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Author: William A. Stein
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%-- Title.
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\begin{document}
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\author{William Stein and Christian Wuthrich}
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\title{Computions About Tate-Shafarevich Groups Using Iwasawa Theory}
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\maketitle
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\abstract{We explain how to combine deep results from Iwasawa theory
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with explicit computation to obtain information about
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$p$-parts of Shafarevich-Tate groups of elliptic curves over $\QQ$.
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This method provides a practical way to compute
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$\Sha(E/\QQ)[p]$ in many cases when traditional $p$-descent methods
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are completely impractical.}
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\section{Introduction}\label{ranksha_sec}
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\william{Be sure to cite \cite{colmez}, perin-riou, etc.}
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\william{In sections 3--5, it would be good to have an actual
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short (!) illustrative example in each section.}
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Let $E$ be an elliptic curve defined over $\QQ$ and let
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\begin{equation}\label{w_eq}
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y^2 \, + \, a_1\, x\,y\, + \,a_3\,y \,=\, x^3 \, + \, a_2\, x^2\, + \,a_4\,x\, + \,a_6
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\end{equation}
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be a choice of global minimal Weierstrass equation for $E$.
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The Mordell proved that the set of rational points
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$E(\QQ)$ is an abelian group of finite rank $r=\rk(E(\QQ))$.
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Birch and Swinnerton-Dyer then conjectured that
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$
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r = \ord_{s=1} L(E,s),
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$
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where $L(E,s)$ is the Hasse-Weil $L$-function of $E$
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(see Conjecture~\ref{bsd_con} below).
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We call $r_{\an} = \ord_{s=1} L(E,s)$ the analytic
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rank of $E$.
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There is no known provably correct general
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algorithm to compute $r$, but one can computationally
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obtain upper and lower bounds in any particular case.
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One way to give a lower bound on $r$ is to search for linearly independent points
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of small height via the method of descent, which involves searching for points of
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even smaller height on a collection of auxiliary curves.
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Complex and $p$-adic Heegner points constructions can also be used in some
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cases to bound the rank from below.
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To give a computable upper bound on the rank $r$,
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apart from the case of analytic ranks $0$ and $1$ when Kolyvagin's work on Euler systems
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can be applied, the only general way of obtaining an upper bound is by doing an $n$-descent
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for some integer $n>1$. The 2-descents implemented by J. Cremona~\cite{cremona}
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and Denis Simon \cite{simon}, and the $3$ and $4$ descents in Magma, are particularly
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powerful. But they may fail in practice to compute the exact rank
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due to the presence of $2$ or $3$-torsion elements in the Tate-Shafarevich group.
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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
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\begin{equation*}
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0\rTo \Sha(E/\QQ) \rTo \HH^1(\QQ,E)\rTo \prod_\vu\HH^1(\QQ_\vu,E)
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\end{equation*}
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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.
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\begin{conjecture}{Shafarevich and Tate}\label{consha_con}
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The group $\Sha(E/\QQ)$ is finite.
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\end{conjecture}
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These two invariants, the rank $r$ and the Tate-Shafarevich group $\Sha(E/\QQ)$ are encoded in the Selmer group.
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Let $E(p)$ 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
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the following exact sequence:
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\begin{equation*}
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0\rTo \Sel_p(E/\QQ)\rTo \HH^1(\QQ,E(p))\rTo \prod_\vu \HH^1(\QQ_\vu,E)\, .
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\end{equation*}
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Likewise, for any positive integer $n$, the $n$-Selmer group is defined by
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the exact sequence
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$$0 \to \Sel^{(n)}(E/\QQ) \to \HH^1(\QQ,E[n])\rTo \prod_\vu \HH^1(\QQ_\vu,E)$$
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where $E[n]$ is the subgroup of elements of order dividing $n$ in $E$.
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It follows from the Kummer sequence that
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there are short exact sequences
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$$
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0\rTo E(\QQ)/n E(\QQ) \rTo \Sel^{(n)}(E/\QQ)\rTo \Sha(E/\QQ)[n]\rTo 0\,.
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$$
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and
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\begin{equation*}
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0\rTo E(\QQ)\otimes \QZ \rTo \Sel_p(E/\QQ)\rTo \Sha(E/\QQ)(p)\rTo 0\,.
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\end{equation*}
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If the Tate-Shafarevich group is finite, then the $\ZZ_p$-corank
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of $\Sel_p(E/\QQ)$ is equal to the rank $r$ of $E(\QQ)$.
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The finiteness of $\Sha(E/\QQ)$ is only known for curves of analytic rank $0$ and $1$
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in which case computation of Heegner points and Kolyvagin's work on Euler systems
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gives an explicit computable multiple of its order.
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The group $\Sha(E/\QQ)$ is not known to be finite for even a single elliptic curve
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with $r_{\an}\geq 2$. In such cases, the best one can do using current techniques
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is hope to bound the $p$-part $\Sha(E/\QQ)(p)$ of $\Sha(E/\QQ)$, for specific
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primes $p$. Even this might not a priori be possible, since it is not known that
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$\Sha(E/\QQ)(p)$ is finite. However, if it were the case that $\Sha(E/\QQ)(p)$
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is finite (as Conjecture~\ref{consha_con} asserts), then this could be verified
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by computing Selmer groups $\Sel^{(p^n)}(E/\QQ)$ for
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sufficiently many $n$ (see, e.g., \cite{stoll}). Note that practical
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computation of $\Sel^{(p^n)}(E/\QQ)$
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is prohibitively difficult for all but a few very small $p^n$.
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The algorithm in this paper gives another method for computing an upper
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bound on the order of $\Sha(E/\QQ)(p)$, for most primes $p$.
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We will exclude $p=2$, since traditional descent methods work well
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at $p=2$, and Iwasawa theory is not as well developed for $p=2$.
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We also exclude primes $p$ such that $E$ has additive reduction
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at $p$ (see Section~\ref{sec:additive}).
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The algorithm requires that the full Mordell-Weil group $E(\QQ)$ is known.
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%% -------------------------------------------------------------------------
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\section{The Birch and Swinnerton-Dyer conjecture}\label{bsd_sec}
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If Conjecture~\ref{bsd_con} below were true, it would yield
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an algorithm to compute both the rank $r$ and the order
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of $\Sha(E/\QQ)$.
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Let $E$ be an elliptic curve over $\QQ$, and
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let $L(E,s)$ be the Hasse-Weil $L$-function associated to the $\QQ$-isogeny class of $E$.
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According to \cite{bcdt} (which completes work initiated in \cite{}), the function
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$L(E,s)$ is holomorphic on the whole complex plane.
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Let $\omegaE$ be the invariant differential $dx/(2y+a_1 x+a_3)$ of
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a minimal Weierstrass equation~\eqref{w_eq} of $E$. We write
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$\OmegaE=\int_{E(\RR)} \omegaE \in \RR_{>0}$ for the N\'eron period of $E$.
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\begin{conjecture}{Birch and Swinnerton-Dyer}\label{bsd_con}
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\begin{enumerate}
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\item The order of vanishing of the Hasse-Weil function $L(E,s)$ at $s=1$ is equal to the rank $r=\rk(E(\QQ))$.
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\item The leading term $\Lstar$ of the Taylor expansion of $L(E,s)$ at $s=1$ satisfies
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\begin{equation}\label{bsd_eq}
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\frac{\Lstar}{\OmegaE} = \frac{\prod_\vu c_\vu\cdot \#\Sha(E/\QQ)}{(\#E(\QQ)_{\tors})^2}\cdot\Reg(E/\QQ)
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\end{equation}
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where the Tamagawa numbers are denoted by $c_\vu$ and $\Reg(E/\QQ)$
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is the regulator of $E$, i.e., the discriminant of the N\'eron-Tate canonical
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height pairing on $E(\QQ)$.
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\end{enumerate}
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\end{conjecture}
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\begin{prop}
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If Conjecture~\ref{bsd_con} is true, then there is an algorithm to compute~$r$
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and $\#\Sha(E/\QQ)$.
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\end{prop}
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\begin{proof}
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The proof is well known, but we repeat it here since it illustrates several key ideas.
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By naively searching for points in $E(\QQ)$ we obtain a lower bound on $r$,
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which is closer and closer to the true rank $r$, the longer we run the search.
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At some point this lower bound will equal $r$, but without using further information
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we do not know when that will occur. As explained, e.g., in \cite{cremona:algs},
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we can for any $k$ compute $L^{(k)}(E,1)$ to any desired precision.
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Such computations yield upper bounds on $r_{\an}$. In particular, if
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we compute $L^{(k)}(E,1)$ and it is nonzero (to the precision of our computation),
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then $r_{\an} < k$. Eventually this method will also converge to give an upper
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bound on $r_{\an}$, though again without further information we do not know
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when our computed upper bound on $r_{\an}$ equals to the true value
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of $r_{\an}$. However, if we know Conjecture~\ref{bsd_con}, we know that
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$r = r_{\an}$, hence at some point the lower bound on $r$ computed using
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point searches, will equal the upper bound on $r_{\an}$ computed using
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the $L$-series. At this point, by Conjecture~\ref{bsd_con} we know the
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true value of $r$.
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Once $r$ is known, one can compute $E(\QQ)$ via a point search (and
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saturation \cite{cremona??}), hence we can approximate $\Reg(E/\QQ)$
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to any desired precision. All other quantities in \ref{bsd_eq} can also be
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approximated to any desired precision. Solving for $\#\Sha(E/\QQ)$
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in \ref{bsd_eq} and computed all other quantities to large enough precision
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to determine $\#\Sha(E/\QQ)$ then determines $\#\Sha(E/\QQ)$, as claimed.
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\end{proof}
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Note that the conjecture~\eqref{bsd_eq} is also invariant under isogenies
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defined over~$\QQ$ (see Cassels~\cite{cassels}).
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%% -------------------------------------------------------------------------
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\section{The $p$-adic $L$-function}\label{lp_sec}
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We will assume for the rest of this article that $E$ does not admit complex multiplication (CM),
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though CM curves are an area of active research for these methods (\cite{rubin, etc}).
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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.
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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
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$f$ the newform associated to $E$. Manin conjectured that $c_{\pi}=1$, and much
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work has been done toward this conjecture (\cite{edixhoven, me}).
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Given a rational number $r$, consider
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the image $\pi_{*}(\{r\})$ in $H_1(E,\RR)$ of the path joining $r$ to $i\,\infty$ in the upper half plane.
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Define
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\begin{equation*}
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\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 )
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\end{equation*}
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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.
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By a theorem of Manin~\cite{manin}, we know that $\lambda^{+}(r)$ belongs to
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$\QQ\cdot \OmegaE$. We define the modular symbol $[r]^{+}\in\QQ$ to be
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\begin{equation*}
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[r]^{+} \cdot \OmegaE = \lambda^{+}(r)
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\end{equation*}
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for all $r\in\QQ$.
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In particular we have $[0]^{+}=L(E,1)\cdot \OmegaE^{-1}$.
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The quantity $[r]^{+}$ can be computed either
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algebraically using modular symbols and linear algebra
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(\cite{cremona:algs}) or numerically, by approximating both
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$\OmegaE$ using the Gauss arithmetic-geometry mean
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and $\lambda^{+}(r)$ by summing a
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rapidly convergent series, and
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bounding the denominator of $\lambda^{+}(r)/\OmegaE$
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using results about modular symbols.\william{This is probably way too vague -- I'm being lazy.}
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Let $p$ be a prime of semistable reduction. We write\footnote{%
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The context should make it clear if we speak about $a_p$ or $a_2$ and $a_3$ as in~\eqref{w_eq}.
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} $a_p$ for the trace of Frobenius.
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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.
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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.
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In either multiplicative case, we have to take $\alpha=a_p$.
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Define a measure on $\ZZ_p^\times$ with values in $\QQ(\alpha)$ by
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\begin{equation*}
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\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]^{+}
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\end{equation*}
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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 $\mu_{\alpha}$.
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Any invertible element $x$ of $\ZZ_p^{\times}$ can be written as $\omega(x)\cdot \langle x\rangle$ where $\omega(x)$ is
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a $(p-1)$st
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root of unity and $\langle x\rangle$ belongs to $1+2p\ZZ_p$. We define the analytic $p$-adic $L$-function by
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\begin{equation*}
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L_\alpha (E,s) = \int_{\ZZ_p^\times} \langle x\rangle^{s-1} \, d\mu_{\alpha}(x)
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\quad\text{ for all $s\in\ZZ_p$.}
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\end{equation*}
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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}).
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Let $\Ginf$ be the Galois group of the cyclotomic extension $\QQ(\mu_{p^\infty})$
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obtained by adjoining to $\QQ$ all $p$-power roots of unity. By $\kappa$ we denote the cyclotomic character $\Ginf\rTo \ZZ_p^\times$.
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Because the cyclotomic character is an isomorphism,
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choosing a topological generator $\gamma$ in $\Gamma = \Ginf^{4(p-1)}$ amounts to picking
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an element $\kappa(\gamma)$ in $1+2p\ZZ_p^\times$.
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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
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\begin{equation}\label{eqn:Lpser}
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\LL_{\alpha}(E,T) = \int_{\ZZ_p^\times} (1+T)^{\frac{\log(x)}{\log(\kappa(\gamma))}} d\mu_\alpha(x)\,.
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\end{equation}
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As in~\cite{pollack}, we define
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the polynomial\william{$\log(a)$ below doesn't make
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sense without further explanation,
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since $a\in (\ZZ/p^k\ZZ)^*$. Likewise for the substitution
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$a=\omega(b)\cdot \kappa(\gamma)^j$ below. Presumably we are making a fixed choice of lifts to $\ZZ_p^*$?}
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\begin{align*}
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P_n &= \sum_{a\in(\ZZ/p^k\ZZ)^{\times}}
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\left[\frac{a}{p^k}\right]^{+} \cdot (1+T)^{ \frac{\log(a)}{\log(\kappa(\gamma))}} \\
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& = \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,
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\end{align*}
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where we changed the summation by putting $ a = \omega(b) \cdot \kappa(\gamma)^j$.
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Then the approximation as a Riemann sum of the above integral for $\LL_{\alpha}(E,T)$ can be written as
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\begin{equation*}
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\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)\,.
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\end{equation*}
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\subsection{The $p$-adic multiplier}
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For a prime of good reduction, we define the $p$-adic multiplier by
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\begin{equation}\label{epsp1}
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\epsilon_p = \left(1-\tfrac{1}{\alpha}\right)^2 \,.
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\end{equation}
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For a prime of bad multiplicative reduction, we put
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\begin{equation*}
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\epsilon_p = \left(1-\tfrac{1}{\alpha}\right) =\begin{cases} 0\quad &\text{if $p$ is split multiplicative and } \\
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2 &\text{ if $p$ is nonsplit.}
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\end{cases}
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\end{equation*}
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\subsection{Interpolation property}
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The $p$-adic $L$-function constructed above satisfies a desired interpolation property with respect to the complex $L$-function. For instance, we have that
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\begin{equation*}
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\LL_{\alpha}(E,0) = L_{\alpha}(E,1) = \int_{\ZZ_p^\times} d\mu_{\alpha} = \epsilon_p \cdot\frac{L(E,1)}{\OmegaE}\,.
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\end{equation*}
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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
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\begin{equation*}
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\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}\,.
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\end{equation*}
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Here $G(\chi^{-1})$ is the Gauss sum and $L(E,\chi^{-1},1)$ is the Hasse-Weil $L$-function of $E$ twisted by $\chi^{-1}$.
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\subsection{The good ordinary case}
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Suppose now that the reduction of the elliptic curve at the prime $p$ is good and ordinary, so
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$a_p$ is not divisible by $p$.
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As mentioned before, in this case there is
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a unique choice of root
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$\alpha$ of the characteristic polynomial
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$x^2 - a_p x + p$ 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}
423
We 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*}
427
where $\qE$ denotes the Tate period of $E$ over $\QQ_p$.
428
This will replace the interpolation formula.
429
Note 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}
433
We 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
454
\subsection{Additive case}\label{sec:additive}%
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*}
541
where $u(P)$ is the unique element of $\ZZ_p^\times$ mapping to $P$ under the
542
Tate 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 $E(\QQ_p)$.
573
The restricted height is then
574
independent of the chosen element $\nu$ in $D_p(E)$. We call its regulator the fine
575
regulator, which is an element of $\QQ_p$ defined up to multiplication by a
576
unit 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.
588
This normalization
589
can 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
677
The following relatively old result is due to
678
Schneider~\cite{schneider2} and
679
Perrin-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
715
independent 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
733
The main conjecture links the two $p$-adic power series (\ref{eqn:Lpser}) and (\ref{eqn:fE})
734
of the previous sections. We formulate everything now simultaneously for the
735
ordinary and the supersingular case, even if they are of quite different nature.
736
We 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
743
Much is now known about this conjecture.
744
To 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*}
748
of the absolute Galois group of $\QQ$.
749
Serre proved that $\bar\rho_p$ is almost always
750
surjective (note that by hypothesis $E$ does not have complex multiplication)
751
and that for semistable curves surjectivity can only fail when there
752
is 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
758
The 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
760
In particular the theorem applies to all odd primes $p$ if $E$ is a semistable curve.
761
For 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
766
Greenberg 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}
770
Suppose the Hasse-Weil $L$-function $L(E,s)$ does not vanish at $s=1$. In this case
771
Kolyvagin proved that $E(\QQ)$ and $\Sha(E/\QQ)$ are finite. In particular
772
Conjecture~\ref{consha_con} is valid; also, Conjectures~\ref{conreg_con}
773
and~\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} = [0]^{+},
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 [0]^{+} \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= [0]^{+} \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}
875
Let $E/\QQ$ be an elliptic curve.
876
Suppose 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
878
For 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}
883
By 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*}
887
by 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
890
depends 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
891
conjecture to read off this precision. Thus given current theorems, we would never
892
know when we're done. So this section is not about an algorithm -- or it is about
893
an algorithm that is conditional on knowing the $p$-adic BSD conjecture. Please clarify.}
894
895
896
\subsection{The Tate-Shafarevich group}
897
Suppose 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}
903
Note that these conditions apply to all but finitely many primes $p$.
904
905
Suppose 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
907
Using 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
909
We compute the leading coefficient $\LL_p^{\ast}(E,0)$ of the analytic $p$-adic $L$-function.
910
If 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}
913
If $E$ 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*}
919
The 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}
928
Suppose 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*}
934
where the convention on $\ord_p(d(T))$ for an element $d(T)\in\QQ_p[\![T]\!]\otimes D_p(E)$ is as before.
935
Again 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