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\begin{document}
\author{William Stein and Christian Wuthrich}
\title{Computions About Tate-Shafarevich Groups Using Iwasawa Theory}
\maketitle
\abstract{We explain how to combine deep results from Iwasawa theory
with explicit computation to obtain information about
$p$-parts of Shafarevich-Tate groups of elliptic curves over $\QQ$.
This method provides a practical way to compute
$\Sha(E/\QQ)[p]$ in many cases when traditional $p$-descent methods
are completely impractical.}
\section{Introduction}\label{ranksha_sec}
\william{Be sure to cite \cite{colmez}, perin-riou, etc.}
\william{In sections 3--5, it would be good to have an actual
short (!) illustrative example in each section.}
Let $E$ be an elliptic curve defined over $\QQ$ and let
\begin{equation}\label{w_eq}
y^2 \, + \, a_1\, x\,y\, + \,a_3\,y \,=\, x^3 \, + \, a_2\, x^2\, + \,a_4\,x\, + \,a_6
\end{equation}
be a choice of global minimal Weierstrass equation for $E$.
The Mordell proved that the set of rational points
$E(\QQ)$ is an abelian group of finite rank $r=\rk(E(\QQ))$.
Birch and Swinnerton-Dyer then conjectured that
$
r = \ord_{s=1} L(E,s),
$
where $L(E,s)$ is the Hasse-Weil $L$-function of $E$
(see Conjecture~\ref{bsd_con} below).
We call $r_{\an} = \ord_{s=1} L(E,s)$ the analytic
rank of $E$.
There is no known provably correct general
algorithm to compute $r$, but one can computationally
obtain upper and lower bounds in any particular case.
One way to give a lower bound on $r$ is to search for linearly independent points
of small height via the method of descent, which involves searching for points of
even smaller height on a collection of auxiliary curves.
Complex and $p$-adic Heegner points constructions can also be used in some
cases to bound the rank from below.
To give a computable upper bound on the rank $r$,
apart from the case of analytic ranks $0$ and $1$ when Kolyvagin's work on Euler systems
can be applied, the only general way of obtaining an upper bound is by doing an $n$-descent
for some integer $n>1$. The 2-descents implemented by J. Cremona~\cite{cremona}
and Denis Simon \cite{simon}, and the $3$ and $4$ descents in Magma, are particularly
powerful. But they may fail in practice to compute the exact rank
due to the presence of $2$ or $3$-torsion elements in the Tate-Shafarevich group.
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
\begin{equation*}
0\rTo \Sha(E/\QQ) \rTo \HH^1(\QQ,E)\rTo \prod_\vu\HH^1(\QQ_\vu,E)
\end{equation*}
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.
\begin{conjecture}{Shafarevich and Tate}\label{consha_con}
The group $\Sha(E/\QQ)$ is finite.
\end{conjecture}
These two invariants, the rank $r$ and the Tate-Shafarevich group $\Sha(E/\QQ)$ are encoded in the Selmer group.
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
the following exact sequence:
\begin{equation*}
0\rTo \Sel_p(E/\QQ)\rTo \HH^1(\QQ,E(p))\rTo \prod_\vu \HH^1(\QQ_\vu,E)\, .
\end{equation*}
Likewise, for any positive integer $n$, the $n$-Selmer group is defined by
the exact sequence
$$0 \to \Sel^{(n)}(E/\QQ) \to \HH^1(\QQ,E[n])\rTo \prod_\vu \HH^1(\QQ_\vu,E)$$
where $E[n]$ is the subgroup of elements of order dividing $n$ in $E$.
It follows from the Kummer sequence that
there are short exact sequences
$$
0\rTo E(\QQ)/n E(\QQ) \rTo \Sel^{(n)}(E/\QQ)\rTo \Sha(E/\QQ)[n]\rTo 0\,.
$$
and
\begin{equation*}
0\rTo E(\QQ)\otimes \QZ \rTo \Sel_p(E/\QQ)\rTo \Sha(E/\QQ)(p)\rTo 0\,.
\end{equation*}
If the Tate-Shafarevich group is finite, then the $\ZZ_p$-corank
of $\Sel_p(E/\QQ)$ is equal to the rank $r$ of $E(\QQ)$.
The finiteness of $\Sha(E/\QQ)$ is only known for curves of analytic rank $0$ and $1$
in which case computation of Heegner points and Kolyvagin's work on Euler systems
gives an explicit computable multiple of its order.
The group $\Sha(E/\QQ)$ is not known to be finite for even a single elliptic curve
with $r_{\an}\geq 2$. In such cases, the best one can do using current techniques
is hope to bound the $p$-part $\Sha(E/\QQ)(p)$ of $\Sha(E/\QQ)$, for specific
primes $p$. Even this might not a priori be possible, since it is not known that
$\Sha(E/\QQ)(p)$ is finite. However, if it were the case that $\Sha(E/\QQ)(p)$
is finite (as Conjecture~\ref{consha_con} asserts), then this could be verified
by computing Selmer groups $\Sel^{(p^n)}(E/\QQ)$ for
sufficiently many $n$ (see, e.g., \cite{stoll}). Note that practical
computation of $\Sel^{(p^n)}(E/\QQ)$
is prohibitively difficult for all but a few very small $p^n$.
The algorithm in this paper gives another method for computing an upper
bound on the order of $\Sha(E/\QQ)(p)$, for most primes $p$.
We will exclude $p=2$, since traditional descent methods work well
at $p=2$, and Iwasawa theory is not as well developed for $p=2$.
We also exclude primes $p$ such that $E$ has additive reduction
at $p$ (see Section~\ref{sec:additive}).
The algorithm requires that the full Mordell-Weil group $E(\QQ)$ is known.
%% -------------------------------------------------------------------------
\section{The Birch and Swinnerton-Dyer conjecture}\label{bsd_sec}
If Conjecture~\ref{bsd_con} below were true, it would yield
an algorithm to compute both the rank $r$ and the order
of $\Sha(E/\QQ)$.
Let $E$ be an elliptic curve over $\QQ$, and
let $L(E,s)$ be the Hasse-Weil $L$-function associated to the $\QQ$-isogeny class of $E$.
According to \cite{bcdt} (which completes work initiated in \cite{}), the function
$L(E,s)$ is holomorphic on the whole complex plane.
Let $\omegaE$ be the invariant differential $dx/(2y+a_1 x+a_3)$ of
a minimal Weierstrass equation~\eqref{w_eq} of $E$. We write
$\OmegaE=\int_{E(\RR)} \omegaE \in \RR_{>0}$ for the N\'eron period of $E$.
\begin{conjecture}{Birch and Swinnerton-Dyer}\label{bsd_con}
\begin{enumerate}
\item The order of vanishing of the Hasse-Weil function $L(E,s)$ at $s=1$ is equal to the rank $r=\rk(E(\QQ))$.
\item The leading term $\Lstar$ of the Taylor expansion of $L(E,s)$ at $s=1$ satisfies
\begin{equation}\label{bsd_eq}
\frac{\Lstar}{\OmegaE} = \frac{\prod_\vu c_\vu\cdot \#\Sha(E/\QQ)}{(\#E(\QQ)_{\tors})^2}\cdot\Reg(E/\QQ)
\end{equation}
where the Tamagawa numbers are denoted by $c_\vu$ and $\Reg(E/\QQ)$
is the regulator of $E$, i.e., the discriminant of the N\'eron-Tate canonical
height pairing on $E(\QQ)$.
\end{enumerate}
\end{conjecture}
\begin{prop}
If Conjecture~\ref{bsd_con} is true, then there is an algorithm to compute~$r$
and $\#\Sha(E/\QQ)$.
\end{prop}
\begin{proof}
The proof is well known, but we repeat it here since it illustrates several key ideas.
By naively searching for points in $E(\QQ)$ we obtain a lower bound on $r$,
which is closer and closer to the true rank $r$, the longer we run the search.
At some point this lower bound will equal $r$, but without using further information
we do not know when that will occur. As explained, e.g., in \cite{cremona:algs},
we can for any $k$ compute $L^{(k)}(E,1)$ to any desired precision.
Such computations yield upper bounds on $r_{\an}$. In particular, if
we compute $L^{(k)}(E,1)$ and it is nonzero (to the precision of our computation),
then $r_{\an} < k$. Eventually this method will also converge to give an upper
bound on $r_{\an}$, though again without further information we do not know
when our computed upper bound on $r_{\an}$ equals to the true value
of $r_{\an}$. However, if we know Conjecture~\ref{bsd_con}, we know that
$r = r_{\an}$, hence at some point the lower bound on $r$ computed using
point searches, will equal the upper bound on $r_{\an}$ computed using
the $L$-series. At this point, by Conjecture~\ref{bsd_con} we know the
true value of $r$.
Once $r$ is known, one can compute $E(\QQ)$ via a point search (and
saturation \cite{cremona??}), hence we can approximate $\Reg(E/\QQ)$
to any desired precision. All other quantities in \ref{bsd_eq} can also be
approximated to any desired precision. Solving for $\#\Sha(E/\QQ)$
in \ref{bsd_eq} and computed all other quantities to large enough precision
to determine $\#\Sha(E/\QQ)$ then determines $\#\Sha(E/\QQ)$, as claimed.
\end{proof}
Note that the conjecture~\eqref{bsd_eq} is also invariant under isogenies
defined over~$\QQ$ (see Cassels~\cite{cassels}).
%% -------------------------------------------------------------------------
\section{The $p$-adic $L$-function}\label{lp_sec}
We will assume for the rest of this article that $E$ does not admit complex multiplication (CM),
though CM curves are an area of active research for these methods (\cite{rubin, etc}).
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.
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
$f$ the newform associated to $E$. Manin conjectured that $c_{\pi}=1$, and much
work has been done toward this conjecture (\cite{edixhoven, me}).
Given a rational number $r$, consider
the image $\pi_{*}(\{r\})$ in $H_1(E,\RR)$ of the path joining $r$ to $i\,\infty$ in the upper half plane.
Define
\begin{equation*}
\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 )
\end{equation*}
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.
By a theorem of Manin~\cite{manin}, we know that $\lambda^{+}(r)$ belongs to
$\QQ\cdot \OmegaE$. We define the modular symbol $[r]^{+}\in\QQ$ to be
\begin{equation*}
[r]^{+} \cdot \OmegaE = \lambda^{+}(r)
\end{equation*}
for all $r\in\QQ$.
In particular we have $[0]^{+}=L(E,1)\cdot \OmegaE^{-1}$.
The quantity $[r]^{+}$ can be computed either
algebraically using modular symbols and linear algebra
(\cite{cremona:algs}) or numerically, by approximating both
$\OmegaE$ using the Gauss arithmetic-geometry mean
and $\lambda^{+}(r)$ by summing a
rapidly convergent series, and
bounding the denominator of $\lambda^{+}(r)/\OmegaE$
using results about modular symbols.\william{This is probably way too vague -- I'm being lazy.}
Let $p$ be a prime of semistable reduction. We write\footnote{%
The context should make it clear if we speak about $a_p$ or $a_2$ and $a_3$ as in~\eqref{w_eq}.
} $a_p$ for the trace of Frobenius.
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.
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.
In either multiplicative case, we have to take $\alpha=a_p$.
Define a measure on $\ZZ_p^\times$ with values in $\QQ(\alpha)$ by
\begin{equation*}
\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]^{+}
\end{equation*}
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 $\mu_{\alpha}$.
Any invertible element $x$ of $\ZZ_p^{\times}$ can be written as $\omega(x)\cdot \langle x\rangle$ where $\omega(x)$ is
a $(p-1)$st
root of unity and $\langle x\rangle$ belongs to $1+2p\ZZ_p$. We define the analytic $p$-adic $L$-function by
\begin{equation*}
L_\alpha (E,s) = \int_{\ZZ_p^\times} \langle x\rangle^{s-1} \, d\mu_{\alpha}(x)
\quad\text{ for all $s\in\ZZ_p$.}
\end{equation*}
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}).
Let $\Ginf$ be the Galois group of the cyclotomic extension $\QQ(\mu_{p^\infty})$
obtained by adjoining to $\QQ$ all $p$-power roots of unity. By $\kappa$ we denote the cyclotomic character $\Ginf\rTo \ZZ_p^\times$.
Because the cyclotomic character is an isomorphism,
choosing a topological generator $\gamma$ in $\Gamma = \Ginf^{4(p-1)}$ amounts to picking
an element $\kappa(\gamma)$ in $1+2p\ZZ_p^\times$.
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
\begin{equation}\label{eqn:Lpser}
\LL_{\alpha}(E,T) = \int_{\ZZ_p^\times} (1+T)^{\frac{\log(x)}{\log(\kappa(\gamma))}} d\mu_\alpha(x)\,.
\end{equation}
As in~\cite{pollack}, we define
the polynomial\william{$\log(a)$ below doesn't make
sense without further explanation,
since $a\in (\ZZ/p^k\ZZ)^*$. Likewise for the substitution
$a=\omega(b)\cdot \kappa(\gamma)^j$ below. Presumably we are making a fixed choice of lifts to $\ZZ_p^*$?}
\begin{align*}
P_n &= \sum_{a\in(\ZZ/p^k\ZZ)^{\times}}
\left[\frac{a}{p^k}\right]^{+} \cdot (1+T)^{ \frac{\log(a)}{\log(\kappa(\gamma))}} \\
& = \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,
\end{align*}
where we changed the summation by putting $ a = \omega(b) \cdot \kappa(\gamma)^j$.
Then the approximation as a Riemann sum of the above integral for $\LL_{\alpha}(E,T)$ can be written as
\begin{equation*}
\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)\,.
\end{equation*}
\subsection{The $p$-adic multiplier}
For a prime of good reduction, we define the $p$-adic multiplier by
\begin{equation}\label{epsp1}
\epsilon_p = \left(1-\tfrac{1}{\alpha}\right)^2 \,.
\end{equation}
For a prime of bad multiplicative reduction, we put
\begin{equation*}
\epsilon_p = \left(1-\tfrac{1}{\alpha}\right) =\begin{cases} 0\quad &\text{if $p$ is split multiplicative and } \\
2 &\text{ if $p$ is nonsplit.}
\end{cases}
\end{equation*}
\subsection{Interpolation property}
The $p$-adic $L$-function constructed above satisfies a desired interpolation property with respect to the complex $L$-function. For instance, we have that
\begin{equation*}
\LL_{\alpha}(E,0) = L_{\alpha}(E,1) = \int_{\ZZ_p^\times} d\mu_{\alpha} = \epsilon_p \cdot\frac{L(E,1)}{\OmegaE}\,.
\end{equation*}
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
\begin{equation*}
\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}\,.
\end{equation*}
Here $G(\chi^{-1})$ is the Gauss sum and $L(E,\chi^{-1},1)$ is the Hasse-Weil $L$-function of $E$ twisted by $\chi^{-1}$.
\subsection{The good ordinary case}
Suppose now that the reduction of the elliptic curve at the prime $p$ is good and ordinary, so
$a_p$ is not divisible by $p$.
As mentioned before, in this case there is
a unique choice of root
$\alpha$ of the characteristic polynomial
$x^2 - a_p x + p$ that satisfies $\ord_p(\alpha) < 1$.
Since $\alpha$ is an algebraic integer, this implies
that $\ord_p(\alpha)=0$, so $\alpha$ is a unit
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
\begin{prop}
Let $E$ be an elliptic curve with good ordinary reduction
at a prime $p > 2$. Then the series $\LL_p(E,T)$ belongs to $\ZZ_p[\![T]\!]$.
\end{prop}
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 !!
\subsection{Multiplicative case}
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
\begin{equation*}
\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}
\end{equation*}
where $\qE$ denotes the Tate period of $E$ over $\QQ_p$.
This will replace the interpolation formula.
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
\begin{equation}\label{epsp2}
\Linv_p = \frac{\log_p(\qE)}{\ord_p(\qE)} \neq 0\,.
\end{equation}
We refer to~\cite{colmezlinvariant} for a detailed discussion of the different $\Linv$-invariants and their connections.
\subsection{The supersingular case}
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.
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}.
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 $X^2 - p^{-1}\,a_p \, X + p^{-1}$.
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
\begin{equation*}
\LL_p(T) = G(T)\cdot \omegaE + a_p \cdot H(T)\cdot \omegaE - p\cdot H(T)\cdot \varphi(\omegaE)\,.
\end{equation*}
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{%
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 $(1-\varphi)\cdot (1-p^{-1}\varphi^{-1}) = 1 -\varphi - (\varphi - a_p \cdot p^{-1}) + p^{-1} = N_p\cdot p^{-1}$.
} e.g.
\begin{equation*}
(1-\varphi)^{-2} \, \LL_p(0) = \frac{L(E,1)}{\OmegaE}\cdot \omegaE \quad\in D_p(E)\,.
\end{equation*}
\subsection{Additive case}\label{sec:additive}%
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.
%% -------------------------------------------------------------------------
\section{$p$-adic heights}\label{hp_sec}
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
by a $p$-adic regulator, which is defined using a $p$-adic analogue of the
height pairing. We follow here the generalised version~\cite{prbe}, \cite{pr00},
and \cite{mst}.
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 $\eta=\etaE$.
If $\nu=\omega$, then we define
\begin{equation*}
h_\omega(P)=-\log(P)^2
\end{equation*}
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 $\hat E$.
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
\begin{equation*}
\wp(t) = x(t)+\frac{a_1^2+4\,a_2}{12} \in\QQ((t))
\end{equation*}
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
\begin{equation*}
- \wp(t) = \frac{d}{\omegaE}\left(\frac{1}{\sigma}\cdot\frac{d\sigma}{\omegaE}\right)
\end{equation*}
such that $\sigma(0)=0$ and $\sigma(t(-P))=-\sigma(t(P))$.
This provides us with a series
\begin{equation*}
\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)\!)\,.
\end{equation*}
As a function on the formal group $\hat E(p\ZZ_p)$ it converges for $\ord_p(t) > \tfrac{1}{p-1}$.
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
\begin{equation*}
h_{\eta}(P) = \frac{2}{m^2} \cdot \log_p\left (\frac{\sigma(t(m\cdot P))}{e(m\cdot P)}\right )
\end{equation*}
%\manque{factors correct } YES.
It is proved in~\cite{bernardi} that this function is quadratic and satisfies the parallelogram law.
Finally, if $\nu= a\, \omega+b\,\eta$ then put
\begin{equation*}
h_\nu(P) = a \, h_{\omega}(P) + b\, h_{\eta}(P)\,.
\end{equation*}
This quadratic function induces a bilinear symmetric pairing $\langle\cdot,\cdot\rangle_{\nu}$ with values in $\QQ_p$.
\subsection{The good ordinary case}
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.
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 $(E,\omegaE)$.
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
\begin{align}
\hat h_p (P) &= \frac{1}{b}\cdot h_{\nu_{\alpha}}(P) \notag\\
&= -\frac{a}{b} \cdot z(P)^2 +2\, \log\left (\frac{\sigma(t(P))}{e( P)}\right ) \notag\\
&= 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}
\end{align}
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 $\ZZ_p[\![t]\!]$.
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}.}
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.
\begin{conjecture}{Schneider~\cite{schneider1}}\label{conreg_con}
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.
\end{conjecture}
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}.
\subsection{The multiplicative case}
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}.
Let $\qE$ be the Tate parameter of the elliptic curve over $\QQ_p$, i.e., we
have a homomorphism $\psi\colon \bar\QQ_p^\times \rTo E(\bar\QQ_p)$ whose kernel is
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
\begin{equation*}
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 )
\end{equation*}
Then we use the formula of the good ordinary case to define the canonical $\sigma$
function $\sigma_p(t(P)) = \exp(\frac{e_2}{24} z(P)^2)\cdot \sigma(t(P))$.
If the reduction is nonsplit multiplicative, then we
use the formula~\eqref{hpeq} for the good ordinary case.
Suppose now that the reduction is split multiplicative.
Let $P$ be a point in $E(\QQ)$ having good reduction at all finite places and with trivial reduction at $p$. Then
\begin{equation*}
\hat h_p(P) = 2 \log_p\left ( \frac{\sigma_p(t(P))}{e(P)} \right) + \frac{\log_p(u(P))^2}{\log(\qE)}
\end{equation*}
where $u(P)$ is the unique element of $\ZZ_p^\times$ mapping to $P$ under the
Tate parametrisation~$\psi$.
The $p$-adic regulator is formed as before but with this modified $p$-adic height~$\hat h_p$.
\subsection{The supersingular case}
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 $D_p(E)$.
The main references for this height are~\cite{prbe} and~\cite{pr00}.
Suppose that $\nu = a\,\omega + b\,\eta$ is any element of $D_p(E)$ not lying in $\QQ_p\,\omegaE$.
It can be easily checked that the value of
\begin{equation*}
H_p(P) = \frac{1}{b} \cdot ( h_{\nu}(P) \cdot\omega - h_{\omega}(P)\cdot \nu )\quad\in D_p
\end{equation*}
is independent of the choice of $\nu$. We will call this
the $D_p$-valued height on $E(\QQ)$.
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
of $h_{\nu}$ on a $\ZZ$-basis of the free part of $E(\QQ)$ with respect
to some decomposition $E(\QQ) = F \oplus E(\QQ)_{\tor}$ (since the height
is $0$ on torsion, the choice of decomposition does not matter). Then
\begin{equation*}
\Reg_p(E/\QQ) = \frac{\Reg_{\nu}\cdot \nu'-\Reg_{\nu'}\cdot \nu}{[\nu',\nu]}\quad\in D_p(E)
\end{equation*}
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 $E/\QQ$.
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.
If one restricts any $p$-adic height $h_{\nu}$ to the fine Mordell-Weil group defined in~\cite{wuthfine} to be the kernel
\begin{equation*}
\mathfrak{M}(E/\QQ) = \ker\left(E(\QQ)\otimes \ZZ_p\rTo \widehat{E(\QQ_p)} \right),
\end{equation*}
where $\widehat{E(\QQ_p)}$ is the $p$-adic completion of $E(\QQ_p)$.
The restricted height is then
independent of the chosen element $\nu$ in $D_p(E)$. We call its regulator the fine
regulator, which is an element of $\QQ_p$ defined up to multiplication by a
unit in $\ZZ_p$.
In general, the $D_p$-valued regulator is 0
if and only if the fine regulator vanishes.
\begin{conjecture}{Perrin-Riou~\cite[Conjecture 3.3.7.i]{prfourier93}}\label{conreg_ss_con}
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.
\end{conjecture}
\subsection{Normalisation}
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.
This normalization
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 $E(\QQ)$.
Hence we write
\begin{equation*}
\Reg_{\gamma}(E/\QQ) = \frac{\Reg_p(E/\QQ)}{\kappa(\gamma)^r}
\end{equation*}
%% -------------------------------------------------------------------------
\section{The $p$-adic Birch and Swinnerton-Dyer conjecture}\label{pbsd_sec}
\subsection{The ordinary case}
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 $s=1$.
\begin{conjecture}{Mazur, Tate and Teitelbaum~\cite{mtt}}\label{pbsd_ord_con}
Let $E$ be an elliptic curve with good ordinary reduction or with multiplicative reduction at a prime $p$.
\begin{itemize}
\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 $r+1$.
\item The leading term $\LL_p^{\ast}(E,0)$ satisfies
\begin{equation}\label{pbsd_ord_eq}
\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)
\end{equation}
unless the reduciton is split multiplicative in which case the leading term is
\begin{equation}
\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).
\end{equation}
\end{itemize}
\william{Are the conjectures only up to a $p$-adic unit or are the {\em conjectures}
really on the nose supposed to be true?}
\end{conjecture}
\subsection{The supersingular case}
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
in the $\QQ_p$-vector space $D_p(E)$ of dimension 2. The fact of having two
coordinates was used cleverly by Kurihara and Pollack in~\cite{kuriharapollack} to construct global points via a $p$-adic analytic computation.
We say that an
element $a(T)\cdot\omegaE + b(T)\cdot\etaE$ in $D_p(E)\otimes \QQ_p[\![T]\!]$
has order $d$ at $T=0$ if $d$ is equal to the minimum of the orders of $a(T)$ and $b(T)$.
\begin{conjecture}{Bernardi and Perrin-Riou~\cite{prbe}}\label{pbsd_ss_con}
Let $E$ be an elliptic curve with good supersingular reduction at a prime $p$.
\begin{itemize}
\item The order of vanishing of the $D_p$-valued $L$-function $\LL_p(E,T)$ at $T=0$ is equal to the rank $r$.
\item The leading term $\LL_p^{\ast}(E,0)$ satisfies
\begin{equation}\label{pbsd_ss_eq}
\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)
\end{equation}
\end{itemize}
\end{conjecture}
%% -------------------------------------------------------------------------
\section{Iwasawa theory of elliptic curves}\label{iwasawa_sec}
We suppose from now on that $p>2$.
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 $\ZZ_p[\![\Gamma]\!]$.
We use the fixed topological generator $\gamma$ of $\Gamma$ to identify $\Lambda$ with $\ZZ_p[\![T]\!]$ by sending $\gamma$ to $1+T$.
It is well-known that any finitely generated $\Lambda$-module admits
a decomposition as a direct sum of
elementary $\Lambda$-modules. Denote by $\QQn$ the
$n$\textsuperscript{th} layer of the $\ZZ_p$-extension. As before, we may define the $p$-Selmer group over $\QQn$ by
the exact sequence
\begin{equation*}
0\rTo \Sel_p(E/\QQn)\rTo \HH^1(\QQn,E(p))\rTo \prod_\vu \HH^1(\QQn_\vu,E)
\end{equation*}
\william{Would you be opposed to using the notation $\Sel_p(\QQn, E)$? It's clearer
and easier to read in this case.}
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
\begin{equation*}
X(E/\QQinf) = \Hom\left(\Sel_p(E/\QQinf),\QZ\right)
\end{equation*}
which is a finitely generated $\Lambda$-module (see~\cite{coatessujatha}).
\subsection{The ordinary case}
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
series
\begin{equation}\label{eqn:fE}
\fE(T)\in\ZZ_p[\![T]\!]
\end{equation}
is well-defined up to multiplication by a unit in $\Lambda^{\!\times}$.
In analogy to the zeta-function of a variety over a finite field, one
should think of $\fE(T)$ as a generating function encoding the growth
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
of the rank. Since a nonzero power series with coefficients in
$\ZZ_p$ can only have finitely many zeros, one can
show that the rank of $E(\QQn)$ has to stabilize in
the tower $\QQ_n$. In other words, the Mordell-Weil
group $E(\QQinf)$ is still of finite rank.
\william{Shouldn't we cite Lichtenbaum here too?}
The following relatively old result is due to
Schneider~\cite{schneider2} and
Perrin-Riou~\cite{pr82}. The multiplicative case is due to Jones~\cite{jones89}.
\begin{thm}[Schneider, Perrin-Riou, Jones]\label{perrinriouschneider_thm}
The order of vanishing of $\fE(T)$ at $T=0$ is at least equal to the rank $r$.
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
\begin{equation*}
\epsilon_p\cdot\frac{ \prod_\vu c_\vu\cdot \#\Sha(E/\QQ)(p)}
{(\#E(\QQ)(p))^2}\cdot\Reg_{\gamma}(E/\QQ)
\end{equation*}
unless the reduction is split multiplicative in which case the same formula holds with $\epsilon_p$ replaced by $\Linv_p$.
\end{thm}
\subsection{The supersingular case}
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
$\LL_p^{\pm}(E,T)$ to which will correspond
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}.
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
map $\Col$ from $\Hinfloc$ to a sub-module of $\QQ_p[\![T]\!]\otimes D_p(E)$.
Define the fine Selmer group to be the kernel
\begin{equation*}
\Rel(E/\QQn) = \ker\left ( \Sel(E/\QQn) \rTo E(\QQn_{\mf p})\otimes\QZ\right)\,.
\end{equation*}
It is again a consequence of the work of Kato \william{Give a reference.} that
the Pontryagin dual $Y(E/\QQinf)$ of $\Rel(E/\QQinf)$ is a $\Lambda$-torsion module. Denote by $g_E(T)$ its characteristic series.
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 $\Sigma$.
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?}
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
\begin{equation*}
\fE(T) = g_E(T)\cdot \Col(\cinf)\cdot h_c(T)^{-1} \in \QQ_p[\![T]\!]\otimes D_p(E)
\end{equation*}
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
independent of the choice of $\cinf$. Of course, $\fE(T)$ is again only defined up to multiplication by a unit in $\Lambda^{\!\times}$.
Again we have an Euler-characteristic result due to Perrin-Riou~\cite{prfourier93}:
\begin{thm}[Perrin-Riou]\label{perrinriou_thm}
The order of vanishing of $\fE(T)$ at $T=0$ is at least equal to the rank $r$.
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
\begin{equation*}
\prod_\vu c_\vu\cdot \#\Sha(E/\QQ)(p)\cdot \Reg_{\gamma}(E/\QQ)
\end{equation*}
\end{thm}
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
if the reduction at~$p$ is supersingular.
%% -------------------------------------------------------------------------
\section{The Main Conjecture}\label{mainconjecture_sec}
The main conjecture links the two $p$-adic power series (\ref{eqn:Lpser}) and (\ref{eqn:fE})
of the previous sections. We formulate everything now simultaneously for the
ordinary and the supersingular case, even if they are of quite different nature.
We still assume that $p\neq 2$.
\begin{conjecture}{Main conjecture of Iwasawa theory for elliptic curves}\label{mainconjecture_con}
If $E$ has good or nonsplit multiplicative reduction at $p$, then
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 $T\cdot \Lambda^{\!\times}$.
\end{conjecture}
Much is now known about this conjecture.
To the elliptic curve $E$ we attach the mod-$p$ representation
\begin{equation*}
\bar\rho_p\colon \Gal(\bar \QQ/\QQ)\rTo \Aut(E[p])\cong \Gl_2(\FF_p)
\end{equation*}
of the absolute Galois group of $\QQ$.
Serre proved that $\bar\rho_p$ is almost always
surjective (note that by hypothesis $E$ does not have complex multiplication)
and that for semistable curves surjectivity can only fail when there
is an isogeny of degree $p$ defined over $\QQ$. See~\cite{serregl2} and~\cite{serrewiles}.
\begin{thmkato}\label{katodiv_thm}
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 $d(T)$.
\end{thmkato}
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}.
In particular the theorem applies to all odd primes $p$ if $E$ is a semistable curve.
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:
\begin{thmkato}\label{ncartan_thm}
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 $p^m\cdot\LL_p(E,T)$.
\end{thmkato}
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.
%% -------------------------------------------------------------------------
\section{If the $L$-series does not vanish}\label{rank0_sec}
Suppose the Hasse-Weil $L$-function $L(E,s)$ does not vanish at $s=1$. In this case
Kolyvagin proved that $E(\QQ)$ and $\Sha(E/\QQ)$ are finite. In particular
Conjecture~\ref{consha_con} is valid; also, Conjectures~\ref{conreg_con}
and~\ref{conreg_ss_con} are trivially true in this case.
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~$E$
has split multiplicative reduction.
\subsection{The good ordinary case}
In the ordinary case we have
\begin{equation*}
\epsilon_p^{-1}\cdot \LL_p(E,0) = \frac{L(E,1)}{ \OmegaE} = [0]^{+},
\end{equation*}
which is a nonzero rational number by~\cite{manin}.
Using the theorem\footnote{In the case of analytic
rank 0, the theorem is actually relatively easy and well explained in~\cite{coatessujatha}.}
of Perrin-Riou and Schneider~\ref{perrinriouschneider_thm} in the first line
and Kato's theorem~\ref{katodiv_thm} on the main conjecture in the
second line,\william{What does it mean ``in the first line''? ``In the second line'' ??}
we find that
\begin{align*}
\ord_p \left (\epsilon_p\cdot\frac{ \prod_\vu c_\vu\cdot \#\Sha(E/\QQ)(p)}{(\#E(\QQ)(p))^2}\right) =&
\ord_p(\fE(0)) \\
\leq& \ord_p(\LL_p(E,0)) \\
&= \ord_p \left (\frac{L(E,1)}{ \OmegaE} \right)
+ \ord_p(\epsilon_p)\,.
\end{align*}
Hence, we have the following
upper bound on the $p$-primary part of the Tate-Shafarevich group which is sharp under the assumption of the main conjecture:
\begin{equation}\label{sha_bound_r0_eq}
\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)\,.
\end{equation}
This bound agrees with the Birch and Swinnerton-Dyer conjecture.
\william{This is stronger than what I stated in my previous bsd computation
paper.}
\subsection{The multiplicative case}
If the reduction is not split, then the above holds just the same.\william{Why?}
If instead the reduction is split multiplicative, we have
that $\LL_p(E,0) =0$ and \begin{equation*}
\LL_p'(E,0)=\Linv_p\cdot\frac{L(E,1)}{ \OmegaE} =\Linv_p\cdot [0]^{+} \neq 0\,.
\end{equation*}
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.
\subsection{The supersingular case}
For the supersingular $D_p(E)$-valued series, we have
\begin{equation*}
(1-\varphi)^{-2}\cdot\LL_p(E,0) = \frac{L(E,1)}{ \OmegaE} \cdot \omegaE= [0]^{+} \cdot \omegaE
\end{equation*}
which is a nonzero element of $D_p(E)$.
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 $\fE(T)$.
Again we obtain an inequality by using theorem~\ref{perrinriou_thm}
\begin{align*}
\ord_p \left( \prod_\vu c_\vu\cdot \#\Sha(E/\QQ)(p) \right) =&
\ord_p((1-\varphi)^{-2}\,\fE(0)) \\
\leq& \ord_p((1-\varphi)^{-2}\,\LL_p(E,0)) \\
&= \ord_p \left (\frac{L(E,1)}{ \OmegaE} \right)\,.
\end{align*}
\subsection{Conclusion}
Summarising the above computations, we have
\begin{thm}
Let $E$ be an elliptic curve such that $L(E,1)\neq 0$. Then $\Sha(E/\QQ)$ is finite and
\begin{equation*}
\# \Sha(E/\QQ) \leq C\cdot\frac{L(E,1)}{\OmegaE}\cdot\frac{(\#E(\QQ)_{\tors})^2}{\prod c_\vu}
\end{equation*}
where $C$ is a product of a power of $2$ and of power of
primes of additive reduction and of powers of
primes for which the representation $\bar\rho_p$ is not
surjective and there is no isogeny of degree $p$ on $E$ defined over $\QQ$.
In particular if $E$ is semistable, then $C$ is a power of $2$.
\end{thm}
This improves Corollary~3.5.19 in~\cite{eulersystems}.
%% -------------------------------------------------------------------------
\section{If the $L$-series vanishes to the first order}\label{rank1_sec}
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
points and the theorem of Kolyvagin show again that $\Sha(E/\QQ)$ is finite and that
the rank of $E(\QQ)$ is equal to $1$. Let $P$ be a choice of generator of the free
part of the Mordell-Weil group (modulo torsion).
Suppose that the $p$-adic height $\hat h_p(P)$ is nonzero.
Thanks to a theorem of Perrin-Riou in~\cite{prheegner},
we must have the following equality of rational numbers
\begin{equation*}
\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))}
\end{equation*}
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
$1$ so $T\mid f_E(T)$). Hence we have
\begin{thm}
Let $E/\QQ$ be an elliptic curve with good ordinary reduction at the odd prime $p$.
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 $\QQ$.
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
\begin{equation*}
\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)
\end{equation*}
\end{thm}
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.
%% -------------------------------------------------------------------------
\section{The algorithm}\label{algorithm_sec}
\subsection{The rank}
Let $E/\QQ$ be an elliptic curve.
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 $r=\rk(E(\QQ))$.
For this purpose, we choose a prime $p$ satisfying the following conditions
\begin{itemize}
\item $p > 2$,
\item $E$ has good reduction at $p$.
\end{itemize}
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
\begin{equation*}
b\,\geq \,\ord_{T=0} \LL_p(E,1) \,\geq\, \ord_{T=0} \fE(T)\, \geq\, r
\end{equation*}
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.
\william{The procedure described in this section is {\em NOT} an algorithm. It
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
conjecture to read off this precision. Thus given current theorems, we would never
know when we're done. So this section is not about an algorithm -- or it is about
an algorithm that is conditional on knowing the $p$-adic BSD conjecture. Please clarify.}
\subsection{The Tate-Shafarevich group}
Suppose now that $E$ is an elliptic curve and $p$ is a prime satisfying the following conditions
\begin{itemize}
\item $p > 2$,
\item $E$ has good reduction at $p$.
\item The image of $\bar\rho_p$ is either the full group $\Gl_2(\FF_p)$ or it is contained in a Borel subgroup.
\end{itemize}
Note that these conditions apply to all but finitely many primes $p$.
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.
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}.
We compute the leading coefficient $\LL_p^{\ast}(E,0)$ of the analytic $p$-adic $L$-function.
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.
\subsection{The ordinary case}
If $E$ has ordinary reduction at $p$, good or multiplicative, then
\begin{align*}
\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) \\
\leq& \ord_p(\LL_p^{\ast}(E,0)) + 2\cdot\ord_p(\#(E(\QQ)(p)) - \ord_p (\epsilon_p)\\
&\ - \sum_\vu\ord_p(c_\vu)-\ord_p(\Reg_{\gamma}(E/\QQ))
\end{align*}
The inequality uses Kato's theorem~\ref{katodiv_thm}.
Note that if the main conjecture holds this inequality will be an equality. It should also be mentioned that Grigorov~\cite{grigorov} has found
a way to compute
lower bounds on the order of the Tate-Shafarevich group in certain cases.
One can also use congruences (i.e., visibility) to construct elements
(see \cite{papersonvisibility}).
\subsection{The supersingular case}
Suppose now that $E$ has supersingular reduction at $p$. Then we may use theorem~\ref{perrinriou_thm} and theorem~\ref{katodiv_thm} to obtain
\begin{align*}
\ord_p( \#\Sha(E/\QQ)(p) ) = & \ord_p(
(1-\varphi)^{-2}\,\fE^{\ast}(0)) - \ord_p(\Reg_p(E/\QQ)) - \ord_p(\prod_\vu c_\vu) \\
\leq& \ord_p((1-\varphi)^{-2}\,\LL_p^{\ast}(E,0)) - \ord_p(\Reg_p(E/\QQ)) - \sum_{\vu}\ord_p(c_\vu)
\end{align*}
where the convention on $\ord_p(d(T))$ for an element $d(T)\in\QQ_p[\![T]\!]\otimes D_p(E)$ is as before.
Again the inequality can be replaced by an equality if the main conjecture holds for $E$ at $p$.
%% -------------------------------------------------------------------------
\section{Technical details}\label{tech_sec}
%% -------------------------------------------------------------------------
\section{Numerical results}\label{numerical_sec}
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