\documentclass{article}1\usepackage[british]{babel}2\usepackage{amsmath,amssymb,amsthm,mathrsfs}3\usepackage[mathcal]{euscript}4\usepackage[a4paper,body={14cm,23cm}, includefoot]{geometry}5\usepackage[dvips]{color}6%\usepackage[tight,centredisplay,PostScript=dvips,nohug,TPIC,heads=curlyvee]{diagrams}78\newcommand{\william}[1]{\footnote{[[William: #1]]}\marginpar{\hfill {\sf[[\thefootnote]]}}}910\newcommand{\christian}[1]{\footnote{[[Christian: #1]]}\marginpar{\hfill {\sf[[\thefootnote]]}}}111213%-- Cyrillic14\input cyracc.def \font\tencyr=wncyr10 \def\russe{\tencyr\cyracc}15\def\Sha{\text{\russe{Sh}}}1617%-- nicer <=18\renewcommand{\geq}{\geqslant}19\renewcommand{\leq}{\leqslant}2021%-- Colours22\definecolor{dgreen}{rgb}{0,0.4,0.4}23\definecolor{dblue}{rgb}{0,0,0.7}24\definecolor{red}{rgb}{1,0,0}2526%-- Theorems27\newtheoremstyle{mythm}{11pt}{11pt}{\it\color{dblue}}{}{\bf\color{dblue}}{.}{\newline}{}28\theoremstyle{mythm}29\newtheorem{thm}{Theorem}30\newtheorem{thmkato}[thm]{Kato's Theorem}3132\newtheoremstyle{mypl}{11pt}{11pt}{\it\color{dblue}}{}{\bf\color{dblue}}{.}{ }{}33\theoremstyle{mypl}34\newtheorem{prop}[thm]{Proposition}35\newtheorem{lem}[thm]{Lemma}36\newtheorem{cor}{Corollary}3738\newtheoremstyle{mycon}{11pt}{11pt}{\it\color{dgreen}}{}{\bf\color{dgreen}}{.}{ }{}39\theoremstyle{mycon}40\newtheorem{con}{Conjecture}4142%named versions43\newenvironment{theorem}[1]{\begin{thm} {\bf (#1)}}{\end{thm}}4445\newenvironment{conjecture}[1]{\begin{con} {\bf (#1)}}{\end{con}}46474849%-- Operators50\DeclareMathOperator{\Aut}{Aut}51\DeclareMathOperator{\Hom}{Hom}52\DeclareMathOperator{\HH}{H}53\DeclareMathOperator{\Gl}{GL}54\DeclareMathOperator{\Gal}{Gal}55\DeclareMathOperator{\rk}{rank}56\DeclareMathOperator{\Reg}{Reg}57\DeclareMathOperator{\tors}{tors}58\DeclareMathOperator{\ord}{ord}59\DeclareMathOperator{\Col}{Col}6061%-- Abbrevations62\newcommand{\sss}{\scriptscriptstyle}63\newcommand{\liminj}{\varinjlim}64\newcommand{\limproj}{\varprojlim}65\newcommand{\vu}{\upsilon}66\newcommand{\mf}{\mathfrak}6768%-- Fields69\newcommand{\QQ}{\mathbb{Q}}70\newcommand{\RR}{\mathbb{R}}71\newcommand{\CC}{\mathbb{C}}72\newcommand{\ZZ}{\mathbb{Z}}73\newcommand{\FF}{\mathbb{F}}74%\newcommand{\QZ}{{{}^{\QQ_p}\!/\!{}_{\ZZ_p}}}75\newcommand{\QZ}{{\QQ_p/\ZZ_p}}767778%-- Iwasawa theory79\newcommand{\QQinf}{{}_\infty\QQ}80\newcommand{\Kinf}{{}_\infty K}81\newcommand{\QQn}{{}_n\QQ}82\newcommand{\QQnplusone}{{}_{n+1}\QQ}83\newcommand{\Hinf}{{}_{\infty\!} \HH}84\newcommand{\Hinfloc}{{}_{\infty\!} \HH_{\text{loc}}^1}85\newcommand{\Hinfglob}{{}_{\infty\!} \HH_{\text{glob}}^1}86\newcommand{\Ginf}{{}_{\infty\!} G}87\newcommand{\cinf}{{}_{\infty} c}8889\DeclareMathOperator{\an}{an}90\DeclareMathOperator{\tor}{tor}919293%-- Elliptic Curves94\newcommand{\Lstar}{L^{\ast}(E,1)}95\newcommand{\Sel}{\mathcal{Sel}}96\newcommand{\Rel}{\mathcal{R}}97\newcommand{\Tp}{T_{\! p}}98\newcommand{\Xinf}{X(E/\QQinf)}99100\newcommand{\LL}{\mathcal{L}}101\newcommand{\Linv}{\mathscr{L}}102103%-- Arrows104105\newcommand{\IncTo}{\hookrightarrow}106\newcommand{\rTo}{\longrightarrow}107108%-- small-er indices109110\newcommand{\OmegaE}{\Omega_{\sss E}}111\newcommand{\qE}{q_{\sss E}}112\newcommand{\ZeroE}{O_{\sss E}}113\newcommand{\fE}{f_{\sss E}}114\newcommand{\etaE}{\eta_{\sss E}}115\newcommand{\omegaE}{\omega_{\sss E}}116117118%--119120\newcommand{\manque}[1]{%121\begin{center}122{\color{red}\bfseries\large [\dots #1 \dots ] }123\end{center}124}125126127\newcommand{\note}[1]{{\sf\small[[#1]]}}128%-- Title.129130\begin{document}131132\author{William Stein and Christian Wuthrich}133\title{Computions About Tate-Shafarevich Groups Using Iwasawa Theory}134135\maketitle136137\abstract{We explain how to combine deep results from Iwasawa theory138with explicit computation to obtain information about139$p$-parts of Shafarevich-Tate groups of elliptic curves over $\QQ$.140This method provides a practical way to compute141$\Sha(E/\QQ)[p]$ in many cases when traditional $p$-descent methods142are completely impractical.}143144145\section{Introduction}\label{ranksha_sec}146147\william{Be sure to cite \cite{colmez}, perin-riou, etc.}148\william{In sections 3--5, it would be good to have an actual149short (!) illustrative example in each section.}150151152Let $E$ be an elliptic curve defined over $\QQ$ and let153\begin{equation}\label{w_eq}154y^2 \, + \, a_1\, x\,y\, + \,a_3\,y \,=\, x^3 \, + \, a_2\, x^2\, + \,a_4\,x\, + \,a_6155\end{equation}156be a choice of global minimal Weierstrass equation for $E$.157The Mordell proved that the set of rational points158$E(\QQ)$ is an abelian group of finite rank $r=\rk(E(\QQ))$.159Birch and Swinnerton-Dyer then conjectured that160$161r = \ord_{s=1} L(E,s),162$163where $L(E,s)$ is the Hasse-Weil $L$-function of $E$164(see Conjecture~\ref{bsd_con} below).165We call $r_{\an} = \ord_{s=1} L(E,s)$ the analytic166rank of $E$.167168There is no known provably correct general169algorithm to compute $r$, but one can computationally170obtain upper and lower bounds in any particular case.171One way to give a lower bound on $r$ is to search for linearly independent points172of small height via the method of descent, which involves searching for points of173even smaller height on a collection of auxiliary curves.174Complex and $p$-adic Heegner points constructions can also be used in some175cases to bound the rank from below.176To give a computable upper bound on the rank $r$,177apart from the case of analytic ranks $0$ and $1$ when Kolyvagin's work on Euler systems178can be applied, the only general way of obtaining an upper bound is by doing an $n$-descent179for some integer $n>1$. The 2-descents implemented by J. Cremona~\cite{cremona}180and Denis Simon \cite{simon}, and the $3$ and $4$ descents in Magma, are particularly181powerful. But they may fail in practice to compute the exact rank182due to the presence of $2$ or $3$-torsion elements in the Tate-Shafarevich group.183184The Tate-Shafarevich group, denoted by $\Sha(E/\QQ)$, is a torsion abelian group associated to $E/\QQ$. It is the kernel of the localisation map185\begin{equation*}1860\rTo \Sha(E/\QQ) \rTo \HH^1(\QQ,E)\rTo \prod_\vu\HH^1(\QQ_\vu,E)187\end{equation*}188where 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.189190\begin{conjecture}{Shafarevich and Tate}\label{consha_con}191The group $\Sha(E/\QQ)$ is finite.192\end{conjecture}193194195These two invariants, the rank $r$ and the Tate-Shafarevich group $\Sha(E/\QQ)$ are encoded in the Selmer group.196Let $E(p)$ 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 by197the following exact sequence:198\begin{equation*}1990\rTo \Sel_p(E/\QQ)\rTo \HH^1(\QQ,E(p))\rTo \prod_\vu \HH^1(\QQ_\vu,E)\, .200\end{equation*}201Likewise, for any positive integer $n$, the $n$-Selmer group is defined by202the exact sequence203$$0 \to \Sel^{(n)}(E/\QQ) \to \HH^1(\QQ,E[n])\rTo \prod_\vu \HH^1(\QQ_\vu,E)$$204where $E[n]$ is the subgroup of elements of order dividing $n$ in $E$.205206It follows from the Kummer sequence that207there are short exact sequences208$$2090\rTo E(\QQ)/n E(\QQ) \rTo \Sel^{(n)}(E/\QQ)\rTo \Sha(E/\QQ)[n]\rTo 0\,.210$$211and212\begin{equation*}2130\rTo E(\QQ)\otimes \QZ \rTo \Sel_p(E/\QQ)\rTo \Sha(E/\QQ)(p)\rTo 0\,.214\end{equation*}215If the Tate-Shafarevich group is finite, then the $\ZZ_p$-corank216of $\Sel_p(E/\QQ)$ is equal to the rank $r$ of $E(\QQ)$.217218The finiteness of $\Sha(E/\QQ)$ is only known for curves of analytic rank $0$ and $1$219in which case computation of Heegner points and Kolyvagin's work on Euler systems220gives an explicit computable multiple of its order.221The group $\Sha(E/\QQ)$ is not known to be finite for even a single elliptic curve222with $r_{\an}\geq 2$. In such cases, the best one can do using current techniques223is hope to bound the $p$-part $\Sha(E/\QQ)(p)$ of $\Sha(E/\QQ)$, for specific224primes $p$. Even this might not a priori be possible, since it is not known that225$\Sha(E/\QQ)(p)$ is finite. However, if it were the case that $\Sha(E/\QQ)(p)$226is finite (as Conjecture~\ref{consha_con} asserts), then this could be verified227by computing Selmer groups $\Sel^{(p^n)}(E/\QQ)$ for228sufficiently many $n$ (see, e.g., \cite{stoll}). Note that practical229computation of $\Sel^{(p^n)}(E/\QQ)$230is prohibitively difficult for all but a few very small $p^n$.231232The algorithm in this paper gives another method for computing an upper233bound on the order of $\Sha(E/\QQ)(p)$, for most primes $p$.234We will exclude $p=2$, since traditional descent methods work well235at $p=2$, and Iwasawa theory is not as well developed for $p=2$.236We also exclude primes $p$ such that $E$ has additive reduction237at $p$ (see Section~\ref{sec:additive}).238The algorithm requires that the full Mordell-Weil group $E(\QQ)$ is known.239240241%% -------------------------------------------------------------------------242\section{The Birch and Swinnerton-Dyer conjecture}\label{bsd_sec}243244If Conjecture~\ref{bsd_con} below were true, it would yield245an algorithm to compute both the rank $r$ and the order246of $\Sha(E/\QQ)$.247248Let $E$ be an elliptic curve over $\QQ$, and249let $L(E,s)$ be the Hasse-Weil $L$-function associated to the $\QQ$-isogeny class of $E$.250According to \cite{bcdt} (which completes work initiated in \cite{}), the function251$L(E,s)$ is holomorphic on the whole complex plane.252Let $\omegaE$ be the invariant differential $dx/(2y+a_1 x+a_3)$ of253a minimal Weierstrass equation~\eqref{w_eq} of $E$. We write254$\OmegaE=\int_{E(\RR)} \omegaE \in \RR_{>0}$ for the N\'eron period of $E$.255\begin{conjecture}{Birch and Swinnerton-Dyer}\label{bsd_con}256\begin{enumerate}257\item The order of vanishing of the Hasse-Weil function $L(E,s)$ at $s=1$ is equal to the rank $r=\rk(E(\QQ))$.258\item The leading term $\Lstar$ of the Taylor expansion of $L(E,s)$ at $s=1$ satisfies259\begin{equation}\label{bsd_eq}260\frac{\Lstar}{\OmegaE} = \frac{\prod_\vu c_\vu\cdot \#\Sha(E/\QQ)}{(\#E(\QQ)_{\tors})^2}\cdot\Reg(E/\QQ)261\end{equation}262where the Tamagawa numbers are denoted by $c_\vu$ and $\Reg(E/\QQ)$263is the regulator of $E$, i.e., the discriminant of the N\'eron-Tate canonical264height pairing on $E(\QQ)$.265\end{enumerate}266\end{conjecture}267\begin{prop}268If Conjecture~\ref{bsd_con} is true, then there is an algorithm to compute~$r$269and $\#\Sha(E/\QQ)$.270\end{prop}271\begin{proof}272The proof is well known, but we repeat it here since it illustrates several key ideas.273By naively searching for points in $E(\QQ)$ we obtain a lower bound on $r$,274which is closer and closer to the true rank $r$, the longer we run the search.275At some point this lower bound will equal $r$, but without using further information276we do not know when that will occur. As explained, e.g., in \cite{cremona:algs},277we can for any $k$ compute $L^{(k)}(E,1)$ to any desired precision.278Such computations yield upper bounds on $r_{\an}$. In particular, if279we compute $L^{(k)}(E,1)$ and it is nonzero (to the precision of our computation),280then $r_{\an} < k$. Eventually this method will also converge to give an upper281bound on $r_{\an}$, though again without further information we do not know282when our computed upper bound on $r_{\an}$ equals to the true value283of $r_{\an}$. However, if we know Conjecture~\ref{bsd_con}, we know that284$r = r_{\an}$, hence at some point the lower bound on $r$ computed using285point searches, will equal the upper bound on $r_{\an}$ computed using286the $L$-series. At this point, by Conjecture~\ref{bsd_con} we know the287true value of $r$.288289Once $r$ is known, one can compute $E(\QQ)$ via a point search (and290saturation \cite{cremona??}), hence we can approximate $\Reg(E/\QQ)$291to any desired precision. All other quantities in \ref{bsd_eq} can also be292approximated to any desired precision. Solving for $\#\Sha(E/\QQ)$293in \ref{bsd_eq} and computed all other quantities to large enough precision294to determine $\#\Sha(E/\QQ)$ then determines $\#\Sha(E/\QQ)$, as claimed.295\end{proof}296297Note that the conjecture~\eqref{bsd_eq} is also invariant under isogenies298defined over~$\QQ$ (see Cassels~\cite{cassels}).299300301%% -------------------------------------------------------------------------302\section{The $p$-adic $L$-function}\label{lp_sec}303304We will assume for the rest of this article that $E$ does not admit complex multiplication (CM),305though CM curves are an area of active research for these methods (\cite{rubin, etc}).306307In 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.308309Let $\pi\colon X_0(N)\rTo E$ be the modular parametrisation of $E$ and let $c_{\pi}$ be the Manin constant, i.e., the positive integer satisfying $c_\pi\cdot\pi^{*}\omegaE = 2\pi i f(\tau) d\tau$ with310$f$ the newform associated to $E$. Manin conjectured that $c_{\pi}=1$, and much311work has been done toward this conjecture (\cite{edixhoven, me}).312313Given a rational number $r$, consider314the image $\pi_{*}(\{r\})$ in $H_1(E,\RR)$ of the path joining $r$ to $i\,\infty$ in the upper half plane.315Define316\begin{equation*}317\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 )318\end{equation*}319There 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.320By a theorem of Manin~\cite{manin}, we know that $\lambda^{+}(r)$ belongs to321$\QQ\cdot \OmegaE$. We define the modular symbol $[r]^{+}\in\QQ$ to be322\begin{equation*}323[r]^{+} \cdot \OmegaE = \lambda^{+}(r)324\end{equation*}325for all $r\in\QQ$.326In particular we have $[0]^{+}=L(E,1)\cdot \OmegaE^{-1}$.327The quantity $[r]^{+}$ can be computed either328algebraically using modular symbols and linear algebra329(\cite{cremona:algs}) or numerically, by approximating both330$\OmegaE$ using the Gauss arithmetic-geometry mean331and $\lambda^{+}(r)$ by summing a332rapidly convergent series, and333bounding the denominator of $\lambda^{+}(r)/\OmegaE$334using results about modular symbols.\william{This is probably way too vague -- I'm being lazy.}335336Let $p$ be a prime of semistable reduction. We write\footnote{%337The context should make it clear if we speak about $a_p$ or $a_2$ and $a_3$ as in~\eqref{w_eq}.338} $a_p$ for the trace of Frobenius.339Suppose 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.340Now 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.341In either multiplicative case, we have to take $\alpha=a_p$.342343Define a measure on $\ZZ_p^\times$ with values in $\QQ(\alpha)$ by344\begin{equation*}345\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]^{+}346\end{equation*}347for 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 $\mu_{\alpha}$.348Any invertible element $x$ of $\ZZ_p^{\times}$ can be written as $\omega(x)\cdot \langle x\rangle$ where $\omega(x)$ is349a $(p-1)$st350root of unity and $\langle x\rangle$ belongs to $1+2p\ZZ_p$. We define the analytic $p$-adic $L$-function by351\begin{equation*}352L_\alpha (E,s) = \int_{\ZZ_p^\times} \langle x\rangle^{s-1} \, d\mu_{\alpha}(x)353\quad\text{ for all $s\in\ZZ_p$.}354\end{equation*}355where 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}).356357Let $\Ginf$ be the Galois group of the cyclotomic extension $\QQ(\mu_{p^\infty})$358obtained by adjoining to $\QQ$ all $p$-power roots of unity. By $\kappa$ we denote the cyclotomic character $\Ginf\rTo \ZZ_p^\times$.359Because the cyclotomic character is an isomorphism,360choosing a topological generator $\gamma$ in $\Gamma = \Ginf^{4(p-1)}$ amounts to picking361an element $\kappa(\gamma)$ in $1+2p\ZZ_p^\times$.362With 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 have363\begin{equation}\label{eqn:Lpser}364\LL_{\alpha}(E,T) = \int_{\ZZ_p^\times} (1+T)^{\frac{\log(x)}{\log(\kappa(\gamma))}} d\mu_\alpha(x)\,.365\end{equation}366As in~\cite{pollack}, we define367the polynomial\william{$\log(a)$ below doesn't make368sense without further explanation,369since $a\in (\ZZ/p^k\ZZ)^*$. Likewise for the substitution370$a=\omega(b)\cdot \kappa(\gamma)^j$ below. Presumably we are making a fixed choice of lifts to $\ZZ_p^*$?}371\begin{align*}372P_n &= \sum_{a\in(\ZZ/p^k\ZZ)^{\times}}373\left[\frac{a}{p^k}\right]^{+} \cdot (1+T)^{ \frac{\log(a)}{\log(\kappa(\gamma))}} \\374& = \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,375\end{align*}376where we changed the summation by putting $ a = \omega(b) \cdot \kappa(\gamma)^j$.377Then the approximation as a Riemann sum of the above integral for $\LL_{\alpha}(E,T)$ can be written as378\begin{equation*}379\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)\,.380\end{equation*}381382\subsection{The $p$-adic multiplier}383For a prime of good reduction, we define the $p$-adic multiplier by384\begin{equation}\label{epsp1}385\epsilon_p = \left(1-\tfrac{1}{\alpha}\right)^2 \,.386\end{equation}387For a prime of bad multiplicative reduction, we put388\begin{equation*}389\epsilon_p = \left(1-\tfrac{1}{\alpha}\right) =\begin{cases} 0\quad &\text{if $p$ is split multiplicative and } \\3902 &\text{ if $p$ is nonsplit.}391\end{cases}392\end{equation*}393394\subsection{Interpolation property}395The $p$-adic $L$-function constructed above satisfies a desired interpolation property with respect to the complex $L$-function. For instance, we have that396\begin{equation*}397\LL_{\alpha}(E,0) = L_{\alpha}(E,1) = \int_{\ZZ_p^\times} d\mu_{\alpha} = \epsilon_p \cdot\frac{L(E,1)}{\OmegaE}\,.398\end{equation*}399A 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 $p^n$, then400\begin{equation*}401\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}\,.402\end{equation*}403Here $G(\chi^{-1})$ is the Gauss sum and $L(E,\chi^{-1},1)$ is the Hasse-Weil $L$-function of $E$ twisted by $\chi^{-1}$.404405\subsection{The good ordinary case}406Suppose now that the reduction of the elliptic curve at the prime $p$ is good and ordinary, so407$a_p$ is not divisible by $p$.408As mentioned before, in this case there is409a unique choice of root410$\alpha$ of the characteristic polynomial411$x^2 - a_p x + p$ that satisfies $\ord_p(\alpha) < 1$.412Since $\alpha$ is an algebraic integer, this implies413that $\ord_p(\alpha)=0$, so $\alpha$ is a unit414in $\ZZ_p$. 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} that415\begin{prop}416Let $E$ be an elliptic curve with good ordinary reduction417at a prime $p > 2$. Then the series $\LL_p(E,T)$ belongs to $\ZZ_p[\![T]\!]$.418\end{prop}419Note 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 !!420421\subsection{Multiplicative case}422We 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 that423\begin{equation*}424\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}425\end{equation*}426where $\qE$ denotes the Tate period of $E$ over $\QQ_p$.427This will replace the interpolation formula.428Note that it is now known thanks to~\cite{steph} that $\log_p(\qE)$ is nonzero. Hence we define the $p$-adic $\Linv$-invariant as429\begin{equation}\label{epsp2}430\Linv_p = \frac{\log_p(\qE)}{\ord_p(\qE)} \neq 0\,.431\end{equation}432We refer to~\cite{colmezlinvariant} for a detailed discussion of the different $\Linv$-invariants and their connections.433434\subsection{The supersingular case}435In 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.436437There 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}.438439As 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 $X^2 - p^{-1}\,a_p \, X + p^{-1}$.440441Write $\LL_{\alpha}(E,T)$ as $G(T) + \alpha \cdot H(T)$ with $G(T)$ and $H(T)$ in $\QQ_p[\![T]\!]$. Then we define442\begin{equation*}443\LL_p(T) = G(T)\cdot \omegaE + a_p \cdot H(T)\cdot \omegaE - p\cdot H(T)\cdot \varphi(\omegaE)\,.444\end{equation*}445This 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{%446Perrin-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 $(1-\varphi)\cdot (1-p^{-1}\varphi^{-1}) = 1 -\varphi - (\varphi - a_p \cdot p^{-1}) + p^{-1} = N_p\cdot p^{-1}$.447} e.g.448\begin{equation*}449(1-\varphi)^{-2} \, \LL_p(0) = \frac{L(E,1)}{\OmegaE}\cdot \omegaE \quad\in D_p(E)\,.450\end{equation*}451452453\subsection{Additive case}\label{sec:additive}%454The 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.455456%% -------------------------------------------------------------------------457\section{$p$-adic heights}\label{hp_sec}458459The 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 replaced460by a $p$-adic regulator, which is defined using a $p$-adic analogue of the461height pairing. We follow here the generalised version~\cite{prbe}, \cite{pr00},462and \cite{mst}.463464Let $\nu$ 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 $\eta=\etaE$.465466If $\nu=\omega$, then we define467\begin{equation*}468h_\omega(P)=-\log(P)^2469\end{equation*}470where $\log$ 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 $\hat E$.471472For $\nu=\eta$, 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 by473\begin{equation*}474\wp(t) = x(t)+\frac{a_1^2+4\,a_2}{12} \in\QQ((t))475\end{equation*}476Here $a_1$ 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 equation477\begin{equation*}478- \wp(t) = \frac{d}{\omegaE}\left(\frac{1}{\sigma}\cdot\frac{d\sigma}{\omegaE}\right)479\end{equation*}480such that $\sigma(0)=0$ and $\sigma(t(-P))=-\sigma(t(P))$.481This provides us with a series482\begin{equation*}483\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)\!)\,.484\end{equation*}485As a function on the formal group $\hat E(p\ZZ_p)$ it converges for $\ord_p(t) > \tfrac{1}{p-1}$.486487Given 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 define488\begin{equation*}489h_{\eta}(P) = \frac{2}{m^2} \cdot \log_p\left (\frac{\sigma(t(m\cdot P))}{e(m\cdot P)}\right )490\end{equation*}491%\manque{factors correct } YES.492It is proved in~\cite{bernardi} that this function is quadratic and satisfies the parallelogram law.493494Finally, if $\nu= a\, \omega+b\,\eta$ then put495\begin{equation*}496h_\nu(P) = a \, h_{\omega}(P) + b\, h_{\eta}(P)\,.497\end{equation*}498This quadratic function induces a bilinear symmetric pairing $\langle\cdot,\cdot\rangle_{\nu}$ with values in $\QQ_p$.499500\subsection{The good ordinary case}501Since 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.502503Let $\nu_{\alpha}= a \, \omega + b\,\eta$ 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 $(E,\omegaE)$.504Then, 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 be505\begin{align}506\hat h_p (P) &= \frac{1}{b}\cdot h_{\nu_{\alpha}}(P) \notag\\507&= -\frac{a}{b} \cdot z(P)^2 +2\, \log\left (\frac{\sigma(t(P))}{e( P)}\right ) \notag\\508&= 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}509\end{align}510The function $\sigma_p(t)$, defined by the last line, is called the canonical sigma-function, see~\cite{mt}, it is known to lie in $\ZZ_p[\![t]\!]$.511The $p$-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}.}512513We 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.514515\begin{conjecture}{Schneider~\cite{schneider1}}\label{conreg_con}516The 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.517\end{conjecture}518519Apart 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}.520521\subsection{The multiplicative case}522In 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}.523524Let $\qE$ be the Tate parameter of the elliptic curve over $\QQ_p$, i.e., we525have a homomorphism $\psi\colon \bar\QQ_p^\times \rTo E(\bar\QQ_p)$ whose kernel is526precisely $\qE^\ZZ$. 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 as527\begin{equation*}528e_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 )529\end{equation*}530Then we use the formula of the good ordinary case to define the canonical $\sigma$531function $\sigma_p(t(P)) = \exp(\frac{e_2}{24} z(P)^2)\cdot \sigma(t(P))$.532If the reduction is nonsplit multiplicative, then we533use the formula~\eqref{hpeq} for the good ordinary case.534535Suppose now that the reduction is split multiplicative.536Let $P$ be a point in $E(\QQ)$ having good reduction at all finite places and with trivial reduction at $p$. Then537\begin{equation*}538\hat h_p(P) = 2 \log_p\left ( \frac{\sigma_p(t(P))}{e(P)} \right) + \frac{\log_p(u(P))^2}{\log(\qE)}539\end{equation*}540where $u(P)$ is the unique element of $\ZZ_p^\times$ mapping to $P$ under the541Tate parametrisation~$\psi$.542The $p$-adic regulator is formed as before but with this modified $p$-adic height~$\hat h_p$.543544\subsection{The supersingular case}545In 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 $D_p(E)$.546The main references for this height are~\cite{prbe} and~\cite{pr00}.547548Suppose that $\nu = a\,\omega + b\,\eta$ is any element of $D_p(E)$ not lying in $\QQ_p\,\omegaE$.549It can be easily checked that the value of550\begin{equation*}551H_p(P) = \frac{1}{b} \cdot ( h_{\nu}(P) \cdot\omega - h_{\omega}(P)\cdot \nu )\quad\in D_p552\end{equation*}553is independent of the choice of $\nu$. We will call this554the $D_p$-valued height on $E(\QQ)$.555556On $D_p(E)$ there is a alternating bilinear form $[\cdot,\cdot]$ characterised by the property that $[\omegaE,\etaE]=1$. Write $\Reg_{\nu}\in\QQ_p$ for the regulator557of $h_{\nu}$ on a $\ZZ$-basis of the free part of $E(\QQ)$ with respect558to some decomposition $E(\QQ) = F \oplus E(\QQ)_{\tor}$ (since the height559is $0$ on torsion, the choice of decomposition does not matter). Then560\begin{equation*}561\Reg_p(E/\QQ) = \frac{\Reg_{\nu}\cdot \nu'-\Reg_{\nu'}\cdot \nu}{[\nu',\nu]}\quad\in D_p(E)562\end{equation*}563is 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 $E/\QQ$.564565It 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.566567If one restricts any $p$-adic height $h_{\nu}$ to the fine Mordell-Weil group defined in~\cite{wuthfine} to be the kernel568\begin{equation*}569\mathfrak{M}(E/\QQ) = \ker\left(E(\QQ)\otimes \ZZ_p\rTo \widehat{E(\QQ_p)} \right),570\end{equation*}571where $\widehat{E(\QQ_p)}$ is the $p$-adic completion of $E(\QQ_p)$.572The restricted height is then573independent of the chosen element $\nu$ in $D_p(E)$. We call its regulator the fine574regulator, which is an element of $\QQ_p$ defined up to multiplication by a575unit in $\ZZ_p$.576577In general, the $D_p$-valued regulator is 0578if and only if the fine regulator vanishes.579580\begin{conjecture}{Perrin-Riou~\cite[Conjecture 3.3.7.i]{prfourier93}}\label{conreg_ss_con}581The 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.582\end{conjecture}583584585\subsection{Normalisation}586In 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.587This normalization588can 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 $E(\QQ)$.589Hence we write590\begin{equation*}591\Reg_{\gamma}(E/\QQ) = \frac{\Reg_p(E/\QQ)}{\kappa(\gamma)^r}592\end{equation*}593594%% -------------------------------------------------------------------------595\section{The $p$-adic Birch and Swinnerton-Dyer conjecture}\label{pbsd_sec}596597\subsection{The ordinary case}598The 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 $s=1$.599600\begin{conjecture}{Mazur, Tate and Teitelbaum~\cite{mtt}}\label{pbsd_ord_con}601Let $E$ be an elliptic curve with good ordinary reduction or with multiplicative reduction at a prime $p$.602\begin{itemize}603\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 $r+1$.604\item The leading term $\LL_p^{\ast}(E,0)$ satisfies605\begin{equation}\label{pbsd_ord_eq}606\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)607\end{equation}608unless the reduciton is split multiplicative in which case the leading term is609\begin{equation}610\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).611\end{equation}612\end{itemize}613\william{Are the conjectures only up to a $p$-adic unit or are the {\em conjectures}614really on the nose supposed to be true?}615\end{conjecture}616617\subsection{The supersingular case}618The 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 value619in the $\QQ_p$-vector space $D_p(E)$ of dimension 2. The fact of having two620coordinates was used cleverly by Kurihara and Pollack in~\cite{kuriharapollack} to construct global points via a $p$-adic analytic computation.621622We say that an623element $a(T)\cdot\omegaE + b(T)\cdot\etaE$ in $D_p(E)\otimes \QQ_p[\![T]\!]$624has order $d$ at $T=0$ if $d$ is equal to the minimum of the orders of $a(T)$ and $b(T)$.625626\begin{conjecture}{Bernardi and Perrin-Riou~\cite{prbe}}\label{pbsd_ss_con}627Let $E$ be an elliptic curve with good supersingular reduction at a prime $p$.628\begin{itemize}629\item The order of vanishing of the $D_p$-valued $L$-function $\LL_p(E,T)$ at $T=0$ is equal to the rank $r$.630\item The leading term $\LL_p^{\ast}(E,0)$ satisfies631\begin{equation}\label{pbsd_ss_eq}632\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)633\end{equation}634\end{itemize}635\end{conjecture}636637%% -------------------------------------------------------------------------638\section{Iwasawa theory of elliptic curves}\label{iwasawa_sec}639We suppose from now on that $p>2$.640Let $\QQinf$ 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 $\ZZ_p[\![\Gamma]\!]$.641We use the fixed topological generator $\gamma$ of $\Gamma$ to identify $\Lambda$ with $\ZZ_p[\![T]\!]$ by sending $\gamma$ to $1+T$.642It is well-known that any finitely generated $\Lambda$-module admits643a decomposition as a direct sum of644elementary $\Lambda$-modules. Denote by $\QQn$ the645$n$\textsuperscript{th} layer of the $\ZZ_p$-extension. As before, we may define the $p$-Selmer group over $\QQn$ by646the exact sequence647\begin{equation*}6480\rTo \Sel_p(E/\QQn)\rTo \HH^1(\QQn,E(p))\rTo \prod_\vu \HH^1(\QQn_\vu,E)649\end{equation*}650\william{Would you be opposed to using the notation $\Sel_p(\QQn, E)$? It's clearer651and easier to read in this case.}652with 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 dual653\begin{equation*}654X(E/\QQinf) = \Hom\left(\Sel_p(E/\QQinf),\QZ\right)655\end{equation*}656which is a finitely generated $\Lambda$-module (see~\cite{coatessujatha}).657658\subsection{The ordinary case}659Assume 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 $\Lambda$. The660series661\begin{equation}\label{eqn:fE}662\fE(T)\in\ZZ_p[\![T]\!]663\end{equation}664is well-defined up to multiplication by a unit in $\Lambda^{\!\times}$.665666In analogy to the zeta-function of a variety over a finite field, one667should think of $\fE(T)$ as a generating function encoding the growth668of 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 growth669of the rank. Since a nonzero power series with coefficients in670$\ZZ_p$ can only have finitely many zeros, one can671show that the rank of $E(\QQn)$ has to stabilize in672the tower $\QQ_n$. In other words, the Mordell-Weil673group $E(\QQinf)$ is still of finite rank.674\william{Shouldn't we cite Lichtenbaum here too?}675676The following relatively old result is due to677Schneider~\cite{schneider2} and678Perrin-Riou~\cite{pr82}. The multiplicative case is due to Jones~\cite{jones89}.679\begin{thm}[Schneider, Perrin-Riou, Jones]\label{perrinriouschneider_thm}680The order of vanishing of $\fE(T)$ at $T=0$ is at least equal to the rank $r$.681It 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 as682\begin{equation*}683\epsilon_p\cdot\frac{ \prod_\vu c_\vu\cdot \#\Sha(E/\QQ)(p)}684{(\#E(\QQ)(p))^2}\cdot\Reg_{\gamma}(E/\QQ)685\end{equation*}686unless the reduction is split multiplicative in which case the same formula holds with $\epsilon_p$ replaced by $\Linv_p$.687\end{thm}688689690\subsection{The supersingular case}691692The 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 series693$\LL_p^{\pm}(E,T)$ to which will correspond694two 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}.695696Let $\Tp E$ 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 Coleman697map $\Col$ from $\Hinfloc$ to a sub-module of $\QQ_p[\![T]\!]\otimes D_p(E)$.698699Define the fine Selmer group to be the kernel700\begin{equation*}701\Rel(E/\QQn) = \ker\left ( \Sel(E/\QQn) \rTo E(\QQn_{\mf p})\otimes\QZ\right)\,.702\end{equation*}703It is again a consequence of the work of Kato \william{Give a reference.} that704the Pontryagin dual $Y(E/\QQinf)$ of $\Rel(E/\QQinf)$ is a $\Lambda$-torsion module. Denote by $g_E(T)$ its characteristic series.705706Let $\Sigma$ 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 $\Sigma$.707708Choose 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?}709Write $h_c(T)$ for the characteristic series of $Z_c$. Then we define an algebraic equivalent of the $D_p(E)$-valued $L$-series by710\begin{equation*}711\fE(T) = g_E(T)\cdot \Col(\cinf)\cdot h_c(T)^{-1} \in \QQ_p[\![T]\!]\otimes D_p(E)712\end{equation*}713where by $\Col(\cinf)$ we mean the image of the localisation of $\cinf$ to $\Hinfloc$ under the Coleman map $\Col$. The resulting series $\fE(T)$ is714independent of the choice of $\cinf$. Of course, $\fE(T)$ is again only defined up to multiplication by a unit in $\Lambda^{\!\times}$.715716Again we have an Euler-characteristic result due to Perrin-Riou~\cite{prfourier93}:717718\begin{thm}[Perrin-Riou]\label{perrinriou_thm}719The order of vanishing of $\fE(T)$ at $T=0$ is at least equal to the rank $r$.720It 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 as721\begin{equation*}722\prod_\vu c_\vu\cdot \#\Sha(E/\QQ)(p)\cdot \Reg_{\gamma}(E/\QQ)723\end{equation*}724\end{thm}725726Note 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 unit727if the reduction at~$p$ is supersingular.728729%% -------------------------------------------------------------------------730\section{The Main Conjecture}\label{mainconjecture_sec}731732The main conjecture links the two $p$-adic power series (\ref{eqn:Lpser}) and (\ref{eqn:fE})733of the previous sections. We formulate everything now simultaneously for the734ordinary and the supersingular case, even if they are of quite different nature.735We still assume that $p\neq 2$.736737\begin{conjecture}{Main conjecture of Iwasawa theory for elliptic curves}\label{mainconjecture_con}738If $E$ has good or nonsplit multiplicative reduction at $p$, then739there 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 $T\cdot \Lambda^{\!\times}$.740\end{conjecture}741742Much is now known about this conjecture.743To the elliptic curve $E$ we attach the mod-$p$ representation744\begin{equation*}745\bar\rho_p\colon \Gal(\bar \QQ/\QQ)\rTo \Aut(E[p])\cong \Gl_2(\FF_p)746\end{equation*}747of the absolute Galois group of $\QQ$.748Serre proved that $\bar\rho_p$ is almost always749surjective (note that by hypothesis $E$ does not have complex multiplication)750and that for semistable curves surjectivity can only fail when there751is an isogeny of degree $p$ defined over $\QQ$. See~\cite{serregl2} and~\cite{serrewiles}.752753\begin{thmkato}\label{katodiv_thm}754Suppose 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 $d(T)$.755\end{thmkato}756757The 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}.758759In particular the theorem applies to all odd primes $p$ if $E$ is a semistable curve.760For 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:761\begin{thmkato}\label{ncartan_thm}762Suppose 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 $p^m\cdot\LL_p(E,T)$.763\end{thmkato}764765Greenberg 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.766767%% -------------------------------------------------------------------------768\section{If the $L$-series does not vanish}\label{rank0_sec}769Suppose the Hasse-Weil $L$-function $L(E,s)$ does not vanish at $s=1$. In this case770Kolyvagin proved that $E(\QQ)$ and $\Sha(E/\QQ)$ are finite. In particular771Conjecture~\ref{consha_con} is valid; also, Conjectures~\ref{conreg_con}772and~\ref{conreg_ss_con} are trivially true in this case.773774Let $p>2$ 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$775has split multiplicative reduction.776777\subsection{The good ordinary case}778In the ordinary case we have779\begin{equation*}780\epsilon_p^{-1}\cdot \LL_p(E,0) = \frac{L(E,1)}{ \OmegaE} = [0]^{+},781\end{equation*}782which is a nonzero rational number by~\cite{manin}.783Using the theorem\footnote{In the case of analytic784rank 0, the theorem is actually relatively easy and well explained in~\cite{coatessujatha}.}785of Perrin-Riou and Schneider~\ref{perrinriouschneider_thm} in the first line786and Kato's theorem~\ref{katodiv_thm} on the main conjecture in the787second line,\william{What does it mean ``in the first line''? ``In the second line'' ??}788we find that789\begin{align*}790\ord_p \left (\epsilon_p\cdot\frac{ \prod_\vu c_\vu\cdot \#\Sha(E/\QQ)(p)}{(\#E(\QQ)(p))^2}\right) =&791\ord_p(\fE(0)) \\792\leq& \ord_p(\LL_p(E,0)) \\793&= \ord_p \left (\frac{L(E,1)}{ \OmegaE} \right)794+ \ord_p(\epsilon_p)\,.795\end{align*}796Hence, we have the following797upper bound on the $p$-primary part of the Tate-Shafarevich group which is sharp under the assumption of the main conjecture:798\begin{equation}\label{sha_bound_r0_eq}799\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)\,.800\end{equation}801This bound agrees with the Birch and Swinnerton-Dyer conjecture.802\william{This is stronger than what I stated in my previous bsd computation803paper.}804805\subsection{The multiplicative case}806If the reduction is not split, then the above holds just the same.\william{Why?}807If instead the reduction is split multiplicative, we have808that $\LL_p(E,0) =0$ and \begin{equation*}809\LL_p'(E,0)=\Linv_p\cdot\frac{L(E,1)}{ \OmegaE} =\Linv_p\cdot [0]^{+} \neq 0\,.810\end{equation*}811Since 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.812813\subsection{The supersingular case}814For the supersingular $D_p(E)$-valued series, we have815\begin{equation*}816(1-\varphi)^{-2}\cdot\LL_p(E,0) = \frac{L(E,1)}{ \OmegaE} \cdot \omegaE= [0]^{+} \cdot \omegaE817\end{equation*}818which is a nonzero element of $D_p(E)$.819The $D_p(E)$-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 $\fE(T)$.820Again we obtain an inequality by using theorem~\ref{perrinriou_thm}821\begin{align*}822\ord_p \left( \prod_\vu c_\vu\cdot \#\Sha(E/\QQ)(p) \right) =&823\ord_p((1-\varphi)^{-2}\,\fE(0)) \\824\leq& \ord_p((1-\varphi)^{-2}\,\LL_p(E,0)) \\825&= \ord_p \left (\frac{L(E,1)}{ \OmegaE} \right)\,.826\end{align*}827828\subsection{Conclusion}829Summarising the above computations, we have830\begin{thm}831Let $E$ be an elliptic curve such that $L(E,1)\neq 0$. Then $\Sha(E/\QQ)$ is finite and832\begin{equation*}833\# \Sha(E/\QQ) \leq C\cdot\frac{L(E,1)}{\OmegaE}\cdot\frac{(\#E(\QQ)_{\tors})^2}{\prod c_\vu}834\end{equation*}835where $C$ is a product of a power of $2$ and of power of836primes of additive reduction and of powers of837primes for which the representation $\bar\rho_p$ is not838surjective and there is no isogeny of degree $p$ on $E$ defined over $\QQ$.839840In particular if $E$ is semistable, then $C$ is a power of $2$.841\end{thm}842843This improves Corollary~3.5.19 in~\cite{eulersystems}.844845%% -------------------------------------------------------------------------846\section{If the $L$-series vanishes to the first order}\label{rank1_sec}847848We 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 Heegner849points and the theorem of Kolyvagin show again that $\Sha(E/\QQ)$ is finite and that850the rank of $E(\QQ)$ is equal to $1$. Let $P$ be a choice of generator of the free851part of the Mordell-Weil group (modulo torsion).852Suppose that the $p$-adic height $\hat h_p(P)$ is nonzero.853Thanks to a theorem of Perrin-Riou in~\cite{prheegner},854we must have the following equality of rational numbers855\begin{equation*}856\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))}857\end{equation*}858where, 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 rank859$1$ so $T\mid f_E(T)$). Hence we have860\begin{thm}861Let $E/\QQ$ be an elliptic curve with good ordinary reduction at the odd prime $p$.862Suppose 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 $\QQ$.863If $L(E,s)$ has a simple zero at $s=1$, then the $p$-primary part of $\Sha(E/\QQ)$ is finite and its valuation is bounded by864\begin{equation*}865\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)866\end{equation*}867\end{thm}868In 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.869870%% -------------------------------------------------------------------------871\section{The algorithm}\label{algorithm_sec}872873\subsection{The rank}874Let $E/\QQ$ be an elliptic curve.875Suppose we are in the situation that we have found $n$ linearly independent points. We wish to prove that $n$ is equal to the rank $r=\rk(E(\QQ))$.876877For this purpose, we choose a prime $p$ satisfying the following conditions878\begin{itemize}879\item $p > 2$,880\item $E$ has good reduction at $p$.881\end{itemize}882By 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 $T=0$. Then883\begin{equation*}884b\,\geq \,\ord_{T=0} \LL_p(E,1) \,\geq\, \ord_{T=0} \fE(T)\, \geq\, r885\end{equation*}886by 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.887888\william{The procedure described in this section is {\em NOT} an algorithm. It889depends 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 that890conjecture to read off this precision. Thus given current theorems, we would never891know when we're done. So this section is not about an algorithm -- or it is about892an algorithm that is conditional on knowing the $p$-adic BSD conjecture. Please clarify.}893894895\subsection{The Tate-Shafarevich group}896Suppose now that $E$ is an elliptic curve and $p$ is a prime satisfying the following conditions897\begin{itemize}898\item $p > 2$,899\item $E$ has good reduction at $p$.900\item The image of $\bar\rho_p$ is either the full group $\Gl_2(\FF_p)$ or it is contained in a Borel subgroup.901\end{itemize}902Note that these conditions apply to all but finitely many primes $p$.903904Suppose 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.905906Using the explicit basis of $E(\QQ)$ we can compute the $p$-adic regulator of $E$ over $\QQ$ using the efficient algorithm in~\cite{mst}.907908We compute the leading coefficient $\LL_p^{\ast}(E,0)$ of the analytic $p$-adic $L$-function.909If 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.910911\subsection{The ordinary case}912If $E$ has ordinary reduction at $p$, good or multiplicative, then913\begin{align*}914\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) \\915\leq& \ord_p(\LL_p^{\ast}(E,0)) + 2\cdot\ord_p(\#(E(\QQ)(p)) - \ord_p (\epsilon_p)\\916&\ - \sum_\vu\ord_p(c_\vu)-\ord_p(\Reg_{\gamma}(E/\QQ))917\end{align*}918The inequality uses Kato's theorem~\ref{katodiv_thm}.919920Note that if the main conjecture holds this inequality will be an equality. It should also be mentioned that Grigorov~\cite{grigorov} has found921a way to compute922lower bounds on the order of the Tate-Shafarevich group in certain cases.923One can also use congruences (i.e., visibility) to construct elements924(see \cite{papersonvisibility}).925926\subsection{The supersingular case}927Suppose now that $E$ has supersingular reduction at $p$. Then we may use theorem~\ref{perrinriou_thm} and theorem~\ref{katodiv_thm} to obtain928\begin{align*}929\ord_p( \#\Sha(E/\QQ)(p) ) = & \ord_p(930(1-\varphi)^{-2}\,\fE^{\ast}(0)) - \ord_p(\Reg_p(E/\QQ)) - \ord_p(\prod_\vu c_\vu) \\931\leq& \ord_p((1-\varphi)^{-2}\,\LL_p^{\ast}(E,0)) - \ord_p(\Reg_p(E/\QQ)) - \sum_{\vu}\ord_p(c_\vu)932\end{align*}933where the convention on $\ord_p(d(T))$ for an element $d(T)\in\QQ_p[\![T]\!]\otimes D_p(E)$ is as before.934Again the inequality can be replaced by an equality if the main conjecture holds for $E$ at $p$.935936%% -------------------------------------------------------------------------937\section{Technical details}\label{tech_sec}938939%% -------------------------------------------------------------------------940\section{Numerical results}\label{numerical_sec}941942943%% -------------------------------------------------------------------------944945\bibliographystyle{amsalpha}946\bibliography{shark}947948\end{document}949950