adic Birch and Swinnerton-Dyer conjecture tells us exactly what the needed precision should be'', but my understanding is that we do not know enough of that
891conjecture to read off this precision.  Thus given current theorems, we would never
892know when we're done.  So this section is not about an algorithm -- or it is about
893an algorithm that is conditional on knowing the $p$-adic BSD conjecture. Please clarify.}
894
895
896\subsection{The Tate-Shafarevich group}
897Suppose now that $E$ is an elliptic curve and $p$ is a prime satisfying the following conditions
898\begin{itemize}
899  \item $p > 2$,
900  \item $E$ has good reduction at $p$.
901  \item The image of $\bar\rho_p$ is either the full group $\Gl_2(\FF_p)$ or it is contained in a Borel subgroup.
902\end{itemize}
903Note that these conditions apply to all but finitely many primes $p$.
904
905Suppose further that the rank computation presented in the previous part of the algorithm was successful (for any prime not necessarily $p$). We may assume that we are able to compute a basis of the full Mordell-Weil group $E(\QQ)$ modulo torsion.
906
907Using the explicit basis of $E(\QQ)$ we can compute the $p$-adic regulator of $E$ over $\QQ$ using the efficient algorithm in~\cite{mst}.
908
909We compute the leading coefficient $\LL_p^{\ast}(E,0)$ of the analytic $p$-adic $L$-function.
910If the order of vanishing of $\LL_p(E,T)$ at $T=0$ is equal to $r$ then we know already that the $p$-primary part of the Tate-Shafarevich group is finite. Moreover, we get an upper bound.
911
912\subsection{The ordinary case}
913If $E$ has ordinary reduction at $p$, good or multiplicative, then
914\begin{align*}
915 \ord_p( \#\Sha(E/\QQ)(p) ) = &  \ord_p(\fE^{\ast}(0)) + \ord_p\left(\frac{(\#E(\QQ)(p))^2}{\epsilon_p\cdot \prod_\vu c_\vu}\cdot\frac{1}{\Reg_{\gamma}(E/\QQ)}\right)  \\
916  \leq& \ord_p(\LL_p^{\ast}(E,0)) + 2\cdot\ord_p(\#(E(\QQ)(p)) -  \ord_p (\epsilon_p)\\
917  &\ - \sum_\vu\ord_p(c_\vu)-\ord_p(\Reg_{\gamma}(E/\QQ))
918\end{align*}
919The inequality uses Kato's theorem~\ref{katodiv_thm}.
920
921 Note that if the main conjecture holds this inequality will be an equality. It should also be mentioned that Grigorov~\cite{grigorov} has found
922 a way to compute
923 lower bounds on the order of the Tate-Shafarevich group in certain cases.
924 One can also use congruences (i.e., visibility) to construct elements
925 (see \cite{papersonvisibility}).
926
927\subsection{The supersingular case}
928Suppose now that $E$ has supersingular reduction at $p$. Then we may use theorem~\ref{perrinriou_thm} and theorem~\ref{katodiv_thm} to obtain
929\begin{align*}
930 \ord_p( \#\Sha(E/\QQ)(p) ) = &  \ord_p(
931 (1-\varphi)^{-2}\,\fE^{\ast}(0)) - \ord_p(\Reg_p(E/\QQ)) - \ord_p(\prod_\vu c_\vu)  \\
932  \leq& \ord_p((1-\varphi)^{-2}\,\LL_p^{\ast}(E,0))  - \ord_p(\Reg_p(E/\QQ))  - \sum_{\vu}\ord_p(c_\vu)
933\end{align*}
934where the convention on $\ord_p(d(T))$ for an element $d(T)\in\QQ_p[\![T]\!]\otimes D_p(E)$ is as before.
935Again the inequality can be replaced by an equality if the main conjecture holds for $E$ at $p$.
936
937%% -------------------------------------------------------------------------
938\section{Technical details}\label{tech_sec}
939
940%% -------------------------------------------------------------------------
941\section{Numerical results}\label{numerical_sec}
942
943
944%% -------------------------------------------------------------------------
945
946\bibliographystyle{amsalpha}
947\bibliography{shark}
948
949\end{document}
950