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\title{Computing $p$-Adic Cyclotomic Heights}
\date{{\bf Notes for a Talk} at Harvard on 2004-12-08}
\author{William Stein}
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\begin{document}
\maketitle
\tableofcontents
\section{Introduction}
Let $E$ be an elliptic curve over~$\Q$ given by a
minimal Weierstrass equation and suppose
$$P=(x,y)=\left(\frac{a}{d^2},\frac{b}{d^3}\right)\in E(\Q),$$ with $a,b,d\in\Z$ and
$\gcd(a,d)=\gcd(b,d)=1$. The
\defn{naive height} of $P$ is
$$\tilde{h}(P) = \log\max\{|a|,d^2\},$$
and the \defn{canonical height} of $P$ is
$$
h(P) = \lim_{n\to\infty} \frac{h(2^n P)}{4^n}.
$$
This definition is not good for computation, because
$2^n P$ gets huge very quickly, and computing
$2^n P$ exactly, for~$n$ large, is not reasonable.
%The canonical height is quadratic, in the sense that
%$h(mP) = m^2 h(P)$ for all integer $m$.
In \cite[\S3.4]{cremona:algs}, Cremona describes an efficient method
(due mostly to Silverman) for computing $h(P)$. One defines
\defn{local heights} $\hat{h}_p:E(\Q)\to\R$, for all primes~$p$, and
$\hat{h}_\infty:E(\Q)\to\R$ such that $$h(P) = \hat{h}_\infty(P) +
\sum \hat{h}_p(P).$$
The local heights $\hat{h}_p(P)$ are easy to
compute explicitly. For example, when $p$ is a prime of good
reduction, $\hat{h}_p(P) = \max\{0,-\ord_p(x)\}\cdot \log(p)$.
{\em This talk is {\bf NOT} about local heights $\hat{h}_p$, and we will
not mention them any further.} Instead, this talk is about a canonical global
$p$-adic height function
$$h_p : E(\Q)\to\Q_p.$$
These height functions are genuine height functions; e.g., $h_p$
is a quadratic function, i.e, $h_p(mP) = m^2 h(P)$ for all~$m$.
They appear when defining the $p$-adic regulators that appear in
$p$-adic analogues of the Birch and Swinnerton-Dyer
conjecture, in work of Mazur, Tate, Teitelbaum, Greenberg, Schneider, Perrin-Riou
and many other people.
\vspace{3ex}
\noindent{\bf Acknowledgement:} Discussions
with Mike Harrison, Nick Katz, and Christian Wuthrich. \\
This is joint work with Barry Mazur and John Tate.
\section{The $p$-Adic Cyclotomic Height Pairing}
Let $E$ be an elliptic curve over~$\Q$ and suppose $p\geq 5$ is a prime
such that $E$ has good ordinary reduction at $p$. Suppose $P\in E(\Q)$
is a point that reduces to $0\in E(\F_p)$ and to the connected
component of $\mathcal{E}_{\F_\ell}$ at all bad primes $\ell$.
We will define functions $\log_p$, $\sigma$, and $d$ below.
In terms of these functions, the $p$-adic height of $P$ is
\begin{equation}\label{eqn:heightdef}
h_p(P) = \frac{1}{p}\cdot \log_p\left(\frac{\sigma(P)}{d(P)}\right) \in \Q_p.
\end{equation}
The function $h_p$ satisfies $h_p(nP) = n^2 h_p(P)$ for all integers~$n$,
so it extends to a function on the full Mordell-Weil group $E(\Q)$.
Setting $$\langle P, Q\rangle_p = \frac{1}{2}\cdot (h_p(P+Q)-h_p(P)-h_p(Q)),$$
we obtain a pairing on
$E(\Q)_{/\tor}$, and the $p$-adic regulator is the discriminant
of this pairing (which is well defined up to sign). We have
the following standard conjecture about this height pairing.
\begin{conjecture}
The pairing $\langle -, -\rangle_p$ is nondegenerate.
\end{conjecture}
Investigations into $p$-adic analogues of the Birch and
Swinnerton-Dyer conjecture for curves of positive rank inevitably lead
to questions about this height pairings, which motivate our interest
in computing it.
% \begin{remark}
% There is also an anticyclotomic height pairing on
% $E(K)\times E(K)$, where $K$ is a quadratic imaginary field.
% It has been studied by Perrin-Riou, Bernardi,
% \end{remark}
We now define each of the undefined quantities in
(\ref{eqn:heightdef}). The function $\log_p:\Q_p^* \to \Q_p$ is the
unique group homomorphism with $\log_p(p)=0$ that extends the homomorphism
$\log_p:1+p\Z_p \to \Q_p$ defined by the usual power series of $\log(x)$ about $1$. Thus
if $x\in\Q_p^*$, we have
$$\log_p(x) = \frac{1}{p-1}\cdot \log_p(u^{p-1}),$$
where $u =
p^{-\ord_p(x)} \cdot x$ is the unit part of~$x$, and the usual
series for $\log$ converges on $u^{p-1}$.
The denominator $d(P)$ is the positive square root of the denominator of the
$x$-coordinate of~$P$.
The $\sigma$ function is the most mysterious quantity in
(\ref{eqn:heightdef}), and it turns out the mystery is closely related
to the difficulty of computing the $p$-adic number $\E_2(E,\omega)$,
where $\E_2$ is the $p$-adic weight $2$ Eisenstein series. There are
{\em many} ways to define or characterize $\sigma$, e.g.,
\cite{mazur-tate:sigma} contains $11$ different characterizations!
Let $$x(t) = \frac{1}{t^2} + \cdots \in \Z((t))$$
be the formal power
series that expresses $x$ in terms of $t=-x/y$ locally near $0\in E$.
Then Mazur and Tate prove there is exactly one function $\sigma(t)\in
t\Z_p[[t]]$ and constant $c\in \Z_p$ that satisfy the equation
\begin{equation}\label{eqn:sigmadef}
x(t)
+ c = -\frac{d}{\omega}\left( \frac{1}{\sigma}
\frac{d\sigma}{\omega}\right).
\end{equation}
This defines $\sigma$, and,
unwinding the meaning of the expression on the right, it leads to an
algorithm to compute $\sigma(t)$ to any desired precision,
which we now sketch.
If we expand (\ref{eqn:sigmadef}), we can view $c$ as a formal
variable and solve for $\sigma(t)$ as a power series with coefficients
that are polynomials in $c$. Each coefficient of $\sigma(t)$ must be
in $\Z_p$, so when there are denominators in the polynomials in $c$,
we obtain conditions on $c$ modulo powers of $p$. Taking these
together for {\em many} coefficients eventually yields enough information to
get $c\pmod{p^n}$, for a given $n$, hence $\sigma(t) \pmod{p^n}$.
However, this algorithm is {\em extremely inefficient} and its
complexity is unclear. Cristian Wuthrich, who has probably done
more computations with this method than anyone else (and has a nice
PARI implementation), told me the
following in email (Oct 2004):
\begin{quote}
``I believe that in the integrality algorithm, approximately $p^n$
coefficients of the sigma function have to be computed to get $c$ up to
$p^n$ (which gives the height up to $p^{n+1}$). i.e. it is hopelessly
ineffective for $p>100$.''
\end{quote}
For the last 15 or 20 years, the above unsatisifactory algorithm has
been the standard one for computing $p$-adic heights, e.g., when
investigating $p$-adic analogues of the BSD conjecture.
\begin{center}
{\em Due to a fortuitous combination of events, the
situation recently improved...}
\end{center}
\section{Using Cohomology to Compute $\sigma$}
Suppose that $E$ is an elliptic curve over $\Q$ given by a Weierstrass equation
$$
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.
$$
Let $x(t)$ be the formal series as before, and set
$$\wp(t) = x(t) + \frac{a_1^2 + 4a_2}{12}\in\Q((t)).$$
One can show that the function $\wp$ satisfies $(\wp')^2 = 4\wp^3 - g_2 \wp - g_3$, etc.; it's
the analogue of the usual complex $\wp$-function.
In \cite{mazur-tate:sigma}, Mazur and Tate prove that
$$
x(t) + c = \wp(t) + \frac{1}{12}\cdot \E_2(E,\omega),
$$
where $\E_2(E,\omega)$ is the value of the Katz $p$-adic
weight~$2$ Eisenstein series at $(E,\omega)$, and the equality is of
elements of $\Q_p((t))$. Thus computing the mysterious $c$ is equivalent
to computing the $p$-adic number $\E_2(E,\omega)\in\Z_p$.
``The'' weight~$2$ Eisenstein appears in many ways.
In the context of clssical modular forms, the function
$$
E_2(z) = 1 - 24\sum_{n=1}^{\infty} \sigma_1(n) q^n
$$
is holomorphic on $\h$, but is not a modular form of
level~$1$. There exists a nonzero constant~$A$ such that
$$
F_2(z) = E_2(z) + \frac{A}{\pi y} \qquad\qquad y = \Im(z)
$$
is not holomorphic, but one can show that it
transforms like a modular form of level~$1$ and weight~$2$.
Thus for any integer $N>1$, the difference
$$F_2(z) - N F_2(Nz) = E_2(z) - N E_2(Nz)$$
is a modular form for~$\Gamma_0(N)$.
However, in the context of Katz's $p$-adic modular forms (i.e.,
functions on pairs $(E,\omega)$),
there is a $p$-adic Eisenstein series
$\E_2$ of level~$1$. It's $q$-expansion is
$$
\E_2(\Tate(q),\omega_{\can}) = 1 - 24\sum_{n=1}^{\infty} \sigma_1(n)q^n,
$$
where $\Tate(q)$ is the Tate curve over $\Q_p$
with parameter $q$ and $\omega_{\can}$ is the canonical
nonvanishing differential on the Tate curve.
This summer, Mazur, Tate, and I explored many ideas for computing
$\E_2(E,\omega)$ explicitly, where $E$ is a curve with good ordinary
reduction at~$p$.
Perhaps the difficulty of computing $\E_2(E,\omega)$ is somehow intrinsic
to the theory?
%There are many strategies for trying to compute
%$\E_2(E,\omega)$, and many lead to frustratingly slow algorithms.
\subsection{Katz's Email}
This section contains an email that Nick Katz sent out in
response to a query from Barry Mazur. It is the basis of the algorithm
we will describe later.
\begin{verbatim}
Date: Thu, 8 Jul 2004 13:53:13 -0400
From: Nick Katz
Subject: Re: convergence of the Eisenstein series of weight two
To: mazur@math.harvard.edu, nmkatz@Math.Princeton.EDU
Cc: tate@math.utexas.edu, was@math.harvard.edu
\end{verbatim}
(I have edited the email below, to better fit the style of these
notes.)
It seems to me you want to use the interpretation of $P=\E_2$ as the
``{\em direction of the unit root subspace}''; that should make it fast to
compute. Concretely, suppose we have a pair $(E, \omega)$ over $\Z_p$, and
to fix ideas $p$ is not $2$ or $3$. Then we write a Weierstrass
equation for $E$,
$$
y^2 = 4x^3 - g_2x - g_3,
$$
so that $\omega=dx/y$, and we denote by
$\eta$ the differential $xdx/y$.
Then $\omega$ and $\eta$ form a $\Z_p$ basis of
$$\H^1 = \H^1_{\dR},$$ and
the key step is to compute the matrix of
absolute Frobenius. Here this map is $\Z_p$-linear, since
we are working over $\Z_p$; otherwise, if we were working
over the Witt vectors of an $\F_q$, the map would only
be $\sigma$-linear.
This calculation goes fast, because the matrix of Frobenius lives
over the entire $p$-adic moduli space, and we are back in the {\bf glory days
of Washnitzer-Monsky cohomology} (of the open curve $E - {\cO}$).
Okay, now suppose we have computed the matrix of $\Frob$ in the
basis $\omega, \eta$. The unit root subspace is a direct factor, call
it $U$, of the $\H^1$, and we know that a complimentary direct factor is
the $\Z_p$ span of $\omega$.
We also know that $\Frob(\omega)$ lies
in $p\H^1$, and this tells us that, $\!\!\!\!\mod{p^n}$,
the subspace~$U$ is the span of
$\Frob^n(\eta)$.
What this means concretely is that if we write,
for each $n$,
$$
\Frob^n(\eta) = a_n\omega + b_n\eta,
$$
then $b_n$ is a unit (congruent modulo $p$ to the $n$th power of the
Hasse invariant) and that $P$ is $-12a_n/b_n$.
See my Antwerp appendix and also my paper {\em $p$-adic
interpolation of real analytic Eisenstein series}.
So in terms of speed of convergence, {\em once} you have $\Frob$, you
have to iterate it $n$ times to calculate $P \pmod{p^n}$.
\subsection{The Algorithms}
The following algorithms culminate in an algorithm for computing
$h_p(P)$ that incorporates Katz's ideas with the discussion elsewhere
in this talk. I have computed $\sigma$ and $h_p$ in numerous
cases using the algorithm described below, and using my
implementations of the ``integrality'' algorithm described above
and also Wuthrich's algorithm, and the results match.
Tate has also done several computations of $h_p$ using other
methods, and again the results match.
Note: The analysis
of some of the necessary precision is not complete below.
Kedlaya's algorithm is an algorithm for computing zeta functions
of hyperelliptic curves over finite fields. An intermediate step
in his algorithm is computation of the matrix of absolute Frobenius
on $p$-adic de Rham cohomology. In Kedlaya's papers, he determines
the precision of various objects needed to compute this matrix
to a given precision.
The first algorithm computes the value $\E_2(E,\omega)$ using
Kedlaya's algorithm and the method suggested by Katz in the email
above.
\begin{algorithm}{Evaluation of $\E_2(E,\omega)$}\label{alg:e2}
Given an elliptic curve over~$\Q$ and prime~$p$, this algorithm
computes $\E_2(E,\omega)\in \Q_p$. We
assume that Kedlaya's algorithm is available for computing a
presentation of the $p$-adic Monsky-Washnitzer cohomology of
$E-\{\cO\}$ with Frobenius action.
\begin{steps}
\item Let $c_4$ and $c_6$ be the $c$-invariants of a minimal model
of~$E$. Set
$$a_4\set -\frac{c_4}{2^4\cdot 3}\qquad\text{and}\qquad
a_6 \set -\frac{c_6}{2^5\cdot 3^3}.$$
\item Apply Kedlaya's algorithm to the hyperelliptic curve
$y^2=x^3 + a_4x + a_6$ (which is isomorphic to $E$) to obtain the matrix
$M$ of the action of absolute Frobenius
on the basis
$$\omega=\frac{dx}{y}, \qquad \eta=\frac{xdx}{y}$$
to precision $O(p^n)$. We view $M$ as acting
from the left.
\item
We know $M$ to precision $O(p^n)$.
Compute the $n$th power of $M$ and let
$\vtwo{a}{b}$ be the second column of $M^n$.
Then $\Frob^n(\eta) = a\omega + b\eta$.
\item Output $M$ and $-12a/b$ (which is $\E_2(E,\omega)$), then terminate.
\end{steps}
\end{algorithm}
The next algorithm uses Algorithm~\ref{alg:e2} to compute $\sigma(t)$.
\begin{algorithm}{The Canonical $p$-adic Sigma Function}\label{alg:sigma}
Given an elliptic curve~$E$ and a good ordinary prime~$p$, this
algorithm computes $\sigma(t)\in\Z_p[[t]]$ modulo $(p^n, t^m)$ for
any given positive integers $n,m$.
%(I have {\em not} figured out
% exactly what precision each object below must be computed to.)
\begin{steps}
\item Using Algorithm~\ref{alg:e2}, compute $e_2 = \E_2(E,\omega)\in
\Z_p$ to precision $O(p^n)$.
\item Compute the formal expansion of $x = x(t) \in \Q[[t]]$
in terms of the local parameter $t=-x/y$ at infinity
to precision $O(t^m)$.
\item Compute the formal logarithm $z(t)=t + \cdots \in \Q((t))$ to precision
$O(t^m)$ using that $$\ds z(t) = \int \frac{dx/dt}{(2y(t)+a_1x(t) + a_3)},$$
where $x(t)=t/w(t)$ and $y(t)=-1/w(t)$ are the formal $x$
and $y$ functions, and $w(t)$ is given by the explicit inductive
formula in \cite[Ch.~7]{silverman:aec}. (Here $t=-x/y$ and $w=-1/y$ and
we can write $w$ as a series in $t$.)
\item Using a power series ``reversion'' (functional inverse)
algorithm, find the unique power series $F(z)\in\Q[[z]]$ such that
$t=F(z)$. Here $F$ is the reversion of $z$, which exists because
$z(t) = t + \cdots$.
\item Set $\wp(t) \set x(t) + (a_1^2 + 4a_2)/12 \in \Q[[t]]$ (to precision
$O(t^m)$), where the
$a_i$ are the coefficients of the Weierstrass equation of $E$.
Then compute the series $\wp(z) = \wp(F(z))\in \Q((z))$.
\item Set $\ds g(z)\set \frac{1}{z^2} - \wp(z) + \frac{e_2}{12}\in\Q_p((z))$.
[Warning:
The theory suggests the last term should be $-e_2/12$ but the calculations do not
work unless I use the above formula. There are probably two
normalizations of $E_2$ in the references.]
\item Set
$\ds \sigma(z) \set z\cdot \exp\left(\int \int g(z) \dz \dz\right)
\in \Q_p[[z]]$.
\item Set $\sigma(t) \set \sigma(z(t))\in t\cdot \Z_p[[t]]$, where $z(t)$
is the formal logarithm computed above. Output $\sigma(t)$
and terminate.
\end{steps}
\end{algorithm}
\begin{remark}
The trick of changing from $\wp(t)$ to $\wp(z)$ is essential so that
we can solve a certain differential equation using just operations
with power series.
\end{remark}
The final algorithm uses $\sigma(t)$ to compute the $p$-adic height.
\begin{algorithm}{$p$-adic Height}
Given an elliptic curve~$E$ over $\Q$, a good ordinary prime~$p$,
and an element $P\in E(\Q)$, this algorithm computes the
$p$-adic height $h_p(P) \in \Q_p$ to precision $O(p^n)$.
%(I will ignore the precision below.)
\begin{steps}
\item{}[Prepare Point] Compute an integer $m$ such that
$mP$ reduces to $\cO\in E(\F_p)$ and to the connected
component of $\mathcal{E}_{\F_\ell}$ at all bad primes $\ell$.
For example,~$m$ could be the least common multiple of the Tamagawa numbers
of $E$ and $\#E(\F_p)$. Set $Q\set mP$ and write $Q=(x,y)$.
\item{}[Denominator] Let $d$ be the positive integer square root of the
denominator of $x$.
\item{}[Compute $\sigma$] Compute $\sigma(t)$ using
Algorithm~\ref{alg:sigma}, and set $s \set \sigma(-x/y) \in \Q_p$.
\item{}[Logs] Compute
$\ds h_p(Q) \set \frac{1}{p}\log_p\left(\frac{s}{d}\right)$, and
$\ds h_p(P) \set \frac{1}{m^2} \cdot h_p(Q)$. Output $h_p(P)$ and terminate.
\end{steps}
\end{algorithm}
\section{Future Directions}
In this section we discuss various directions for future
investigation.
\subsection{Log Convergence}
Suppose $E_t$ is an elliptic curves over $\Q(t)$. It might be
interesting to obtain formula for $\E_2(E_t)$ as
an element of $\Q_p((t))$. This might shed light on the
analytic behavior of the $p$-adic modular form $\E_2$, and on Tate's
recent experimental observations about the behavior of the
$(1/j)$-expansion of the weight~$0$ modular function $\E_2 E_4/E_6$.
More precisely, Tate computed the expansion of $\E_2 E_4/E_6$
in powers of $1/j$ for $p=2,3,5$, and observed very
slow convergence. The rest of this
section is very closely based on an email from Tate about
his observation.
Here's a very small result concerning the $p$-adic nature of $\E_2$
for $p=2,3,5$. For the primes $p\leq 5$ we can test the convergence of
a weight $0$ level $1$ $p$-adic modular function~$f$ (with poles only
at infinity) by expanding in powers of $z=1/j$. Say
$f=\sum_{n=1}^{\infty} a_nz^n$. If $f=zdg/dz$ for some formal series
$g=\sum b_nz^n$ with $p$-integral coefficients $b_n$, then $a_n =
nb_n$, so for example $a_{p^m}=p^mb_{p^m}$ is divisible by $p^m$,
which is a tiny hint of~$f$ having ``logarithmic'' $p$-adic
convergence.
\begin{theorem}
The form
$$
f=\frac{E_2 E_4}{E_6} - 1
$$
has this property, with $g=3\log(E_4)$
divisible by $720$ in $\Z_2$, $\Z_3$ and $\Z_5$.
\end{theorem}
I leave the proof as an exercise. The idea is that by well-known
formulas, if $P=E_2$, $Q=E_4$, and $R=E_6$, then
$$
q\frac{dg}{dq} = 3q\frac{d\log(Q)}{dq} = 3q\frac{dQ}{Qdq} = P - \frac{R}{Q}
$$
and
$$
q\frac{dz}{zdq} = \frac{R}{Q}.
$$
Now divide the first equality by the second to get the result. Note
that for $p=2$ and~$3$, the result for $n=p^m$ seems just right. For
$f=PQ/R$,
it gives
$$
v_2(a_{2^m}) \geq m+v_2(720)=m+4,
$$
and similarly
$$
v_3(a_{3^m}) \geq m+2,
$$
and those inequalities are equalities for $2^m$ and $3^m <200$.
For the record, in case it might give a clue to what is going on,
experimentally we have, for $n<200$:
$$
v_2(a_n) = l_2(n) + 3s_2(n),
$$
where $l_2(n)=1+\lfloor \log_2(n) \rfloor$
and $s(n)$ is the sum of the digits of $n$ written
in base $2$. Similarly for $n<200$,
$$
v_3(a_n) = l_3(n) + s_3(n).
$$
For $p=5$ it seems that at least $v_5(a_n)\geq l_5(n)$; in fact, even
$$
v_5(a_n) \geq l_5(2n),
$$
with likely equality for $2n=5^m-1$ and $5^m+1$.
\subsection{Connections with $p$-adic Birch and Swinnerton-Dyer}
It would also be interesting to do many more computations in support of
$p$-adic analogues of the BSD conjectures of \cite{mtt}, especially
when $E/\Q$ has large rank. Substantial theoretical work has been
done toward these $p$-adic conjectures, and this work may be useful to
algorithms for computing information about Shafarevich-Tate and Selmer
groups of elliptic curves. For example, in \cite{pr:exp}, Perrin-Riou
uses her results about the $p$-adic BSD conjecture in the
supersingular case to prove that $\Sha(E/\Q)[p]=0$ for certain~$p$ and
elliptic curves~$E$ of rank $>1$, for which the work of Kolyvagin and
Kato does not apply. Mazur and Rubin (with my computational input)
are also obtaining results that could be viewed as fitting into this
program.
I have been involved with Andrei Jorza and Stephen Patrikas on a
project to verify the full Birch and Swinnerton-Dyer conjecture for
all elliptic curves of conductor $\leq 1000$ and rank $\leq 1$. There
are many examples in which the rank is $1$ and the upper bound coming
from Kolyvagin's Euler system is divisible by a prime $p\geq 7$, which
also divides a Tamagawa number. The results of Kolyvagin and Kato do
not give a sufficiently tight upper bound on $\Sha(E/\Q)$. However,
discussions with Greenberg, Pollack, Grigorov,
and Perrin-Riou have convinced me that it might be possible in many cases to do
appropriate computations of $p$-adic heights and derivatives of
$p$-adic $L$-functions, combined with results of Kato and Schneider,
and obtain a sufficiently strong upper bounds on $\#\Sha(E/\Q)$.
\subsection{Optimization}
I would like to optimize the implementation of the algorithm.
Probably the most time-consuming step is computation of
$\E_2(E,\omega)$ using Kedlaya's algorithm. My current implementation
uses Michael Harrison's implementation of Kedlaya's algorithm for
$y^2=f(x)$, with $f(x)$ of arbitrary degree. (Michael Harrison was a
Coates student who was in industry for many years, and is now back.)
Perhaps implementing just what is needed for elliptic curves from
Kedlaya's algorithm would be more efficient. Also, Harrison tells me
his implementation isn't nearly as optimized as it might be.
\subsection{Natural Generalizations}
\begin{enumerate}
\item It might be possible to compute $p$-adic heights on Jacobians of
hyperelliptic curves.
\item Formulate everything above over number fields, and extend to the case
of additive reduction.
\item What about when $p$ is a prime of supersingular reduction?
\end{enumerate}
\section{Examples}
In this section I show you examples of how to use the MAGMA package
I wrote for computing with $p$-adic heights, and give you a sense
for how efficient it is.
\begin{verbatim}
> function EC(s) return EllipticCurve(CremonaDatabase(),s); end function;
> E := EC("37A");
> Attach("kedlaya.m"); // get this from me
> Attach("padic_height.m"); // get this from me
> P := good_ordinary_primes(E,100); P;
[ 5, 7, 11, 13, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73,
79, 83, 89, 97 ]
> for p in P do time print p, regulator(E,p,10); end for;
5 22229672 + O(5^11)
Time: 0.040
7 317628041 + O(7^11)
...
89 15480467821870438719 + O(89^10)
Time: 1.190
97 -11195795337175141289 + O(97^10)
Time: 1.490
> E := EC("389A");
> P := good_ordinary_primes(E,100); P;
[ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97 ]
> for p in P do time print p, regulator(E,p,10); end for;
5 -3871266 + O(5^11)
Time: 0.260
7 483898350 + O(7^11)
...
89 9775723521676164462 + O(89^10)
Time: 1.330
97 -13688331881071698338 + O(97^10)
Time: 1.820
> E := EC("5077A");
> P := good_ordinary_primes(E,100); P;
[ 5, 7, 11, 13, 17, 19, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97 ]
> for p in P do time print p, regulator(E,p,10); end for;
5 655268*5^-2 + O(5^7)
Time: 0.800
7 -933185758 + O(7^11)
...
89 -3325438607428779200 + O(89^10)
Time: 1.910
97 -5353586908063282167 + O(97^10)
Time: 2.010
--------
> E := EC("37A");
> time regulator(E,5,50);
115299522541340178416234094637464047 + O(5^51)
Time: 1.860
> Valuation(115299522541340178416234094637464047 - 22229672,5);
9
> time regulator(E,97,50);
-5019271523950156862996295340254565181870308222348277984940964806\
97957622583267105973403430183075091 + O(97^50)
Time: 31.7
\end{verbatim}
\bibliography{biblio}
\end{document}