%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% Use latex to typeset this document. %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Documenta-LaTeX.sty (2e)
%
% Version 0.5
%
% Copyright (C) 2000-2006 Ulf Rehmann
% rehmann@mathematik.uni-bielefeld.de
%
% ------------------------------ PROLOGUE -----------------------------
%
% This prologue is to be prepended to your LaTeX file in order to
% give it the layout required by Documenta Mathematica.
% Please edit the \documentclass line below, add \usepackage lines
% according to your needs and
% go to the line starting with \Title below, fill in title,
% authors' names and so on and follow the further instructions there.
%
% Choose one of the commands \documentclass[]{} or \documentstyle[]{}
% depending on the version of LaTeX you use:
%
%
% If you use LaTeX 2e use the following line (or similar):
\documentclass[10pt,twoside]{article}
% delete if this gives you any problems.
%\usepackage[active]{srcltx}
%*************Declaration Section******************
\usepackage{float}
%\usepackage[hypertex]{hyperref}
%\usepackage[hyperindex,pdfmark]{hyperref}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\hoffset=-0.05\textwidth
\textwidth=1.10\textwidth
\voffset=-0.05\textheight
\textheight=1.10\textheight
%\bibliographystyle{amsalpha}
\newcommand{\edit}[1]{\footnote{[[#1]]}\marginpar{\hfill {\sf[[\thefootnote]]}}}
\renewcommand{\edit}[1]{}
\newcommand{\editwas}[1]{\footnote{[[#1]]}\marginpar{\hfill {\sf[[\thefootnote]]}}}
\renewcommand{\editwas}[1]{}
\newcommand{\an}{{\rm an}}
\newcommand{\e}{\mathbf{e}}
\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
\newcommand{\textcyr}[1]{%
{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
\selectfont #1}}
\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
\newcommand{\Gam}{\Gamma}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\ds}{\displaystyle}
\newcommand{\M}{\mathcal{M}}
\newcommand{\E}{\mathbf{E}}
\newcommand{\q}{\mathbf{q}}
\newcommand{\ra}{\rightarrow}
\newcommand{\la}{\leftarrow}
\newcommand{\con}{\equiv}
\newcommand{\set}[1]{\{#1\}}
\newcommand{\alp}{\alpha}
\newcommand{\vtwo}[2]{\left(\begin{matrix}#1\\#2\end{matrix}\right)}
\newcommand{\eps}{\varepsilon}
\newcommand{\A}{\mathbf A}
\newcommand{\Q}{\mathbf{Q}}
\newcommand{\C}{\mathbf{C}}
\newcommand{\D}{{\mathbf D}}
\newcommand{\Z}{\mathbf{Z}}
\newcommand{\F}{\mathbf{F}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\et}{\text{\rm\'et}}
\newcommand{\h}{\mathfrak{h}}
\newcommand{\p}{\mathfrak{p}}
\newcommand{\m}{\mathfrak{m}}
\renewcommand{\l}{\ell}
\renewcommand{\t}{\tau}
\renewcommand{\a}{\mathfrak{a}}
\renewcommand{\H}{\HH}
\DeclareMathOperator{\new}{new}
\DeclareMathOperator{\tor}{tor}
\DeclareMathOperator{\Frob}{Frob}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\ab}{ab}
\DeclareMathOperator{\Fr}{Fr}
\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\SL}{SL}
\renewcommand{\set}{=}
\newcommand{\ggeq}{\,\,\,\underline{\gg}\,\,\,}
\newcommand{\isom}{\cong}
%%%% Theoremstyles
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{hypothesis}[theorem]{Hypothesis}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{question}[theorem]{Question}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{alg}[theorem]{Algorithm}
\newtheorem{openproblem}[theorem]{Open Problem}
%\theoremstyle{remark}
\newtheorem{goal}[theorem]{Goal}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{remarks}[theorem]{Remarks}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newenvironment{algorithm}[1]{%
\begin{alg}[#1]\index{algorithm!#1}\sf
}%
{\end{alg}}
\newenvironment{steps}%
{\begin{enumerate}\setlength{\itemsep}{0.1ex}}{\end{enumerate}}
\DeclareMathOperator{\HH}{H}
\renewcommand{\H}{\HH}
\numberwithin{equation}{section}
\numberwithin{figure}{section}
\numberwithin{table}{section}
% If you use LaTeX 2.09 or earlier use the following line (or similar):
%\documentstyle[10pt,twoside]{article}
%
% Please go to the line starting with \Title below, fill in title,
% authors' names and so on and follow the further instructions there.
%
% The final values of the following 4 uncommented lines will be filled in
% by Doc. Math. after the manuscript has been accepted for publication.
%
\def\YEAR{\year}\newcount\VOL\VOL=\YEAR\advance\VOL by-1995
\def\firstpage{585}\def\lastpage{622}
\def\received{September 9, 2005}\def\revised{May 2, 2006}
\def\communicated{}
%
\makeatletter
\def\magnification{\afterassignment\m@g\count@}
\def\m@g{\mag=\count@\hsize6.5truein\vsize8.9truein\dimen\footins8truein}
\makeatother
%%% Choose 10pt/12pt:
%\oddsidemargin1.66cm\evensidemargin1.66cm\voffset1.2cm%10pt
\oddsidemargin1.91cm\evensidemargin1.91cm\voffset1.4cm%10pt
%\magnification1200\oddsidemargin.41cm\evensidemargin.41cm\voffset-.75cm%12pt
%\textwidth12.5cm\textheight19.5cm
\textwidth12.0cm\textheight19.0cm
\font\eightrm=cmr8
\font\caps=cmcsc10 % Theorem, Lemma etc
\font\ccaps=cmcsc10 % Sections
\font\Caps=cmcsc10 scaled \magstep1 % Title
\font\scaps=cmcsc8
%
%-----------------headlines-----------------------------------
%\input german.sty % uncomment if necessary
%\usepackage{german} % uncomment if necessary
\pagestyle{myheadings}
\pagenumbering{arabic}
\setcounter{page}{\firstpage}
\def\bfseries{\normalsize\caps}
\makeatletter
\setlength\topmargin {14\p@}
\setlength\headsep {15\p@}
\setlength\footskip {25\p@}
\setlength\parindent {20\p@}
\@specialpagefalse\headheight=8.5pt
\def\DocMath{}
\def\DocMath{{\def\th{\thinspace}\scaps Documenta Math.}}
\renewcommand{\@oddfoot}{\hfill\scaps Documenta Mathematica
$\cdot$ Extra Volume Coates (\number\YEAR) \number\firstpage--\lastpage\hfill}
\renewcommand{\@evenfoot}{\ifnum\thepage>\lastpage\hfill\scaps
Documenta Mathematica $\cdot$ Extra Volume Coates (\number\YEAR)
\hfill\else\@oddfoot\fi}%
\renewcommand{\@evenhead}{%
\ifnum\thepage>\lastpage\rlap{\thepage}\hfill%
\else\rlap{\thepage}\slshape\leftmark\hfill{\caps\SAuthor}\hfill\fi}%
\renewcommand{\@oddhead}{%
\ifnum\thepage=\firstpage{\DocMath\hfill\llap{\thepage}}%
\else{\slshape\rightmark}\hfill{\caps\STitle}\hfill\llap{\thepage}\fi}%
\makeatother
\def\TSkip{\bigskip}
\newbox\TheTitle{\obeylines\gdef\GetTitle #1
\ShortTitle #2
\SubTitle #3
\Author #4
\ShortAuthor #5
\EndTitle
{\setbox\TheTitle=\vbox{\baselineskip=20pt\let\par=\cr\obeylines%
\halign{\centerline{\Caps##}\cr\noalign{\medskip}\cr#1\cr}}%
\copy\TheTitle\TSkip\TSkip%
\def\next{#2}\ifx\next\empty\gdef\STitle{#1}\else\gdef\STitle{#2}\fi%
\def\next{#3}\ifx\next\empty%
\else\setbox\TheTitle=\vbox{\baselineskip=20pt\let\par=\cr\obeylines%
\halign{\centerline{\caps##} #3\cr}}\copy\TheTitle\TSkip\TSkip\fi%
%\setbox\TheTitle=\vbox{\let\par=\cr\obeylines%
%\halign{\centerline{\caps##} #4\cr}}\copy\TheTitle\TSkip\TSkip%
\centerline{\caps #4}\TSkip\TSkip%
\def\next{#5}\ifx\next\empty\gdef\SAuthor{#4}\else\gdef\SAuthor{#5}\fi%
\ifx\received\empty\relax
\else\centerline{\eightrm Received: \received}\fi%
\ifx\revised\empty\TSkip%
\else\centerline{\eightrm Revised: \revised}\TSkip\fi%
\ifx\communicated\empty\relax
\else\centerline{\eightrm Communicated by \communicated}\fi\TSkip\TSkip%
\catcode'015=5}}\def\Title{\obeylines\GetTitle}
\def\Abstract{\begingroup\narrower
\parskip=\medskipamount\parindent=0pt{\caps Abstract. }}
\def\EndAbstract{\par\endgroup\TSkip}
\long\def\MSC#1\EndMSC{\def\arg{#1}\ifx\arg\empty\relax\else
{\par\narrower\noindent%
2000 Mathematics Subject Classification: #1\par}\fi}
\long\def\KEY#1\EndKEY{\def\arg{#1}\ifx\arg\empty\relax\else
{\par\narrower\noindent Keywords and Phrases: #1\par}\fi\TSkip}
\newbox\TheAdd\def\Addresses{\vfill\copy\TheAdd\vfill
\ifodd\number\lastpage\vfill\eject\phantom{.}\vfill\eject\fi}
{\obeylines\gdef\GetAddress #1
\Address #2
\Address #3
\Address #4
\EndAddress
{\def\xs{6.0truecm}\parindent=0pt
\setbox0=\vtop{{\obeylines\hsize=\xs#1\par}}\def\next{#2}
\ifx\next\empty % 1 address
\setbox\TheAdd=\hbox to\hsize{\hfill\copy0\hfill}
\else\setbox1=\vtop{{\obeylines\hsize=\xs#2\par}}\def\next{#3}
\ifx\next\empty % 2 addresses
\setbox\TheAdd=\hbox to\hsize{\hfill\copy0\hfill\copy1\hfill}
\else\setbox2=\vtop{{\obeylines\hsize=\xs#3\par}}\def\next{#4}
\ifx\next\empty\ % 3 addresses
\setbox\TheAdd=\vtop{\hbox to\hsize{\hfill\copy0\hfill\copy1\hfill}
\vskip20pt\hbox to\hsize{\hfill\copy2\hfill}}
\else\setbox3=\vtop{{\obeylines\hsize=\xs#4\par}}
\setbox\TheAdd=\vtop{\hbox to\hsize{\hfill\copy0\hfill\copy1\hfill}
\vskip20pt\hbox to\hsize{\hfill\copy2\hfill\copy3\hfill}}
\fi\fi\fi\catcode'015=5}}\gdef\Address{\obeylines\GetAddress}
\hfuzz=0.1pt\tolerance=2000\emergencystretch=20pt\overfullrule=5pt
\begin{document}
%%%%% ------------- fill in your data below this line -------------------
%%%%% The following lines \Title ... \EndAddress must ALL be present
%%%%% and in the given order.
\Title
%%%%% Put here the title. Line breaks will be recognized.
Computation of {\sl \large p}-Adic Heights and Log Convergence
\ShortTitle
$p$-Adic Heights and Log Convergence
%%%%% Running title for odd numbered pages, ONE line, please.
%%%%% If none is given, \Title will be used instead.
\SubTitle
%%%%% A possible subtitle goes here.
{\small In celebration of John Coates' 60th birthday}
\Author
%%%%% Put here name(s) of authors. Line breaks will be recognized.
Barry Mazur, William Stein\footnote{This material is based upon work %
supported by the National Science Foundation under Grant %
No. 0555776.}, John Tate
\ShortAuthor
Mazur, Stein, Tate
%%%%%% Running title for even numbered pages, ONE line, please.
%%%%%% If none is given, \Author will be used instead.
\EndTitle
\Abstract
%%%%% Put here the abstract of your manuscript.
%%%%% Avoid macros and complicated TeX expressions, as this is
%%%%% automatically translated and posted as an html file.
This paper is about computational and theoretical questions regarding
$p$-adic height pairings on elliptic curves over a global field $K$.
The main stumbling block to computing them efficiently is in
calculating, for each of the completions $K_v$ at the places~$v$
of~$K$ dividing~$p$, a {\it single quantity}: the value of the
$p$-adic modular form ${\bf E}_2$ associated to the elliptic curve.
Thanks to the work of Dwork, Katz, Kedlaya, Lauder and
Monsky-Washnitzer we offer an efficient algorithm for computing these
quantities, i.e., for computing the value of ${\bf E}_2$ of an
elliptic curve. We also discuss the $p$-adic convergence rate of
canonical expansions of the $p$-adic modular form~${\bf E}_2$ on the
Hasse domain. In particular, we introduce a new notion of log
convergence and prove that $\E_2$ is log convergent.
\EndAbstract
\MSC
%%%%% 2000 Mathematics Subject Classification:
11F33, 11Y40, 11G50
\EndMSC
\KEY
%%%%% Keywords and Phrases:
$p$-adic heights, algorithms, $p$-adic modular forms, Eisenstein series, sigma-functions
\EndKEY
%%%%% All 4 \Address lines below must be present. To center the last
%%%%% entry, no empty lines must be between the following \Address
%%%%% and \EndAddress lines.
%%%%% Address of first Author here
\Address
Barry Mazur
Department of Mathematics
Harvard University
mazur@math.harvard.edu
\Address
John Tate
Department of Mathematics
University of Texas at Austin
tate@math.utexas.edu
\Address
William A. Stein
Department of Mathematics
University of California at San Diego
wstein@ucsd.edu
\Address
\EndAddress
%%
%% Make sure the last tex command in your manuscript
%% before the first \end{document} is the command \Addresses
%%
%%---------------------Here the prologue ends---------------------------------
%%--------------------Here the manuscript starts------------------------------
\parindent15pt
\section{Introduction}\label{sec:intro}
Let $p$ be an odd prime number, and $E$ an elliptic curve over a
global field $K$ that has good ordinary reduction at $p$. Let $L$ be
any (infinite degree) Galois extension with a continuous injective
homomorphism $\rho$ of its Galois group to $\Q_p$. To the data
$(E,K,\rho)$, one associates\footnote{See
\cite{mazur-tate:canonical}, \cite{schneider:height1}
\cite{schneider:height2}, \cite{MR1042777}, \cite{MR1091621},
\cite {MR1263527}, \cite{MR1299736}, \cite{MR2021039},
and \cite{MR2076563}.} a canonical (bilinear, symmetric)
($p$-adic) height pairing
$$
( \ ,\ )_{\rho} : E(K)\times E(K)
\longrightarrow \Q_p.
$$
Such pairings are of great
interest for the arithmetic of~$E$ over~$K$, and they arise
specifically in $p$-adic analogues of the Birch and Swinnerton-Dyer
conjecture.\footnote{See \cite{schneider:height1}, \cite{schneider:height2}
\cite{mazur-tate:canonical}, \cite{mazur-tate:refined},
\cite{perrin-riou:expmath}.
See also the important recent work of Jan Nekov{\'a}{\v{r}}
\cite{nekovar:selmer}.}
The goal of this paper is to discuss some computational questions
regarding $p$-adic height pairings. The main stumbling block to
computing them efficiently is in calculating, for each of the
completions $K_v$ at the places~$v$ of~$K$ dividing~$p$, the value of
the $p$-adic modular form ${\bf E}_2$ associated to the elliptic curve
with a chosen Weierstrass form of good reduction over $K_v$.
We shall offer an algorithm for computing these quantities, i.e., for
computing the value of ${\bf E}_2$ of an elliptic curve (that builds on
the works of Katz and Kedlaya listed in our bibliography) and we also
discuss the $p$-adic convergence rate of canonical expansions of the
$p$-adic modular form ${\bf E}_2$ on the Hasse domain, where for $p\ge
5$ we view ${\bf E}_2$ as an infinite sum of classical modular forms
divided by powers of the (classical) modular form ${\bf E}_{p-1}$,
while for $p\le 5$ we view it as a sum of classical modular forms
divided by powers of~${\bf E}_4$.
We were led to our fast method of computing $\E_2$ by our realization
that the more naive methods, of computing it by integrality or by
approximations to it as function on the Hasse domain, were not
practical, because the convergence is ``logarithmic'' in the sense
that the $n$th convergent gives only an accuracy of~$\log_p(n)$.
We make this notion of log convergence precise in Part~\ref{part2},
where we also prove that ${\bf E}_2$ is log convergent.
The reason why this constant ${\bf E}_2$ enters the calculation is
because it is needed for the computation of the $p$-adic sigma
function \cite{mazur-tate:sigma}, which in turn is the critical
element in the formulas for height pairings.
For example, let us consider the {\it cyclotomic} $p$-adic height
pairing in the special case where $K=\Q$ and $p\geq 5$.
If $G_{\Q}$ is the Galois group of an algebraic closure of $\Q$ over
$\Q$, we have the natural surjective continuous homomorphism $\chi:
G_{\Q} \to \Z_p^*$ pinned down by the standard formula $g(\zeta) =
\zeta^{\chi(g)}$ where $g \in G_{\Q}$ and $\zeta$ is any $p$-power
root of unity. The $p$-adic logarithm $\log_p:\Q_p^* \to (\Q_p,+)$ 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 $1$. Explicitly, if $x\in\Q_p^*$, then
$$\log_p(x) = \frac{1}{p-1}\cdot \log_p(u^{p-1}),$$
where $u =
p^{-\ord_p(x)} \cdot x$ is the unit part of~$x$, and the usual
series for $\log$ converges at $u^{p-1}$.
The composition $(\frac{1}{p}\cdot \log_p)\circ \chi$ is a cyclotomic
linear functional $G_{\Q} \to \Q_p$ which, in the body of our text,
will be dealt with (thanks to class field theory) as the idele class
functional that we denote $\rho_{\Q}^{\rm cycl}$.
Let $\cE$ denote the N\'eron model of~$E$ over~$\Z$. Let $P\in E(\Q)$
be a non-torsion point that reduces to $0\in E(\F_p)$ and to the
connected component of $\cE_{\F_\ell}$ at all primes $\ell$ of bad
reduction for~$E$. Because $\Z$ is a unique factorization domain, any
nonzero point $P=(x(P),y(P)) \in E(\Q)$ can be written uniquely in the
form $(a/d^2, b/d^3)$, where $a,b,d \in \Z$, $\gcd(a,d)=\gcd(b,d)=1$,
and $d>0$. The function $d(P)$ assigns to $P$ this square root~$d$ of
the denominator of $x(P)$.
Here is the formula for the {\it cyclotomic} $p$-adic height of $P$,
i.e., the value of $$h_p(P) := -{\frac{1}{2}}(P,P)_p \in
\Q_p$$ where $(\ ,\ )_p$ is the height pairing attached to
$G_{\Q} \to \Q_p$, the cyclotomic linear functional described above:
\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}
Here $\sigma = \sigma_p$ is the $p$-adic sigma function of
\cite{mazur-tate:sigma} associated to the pair $(E,\omega)$.
\edit{I added this entire paragraph.}
The $\sigma$-function depends only on $(E,\omega)$ and not on a choice of
Weierstrass equation, and behaves like a modular form of weight $-1$, that
is $\sigma_{E,c\omega} =c\cdot \sigma_{E,\omega}$. It is ``quadratic''
the sense that for any $m\in\Z$ and point~$Q$ in the
formal group $E^f(\overline{\Z}_p)$,
we have
\begin{equation}\label{eqn:quad}
\sigma(mQ) = \sigma(Q)^{m^2} \cdot f_m(Q),
\end{equation}
where $f_m$ is the $m$th division polynomial of~$E$ relative
to~$\omega$ (as in \cite[App.~1]{mazur-tate:sigma}).
The $\sigma$-function is ``bilinear''\label{page:bil} in that
for any $P,Q \in E^f(\Z_p)$, we have
\begin{equation}\label{eqn:bil}
\frac{\sigma(P-Q)\cdot \sigma(P+Q)}{\sigma^2(P)\cdot \sigma^2(Q)}
= x(Q) - x(P).
\end{equation}
See \cite[Thm.~3.1]{mazur-tate:sigma} for proofs of the above
properties of~$\sigma$.
The height function~$h_p$ of (\ref{eqn:heightdef}) extends uniquely to
a function on the full Mordell-Weil group $E(\Q)$ that satisfies
$h_p(nQ) = n^2 h_p(Q)$ for all integers~$n$ and $Q \in E(\Q)$. For
$P,Q \in E(\Q)$, setting
%\edit{William: Barry, you had $-h_p(Q)$,
%and I changed it to $h_p(Q)$. Yes, you re right (B.M.)}
$$( P, Q )_p = h_p(P)+h_p(Q) -h_p(P+Q),$$
we obtain a pairing on $E(\Q)$. The {\em $p$-adic regulator} of $E$
is the discriminant of the induced pairing on $E(\Q)_{/\tor}$ (well
defined up to sign), and we have the following standard conjecture
about this height pairing.
\begin{conjecture}
The cyclotomic height pairing $(\ , \ )_p$ is nondegenerate; equivalently,
the $p$-adic regulator is nonzero.
\end{conjecture}
\begin{remark} Height pairings attached to other $p$-adic linear
functionals can be degenerate; in fact, given an elliptic curve
defined over $\Q$ with good ordinary reduction at $p$, and $K$ a
quadratic imaginary field over which the Mordell-Weil group $E(K)$
is of odd rank, the $p$-adic anticyclotomic height pairing for $E$
over $K$ is {\em always} degenerate.
\end{remark}
%We now give definitions of $\log_p$, $d$, and $\sigma$.
The $p$-adic $\sigma$ 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 many ways to define~$\sigma$, e.g., \cite{mazur-tate:sigma}
contains $11$ different characterizations of~$\sigma$! We now
describe a characterization that leads directly to an algorithm (see
Algorithm~\ref{alg:int}) to compute~$\sigma(t)$. Let
\begin{equation}\label{eqn:xt}
x(t) = \frac{1}{t^2} + \cdots \in \Z_p((t))
\end{equation}
be the formal power series that expresses $x$ in terms of the local
parameter $t=-x/y$ at infinity.
%\editwas{I'm worried about whether
%this is in $\Z((t))$ if the Weierstrass equation is very strange.}
The following theorem, which is proved in \cite{mazur-tate:sigma},
uniquely determines~$\sigma$ and $c$.
\begin{theorem}\label{thm:uniqde}
There is exactly one odd function $\sigma(t) = t + \cdots \in
t\Z_p[[t]]$ and constant $c\in \Z_p$ that together satisfy the
differential equation
\begin{equation}\label{eqn:sigmadef}
x(t)
+ c = -\frac{d}{\omega}\left( \frac{1}{\sigma}
\frac{d\sigma}{\omega}\right),
\end{equation}
where $\omega$ is the invariant differential
$dx/(2y+a_1x+a_3)$ associated with our chosen Weierstrass equation
for $E$.
\end{theorem}
%\edit{William: I removed the $1+$ in this formula. It should
% be $1\cdot$ but I don't think it is necessary to signal constant = $1$.}
%\editwas{I'm worried: do we need ``odd''. Also is coeff
%of $t=1$ required?}
\begin{remark}The condition that $\sigma$ is odd and that
the coefficient of $t$ is $1$ are essential.
\end{remark}
In (\ref{eqn:heightdef}),
by $\sigma(P)$ we mean $\sigma(-x/y)$, where $P=(x,y)$. We have thus
given a complete definition of $h_p(Q)$ for any point $Q \in E(\Q)$
and a prime $p\geq 5$ of good ordinary reduction for $E$.
\subsection{The $p$-adic $\sigma$-function}
The differential equation (\ref{eqn:sigmadef}) leads to a slow
algorithm to compute $\sigma(t)$ to any desired precision. This is
Algorithm~\ref{alg:int} below, which we now summarize. 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 we
obtain conditions on~$c$ modulo powers of~$p$. Taking these together
for many coefficients must eventually yield enough information to
compute $c\pmod{p^n}$, for a given $n$, hence $\sigma(t) \pmod{p^n}$.
This integrality algorithm is hopelessly slow in
general.
%inefficient if one wants $n$ or $p$ to
%be at all large. Wuthrich remarks that in his experiments,
%approximately $p^n$ coefficients of $\sigma$ have to be computed to
%get $c$ up to precision $p^n$, so this method is
Another approach to computing~$\sigma$ is to observe that, up to a
constant,~$c$ is closely related to the value of a certain $p$-adic
modular form. More precisely, suppose that $E$ is given by a (not
necessarily minimal) Weierstrass equation
\begin{equation}\label{eqn:weq}
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6,
\end{equation}
and let $\omega = dx/(2y+a_1x + a_3)$.
Let $x(t)$ be as in (\ref{eqn:xt}). Then the series
\begin{equation}\label{eqn:wpx}
\wp(t) = x(t) + \frac{a_1^2 + 4a_2}{12}\in\Q((t))
\end{equation}
satisfies
$(\wp')^2 = 4\wp^3 - g_2 \wp - g_3$.
In \cite{mazur-tate:sigma} we
find\footnote{There is a sign error in \cite{mazur-tate:sigma}.} that
\begin{equation}\label{eqn:wpe2}
x(t) + c = \wp(t) - \frac{1}{12}\cdot \E_2(E,\omega),
\end{equation}
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
$\Q_p((t))$. Using the definition of $\wp(t)$ and solving for $c$, we
find that
\begin{equation}\label{eqn:ce2}
c = \frac{a_1^2 + 4a_2}{12} - \frac{1}{12} \E_2(E,\omega).
\end{equation}
Thus computing $c$ is equivalent
to computing the $p$-adic number $\E_2(E,\omega)$.
Having computed~$c$ to some precision, we then solve
for $\sigma$ in (\ref{eqn:sigmadef}) using Algorithm~\ref{alg:sigma}
below.
%It should now be crystal clear that the nub of the computation of
%$h_p(P)$ is in computing the single number $\E_2(E,\omega)$.
%There are two attitudes towards the computation of $\E_2$:
%\begin{enumerate}
%\item A surprisingly fast procedure for computing it ``one at a
% time'', which will be explained in Part~\ref{part1} (in
%particular, see Section~\ref{sec:e2cohom}).
%\item Procedures that compute it ``all at once'' as a $p$-adic modular
%form. In Part~\ref{part2}, we will provide evidence for thinking
%that this is destined to be a rather slow undertaking.
%\end{enumerate}
\subsection{$p$-adic analogues of the Birch and Swinnerton-Dyer conjecture}
One motivation for this paper is to provide tools for doing
computations in support of $p$-adic analogues of the BSD conjectures
(see \cite{mtt}), especially when $E/\Q$ has rank at least~$2$.
%Substantial theoretical work has been done toward these conjectures,
%which maybe be useful for getting 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 \cite{mazur-rubin:large_selmer} have also obtained results
%that could be viewed as fitting into this program.
Another motivation for this work comes from the study of the fine
structure of Selmer modules. Let $K$ be a number field and $\Lambda$
the $p$-adic integral group ring of the Galois group of the maximal
${\bf Z}_p$-power extension of $K$. Making use of fundamental results
of Nekov{\'a}{\v{r}} \cite{nekovar:selmer} and of Greenberg
\cite{MR1977007} one can construct (see
\cite{mazur-rubin:large_selmer}) for certain elliptic curves defined
over $K$, a skew-Hermitian matrix with coefficients in $\Lambda$ from
which one can read off a free $\Lambda$-resolution of the canonical
Selmer $\Lambda$-module of the elliptic curve in question over $K$. To
compute the entries of this matrix modulo the square of the
augmentation ideal in $\Lambda$ one must know {\it all} the $p$-adic
height pairings of the elliptic curve over $K$. Fast algorithms for
doing this provide us with an important first stage in the
computation of free $\Lambda$-resolutions of Selmer $\Lambda$-modules.
The paper \cite{bsdalg1} is about computational verification of the
full Birch and Swinnerton-Dyer conjecture for specific elliptic
curves~$E$. There are many cases in which the rank of~$E$ is~$1$ and
the upper bound on $\#\Sha(E/\Q)$ coming from Kolyvagin's Euler system
is divisible by a prime $p\geq 5$ that also divides a Tamagawa number.
In such cases, theorems of Kolyvagin and Kato combined with explicit
computation do not give a sufficiently sharp upper bound on
$\#\Sha(E/\Q)$. However, it should be possible in these cases to
compute $p$-adic heights and $p$-adic $L$-functions, and use results
of Kato, Schneider, and others to obtain better bounds on
$\#\Sha(E/\Q)$. Wuthrich and the second author (Stein) are writing a
paper on this.
\subsection{Sample computations}
In Section~\ref{sec:sample} we illustrate our algorithms with curves
of ranks $1, 2, 3, 4 \text{ and } 5$, and two twists of $X_0(11)$ of
rank $2$.
\vspace{2em}
\noindent{\bf Acknowledgement:}
It is a pleasure to thank Nick Katz for feedback that led to
Section~\ref{sec:cohom}. We would also like to thank Mike
Harrison for discussions about his implementation of Kedlaya's
algorithm in Magma, Kiran Kedlaya for conversations about his
algorithm, Christian Wuthrich for feedback about computing $p$-adic
heights, Alan Lauder for discussions about computing $\E_2$ in
families, and Fernando Gouvea for remarks about non-overconvergence of
$\E_2$. We would also like to thank all of the above people for
comments on early drafts of the paper.
Finally, we thank Jean-Pierre Serre for the proof of Lemma~\ref{tate:lem6.6}.
\part{Heights, $\sigma$-functions, and ${\bf E}_2$}\label{part1}
\section{The Formulas }
In this section we give formulas for the $p$-adic height pairing in terms of
the $\sigma$ function. We have already done this over $\Q$ in
Section~\ref{sec:intro}. Let $p$ be an (odd) prime number,~$K$ a number field, and~$E$ an
elliptic curve over~$K$ with good ordinary reduction at all places of
$K$ above $p$. For any non-archimedean place~$w$
of~$K$, let $k_w$ denote the residue class field at~$w$.
\subsection{General global height pairings}
By the {\it idele class ${\Q}_p$-vector space} of
$K$ let us mean
$$
I(K)={\Q}_p\otimes_{\Z}\left\{{\A}_K^*/\left(K^*\cdot
\prod_{v \,\,\nmid\,\, p}{\cal O}_v^*\cdot {\rm C}\right)\right\},
$$
where ${\A}_K^*$ is the group of ideles of $K$, and $ {\rm C}$ denotes
its connected component
containing the identity. Class field theory gives us an
identification $I(K) =\Gamma(K)\otimes_{{\Z}_p}{\Q}_p$, where $
\Gamma(K)$ is the Galois group of the maximal ${\Z}_p$-power extension
of $K$. For every (nonarchimedean) place $v$ of $K$, there is a
natural homomorphism $\iota_v: K_v^* \to I(K)$.
%\edit{William: tthere are some bad breaks here, but I don't think it
% is time to worry specifically about them, since there will surely be
% further corrections and changings of the draft.}
For $K$-rational
points $\alpha, \beta \in E(K)$ we want to give explicit formulas for
an element that we might call the ``universal" $p$-adic height pairing
of~$\alpha$ and~$\beta$; denote it $(\alpha, \beta) \in I(K)$. If
$\rho: I(K) \to {\Q}_p$ is any linear functional, then the {\em
$\rho$-height pairing} is a symmetric bilinear pairing
$$(\ ,\ )_{\rho}: E(K)\times E(K) \to {\Q}_p, $$ defined as the composition
of the universal pairing with the linear functional~$\rho$:
$$
(\alpha, \beta)_{\rho} =\rho(\alpha, \beta)\in {\Q}_p.
$$
We define the {\em $\rho$-height} of a point $\alpha \in E(K)$ by:
$$h_{\rho}(\alpha) = -{\frac{1}{2}} (\alpha,
\alpha)_{\rho}\in {\Q}_p.$$
Of course, any such (nontrivial) linear functional $\rho$ uniquely
determines a ${\Z}_p$-extension, and we sometimes refer to the
$\rho$-height pairing in terms of this ${\Z}_p$-extension. E.g., if
$\rho$ cuts out the cyclotomic ${\Z}_p$-extension, then the
$\rho$-height pairing is a normalization of the {\it cyclotomic height
pairing} that has, for the rational field, already been discussed in
the introduction.
If $K$
is quadratic imaginary, and $\rho$ is the anti-cyclotomic linear
functional, meaning that it is the unique linear functional (up to
normalization) that has the property that $\rho({\bar x}) = -\rho(x)$
where ${\bar x}$ is the complex conjugate of $x$, then we will be presently
obtaining explicit formulas for this anti-cyclotomic height pairing.
We will obtain a formula for $(\alpha, \beta) \in I(K)$ by defining,
for every nonarchimedean place, $v$, of $K$ a ``local height pairing,"
$(\alpha, \beta)_v \in K_v^*$. These local pairings will be very
sensitive to some auxiliary choices we make along the way, but for a
fixed $\alpha$ and $\beta$ the local height pairings $(\alpha,
\beta)_v$ will vanish for all but finitely many places~$v$; the global
height is the sum of the local ones and will be independent of all the
choices we have made.
\subsection{Good representations} \label{sec:good_rep}
Let $\alpha, \beta \in E(K)$. By a {\em good representation} of the
pair $\alpha, \beta$ we mean that we are given a four-tuple of points
$(P,Q,R,S)$ in $E(K)$ (or, perhaps, in $E(K')$ where $K'/K$ is a
number field extension of $K$) such that
\begin{itemize}
\item $\alpha$ is the divisor class of the divisor $[P]-[Q]$ of $E$,
and $\beta$ is the divisor class of the divisor $[R]-[S]$,
\item $P,Q,R,S$ are four distinct points,
\item for each $v \ | \ p$ all four points $P,Q,R,S$ specialize to the
same point on the fiber at~$v$ of the N{\'e}ron model of $E$.
\item
at all places~$v$ of~$K$ the points $P,Q,R,S$ specialize to the same
component of the fiber at~$v$ of the N{\'e}ron model of $E$.
\end{itemize}
We will show how to erase these special assumptions later, but for
now, let us assume all this, fix a choice of a good representation,
$P,Q,R,S$, of $(\alpha, \beta)$ as above, and give the formulas in
this case.
\subsection{Local height pairings when $v \mid p$}
Let $\sigma_v$ be the canonical $p$-adic $\sigma$-function attached to
the elliptic curve~$E$ over~$K_v$ given in Weierstrass form. We may
view $\sigma_v$ as a mapping from $E_1(K_v)$ to $K_v^*$, where
$E_1(K_v)$ is the kernel of the reduction map $E(K_v) \to E(k_v)$, and
$E(k_v)$ denotes the group of points on the reduction of $E$ modulo
$v$. Define $(\alpha, \beta)_v \in K_v^*$ by the formula,
$$ (\alpha, \beta)_v=
\frac{\sigma_v(P-R)\sigma_v(Q-S)}{\sigma_v(P-S) \sigma_v(Q-R)}\
\in\ K_v^*.
$$
The dependence of $\sigma$ on the Weierstrass equation is through the
differential $\omega = dx/(2y+a_1x+a_3)$, and $\sigma_{c\omega}
=c\sigma_\omega$, so this depends upon the choice of $P, Q, R, S$, but
does not depend on the choice of Weierstrass equation for~$E$.
\subsection{ Local height pairings when $v \nmid p$}
First let $x$ denote the ``$x$-coordinate" in some minimal Weierstrass
model for~$A$ at~$v$. Define for a point $T$ in $E(K_v)$ the rational
number $\lambda_v(T)$ to be {\it zero} if $x(T) \in {\cal O}_v$, and to
be $-\frac{1}{2}v(x(T))$ if $x(T) \,\, {\not \in}\,\, {\cal O}_v$.
Next, choose a uniformizer $\pi_v$ of $K_v$ and define:
$$
{\tilde\sigma}_v(T) = \pi_v^{\lambda_v(T)},
$$
the square of which is in $K_v^*.$ We think of ${\tilde\sigma}_v$ as a
rough replacement for $\sigma_v$ in the following sense. The $v$-adic
valuation of ${\tilde\sigma}_v$ is the same as $v$-adic valuation of
the $v$-adic sigma function (if such a function is definable at $v$)
and therefore, even if $\sigma_v$ cannot be defined,
${\tilde\sigma}_v$ is a perfectly serviceable substitute at places~$v$
at which our $p$-adic idele class functionals $\rho$ are necessarily
unramified, and therefore sensitive only to the $v$-adic valuation.
For $v \nmid p$, put:
$$
(\alpha, \beta)_v=
\frac{{\tilde\sigma}_v(P-R){\tilde\sigma}_v(Q-S)}{{\tilde\sigma}_v(P-S) {\tilde\sigma}_v(Q-R)}.
$$
The square of this is in $K_v^*$. However, note that
$\pi_v^{\lambda_v(T)}$ really means $\sqrt{\pi_v}^{2\lambda_v(T)}$,
for a fixed choice of $\sqrt{\pi_v}$ and that the definition of
$(\alpha,\beta)_v$ is independent of the choice of square root and
therefore that $(\alpha,\beta)_v$, not only its square, is in $K_v^*$.
Our local height $(\alpha, \beta)_v$, depends upon the choice of
$P,Q,R,S$ and of the uniformizer $\pi_v$.
\subsection{How the local heights change, when we change our choice of divisors}
Let $\beta \in E(K)$ be represented by both $[R]-[S]$ and $[R']-[S']$.
Let $\alpha \in E(K)$ be represented by $[P]-[Q]$. Moreover let both
four-tuples $P,Q,R,S$ and $P,Q,R',S'$ satisfy the {\it good
representation} hypothesis described at the beginning of
Section~\ref{sec:good_rep}. Since, by hypothesis, $[R]-[S]-[R']+[S']$ is linearly
equivalent to zero, there is a rational function $f$ whose divisor of
zeroes and poles is
$$
(f) = [R]-[S]-[R']+[S'].
$$
If~$v$ is a nonarchimedean place of $K$ define $(\alpha, \beta)_v$ to
be as defined in the previous sections using the choice of four-tuple
of points $P,Q,R,S$, (and of uniformizer $\pi_v$ when $v \nmid p$).
Similarly, define $(\alpha, \beta)'_v$ to be as defined in the
previous sections using the choice of four-tuple of points
$P,Q,R',S'$, (and of uniformizer $\pi_v$ when $v \nmid p$).
\begin{proposition}
\begin{enumerate}
\item
If $v\ | \ p$ then
$$
(\alpha, \beta)_v = \frac{f(P)}{f(Q)}\cdot
(\alpha, \beta)'_v\ \in\ K_v^*.
$$
\item
If $v\ \nmid \ p$ then there is a unit $u$ in the ring of integers of $K_v$ such that
$$
(\alpha, \beta)_v^2 = u\cdot \left(\frac{f(P)}{f(Q)}\cdot
(\alpha, \beta)'_v\right)^2\ \in\ K_v^*.
$$
\end{enumerate}
\end{proposition}
\subsection{ The global height pairing more generally}
We can then form the sum of local terms to define the
global height
$$
(\alpha, \beta) =\ \ \ \frac{1}{2}\sum_v \iota_v((\alpha, \beta)_v^2)\ \in\ I(K).
$$
This definition is independent of
any of the (good representation) choices $P,Q,R,S$ and the $\pi_v$'s made. It is independent of the choice of $\pi_v$'s because the units in the ring of integers of $K_v$ is in the kernel of $\iota_v$ if $v\nmid p$. It is independent of the choice of $P,Q,R,S$ because by the previous proposition, a change (an allowable one, given our hypotheses) of $P,Q,R,S$ changes the value of $(\alpha, \beta)$ by a factor that is a principal idele, which is sent to zero in $I(K)$.
What if, though, our choice of $P,Q,R,S$ does {\it not} have the
property that~$\alpha$ and~$\beta$ reduce to the same point in the
N{\'e}ron fiber at~$v$ for all $v \ | \ p$, or land in the same
connected component on each fiber of the N{\'e}ron model? In this
case the pair $\alpha, \beta$ do not have a {\it good representation}.
But replacing $\alpha, \beta$
by $m\cdot \alpha,n\cdot \beta$ for sufficiently large positive
integers $m,n$ we can guarantee that the pair $m\cdot \alpha, n\cdot
\beta$ does possess a good representation, and obtain formulas for
$(\alpha, \beta)$ by:
$$
(\alpha, \beta)= \frac{1}{mn}(m\cdot \alpha, n\cdot \beta).
$$
Note in passing that to compute the global height pairing
$(\alpha,\alpha)$ for a nontorsion point $\alpha \in E(K)$ that
specializes to $0$ in the N\'eron fiber at $v$ for all $v \ | \ p$,
and that lives in the connected component containing the identity in
all N\'eron fibers, we have quite a few natural choices of {\it good
representations}. For example, for positive integers $m \ne n$, take
$$P= (m+1)\cdot \alpha; \ Q = m\cdot \alpha;\ R= (n+1)\cdot \alpha; \
S = n\cdot \alpha.$$ Then for any $p$-adic idele class functional
$\rho$ the global $\rho$-height pairing $(\alpha, \alpha)_{\rho}$ is
given by
\begin{align*}
&\sum_{v\ | \ p} \rho_v\left\{\frac{
\sigma_v((m-n)\alpha)^2}{\sigma_v((m-n+1)\alpha)\cdot \sigma_v((m-n-1)\alpha)}\right\}
\\
& \qquad\qquad\qquad + \sum_{v\ \nmid \ p} \rho_v
\left\{\frac{{\tilde\sigma}_v((m-n)\alpha)^2}{{\tilde\sigma}_v((m-n+1)\alpha)\cdot {\tilde\sigma}_v((m-n-1)\alpha)}\right\},
\end{align*}
which simplifies to
$$
(2(m-n)^2- (m-n+1)^2 - (m-n-1)^2)\cdot
\left\{ \sum_{v\ | \ p} \rho_v \sigma_v(\alpha)
+ \sum_{v\ \nmid \ p} \rho_v {\tilde\sigma_v}(\alpha)\right
\}.
$$
\edit{I added a subscript of $v$ to the right-most $\tilde{\sigma}$
in this formula.}
Since $(2(m-n)^2- (m-n+1)^2 - (m-n-1)^2) = -2$ we have the formula
$$h_{\rho}(\alpha) = -{\frac{1}{2}}(\alpha,\alpha)_{\rho}$$ quoted earlier.
\subsection{ Formulas for the $\rho$-height}\label{sec:formulas_rhoheight}
For each $v$, let $\sigma_v$ be the canonical $p$-adic
$\sigma$-function of $E$ over $K_v$ given in Weierstrass form.
Suppose $P\in E(K)$ is a (non-torsion) point that reduces to~$0$ in
$E(k_v)$ for each $v\mid p$, and to the connected component of all
special fibers of the N\'eron model of~$E$. Locally at each place~$w$
of~$K$, we have a denominator $d_w(P)$, well defined up to units.
% \begin{remark}
% The local denominator $d_w(P)$ is really nothing more than a
% rough-and-ready substitute for an inverse of $\sigma_w(P)$ up to a
% unit; it can be called upon to serve as such a thing even if we
% cannot define $\sigma_w(P)$ precisely. This is what we use, then,
% instead of the $\sigma$ function at places where we only are
% interested in the $\sigma$ function up to a unit, i.e., where $\psi$
% is unramified.
% \end{remark}
% \edit{I'm worried about the $1/2$ in the definition of the
% $\lambda$'s previously. We should make sure that the normalization
% here is correct; and there as well (Barry); {\bf New note from
% barry: } I'm no longer worried, and as it turned out---see
% above-- there really should have been a $-{\frac{1}{2}}$ but
% please check me; also, it is, pretty much, up to us to decide the
% normalization of $h_{\rho}$ for this paper, since we are, I think,
% defining everything precisely, and I suspect the literature has a
% bunch of different normalizations, so of course we better have a
% self-consistent normalization throughout, but also the ``best"
% one. Is this a good one?}
We have
$$
h_{\rho}(P) = \sum_{v\ |\ p}
\rho_v(\sigma_v(P)) - \sum_{w\,\,\nmid\,\, p}\rho_w(d_w(P)).
$$
Note that $h_{\rho}$ is quadratic because of the quadratic
property of $\sigma$ from (\ref{eqn:quad}), and the
$h_{\rho}$-pairing is then visibly bilinear. See also
property (\ref{eqn:bil}).
%\edit{But maybe one should take a closer look and see
%exactly where to insert the appropriate sentence so that it keeps
%things clear. {\bf Barry:} this is as good a place as any for it.}
\subsection{Cyclotomic $p$-adic heights}
The idele class ${\Q}_p$-vector space $I({\Q})$ attached to
${\Q}$ is canonically isomorphic to ${\Q}_p\otimes {\Z}_p^*$.
Composition of this canonical isomorphism with the mapping $1
\times{\frac{1}{p}}{\rm log}_p$ induces an isomorphism
$$
\rho_{\text{cycl}}^{\Q}:I({\Q})
= {\Q}_p\otimes {\Z}_p^*
\xrightarrow{\,\,\,\,\,\cong\,\,\,\,\,} {\Q}_p.
$$
For $K$ any number field, consider the homomorphism on idele class
${\Q}_p$-vector spaces induced by the norm $N_{K/ {\Q}}:I(K) \to
I({\Q})$, and define
$$
\rho_{\text{cycl}}^K: I(K) \to {\Q}_p
$$
as the composition
$$
\rho_{\text{cycl}}^K = \rho_{\text{cycl}}^{\Q}\circ N_{K/ {\Q}}.
$$
By the {\em cyclotomic height pairing} for an elliptic curve $E$ over
$K$ (of good ordinary reduction at all places $v$ of $K$ above $p$) we
mean the $\rho_\text{cycl}^K$-height pairing $E(K)\times E(K) \to {\Q}_p$. We put
$$
h_p(P)=
h_{\rho_{\text{cycl}}^K }(P)
$$ for short.
Here is an explicit formula for it.
$$
h_p(P) = {\frac{1}{p}} \cdot \left(\sum_{v\mid p}
\log_p(N_{K_v/\Q_p}(\sigma_v(P))) -
\sum_{w\nmid p}\ord_w(d_w(P)) \cdot \log_p(\#k_w)\right).
$$
%\edit{Over $\Q$ we normalized this by putting a factor of $1/p$ out
% front. What do you want to put out front in the general case?
%I'm guessing we want $1/p^{[K:\Q]}$, but could be wrong. Barry: I removed the $1/p^{\rm degree}$ factor here. It shouldn't be there.}
If
we assume that $P$ lies in a sufficiently small (finite index)
subgroup of $E(K)$ (see \cite[Prop.~2]{wuthrich:heightfamily}),
then there will be a global choice of denominator
$d(P)$, and the formula simplifies to
$$
h_p(P) = \frac{1}{p} \cdot \log_p\left( \prod_{v\mid p} N_{K_v/\Q_p}\left(\frac{\sigma_v(P)}{d(P)}
\right) \right).
$$
\subsection{Anti-cyclotomic $p$-adic heights}
Let $K$ be a quadratic imaginary field in which $p$
splits as $(p) = \pi\cdot {\bar \pi}$. Suppose $\rho: \A_K^*/K^* \to
\Z_p$ is a nontrivial {\em anti-cyclotomic} idele class
character, meaning that if ${\bf c}: \A_K^*/K^* \to \A_K^*/K^*$ denotes the involution of the idele class group induced
by complex conjugation $x\mapsto {\bar x}$ in $K$, then $\rho\cdot
{\bf c} = -\rho$.
Then the term
$$
\sum_{v\ |\ p}\rho_v(\sigma_v(P))$$ in the formula for
the $\rho$-height at the end of Section~\ref{sec:formulas_rhoheight} is
just
$$
\sum_{v\ |\ p} \rho_v(\sigma_v(P))
= \rho_{\pi}(\sigma_{\pi}(P)) - \rho_{\pi}(\sigma_{\pi}({\bar P})),
$$
so we have the following formula for
the $\rho$-height of $P$:
$$
h_{\rho}(P)= \rho_{\pi}(\sigma_{\pi}(P)) -
\rho_{\pi}(\sigma_{\pi}({\bar P})) -
\sum_{w\,\,\nmid\,\, p}\rho_w(d_w(P)).
$$
\begin{remark}
The Galois equivariant property of the $p$-adic height pairing implies
that if $P$ is a $\Q$-rational point, its anti-cyclotomic height is $0$.
Specifically, let $K/k$ be any Galois extension of number fields,
with Galois group $G = \Gal(K/k)$. Let $V=V(K)$ be the $\Q_p$-vector
space (say) defined as $(G_K)^{\ab}\otimes {\Q}_p$, so that $V$ is
naturally a $G$-representation space. Let~$E$ be an elliptic curve
over $k$ and view the Mordell-Weil group $E(K)$ as equipped with its
natural $G$-action. Then (if~$p$ is a good ordinary prime for $E$)
we have the $p$-adic height pairing
$$
\langle P, Q\rangle \in V,
$$
for $P,Q \in E(K)$ and we have Galois equivariance,
$$
\langle g\cdot P, g\cdot Q\rangle = g\cdot\langle P, Q\rangle,
$$
for any $g$ in the Galois group.
Put $k = {\bf Q}$, $K/k$ a quadratic imaginary field. Then $V$ is of
dimension two, with $V = V^+ \oplus V^-$ each of the $V^\pm$ being of
dimension one, with the action of complex conjugation, $g \in G$ on
$V^\pm$ being given by the sign; so that $V^+$ corresponds to the
cyclotomic ${\bf Z}_p$-extension and $V^-$ corresponds to the
anticyclotomic ${\bf Z}_p$- extension. In the notation above, the
anticyclotomic height of $P$ and $Q$ is just $\langle g\cdot P, g\cdot
Q\rangle^-$ where the superscript $-$ means projection to $V^-$.
Suppose that $P \in E({\bf Q})$, so that $g\cdot P = P$. Then we have
by Galois equivariance $$\langle P, P\rangle^- = \langle g\cdot P,
g\cdot P\rangle^-= - \langle P, P\rangle^-,$$ so $\langle P,
P\rangle^- = 0$. More generally, the anticyclotomic height is zero as a
pairing on either $E(K)^+\times E(K)^+$ or $E(K)^- \times E(K)^-$ and
can only be nonzero on $E(K)^+\times E(K)^-$. If $E(K)$ is of odd
rank, then the ranks of $E(K)^+$ and $E(K)^-$ must be different, which
obliges the pairing on $E(K)^+\times E(K)^-$ to be either
left-degenerate or right-degenerate (or, of course, degenerate on both
sides). Rubin and the first author conjecture that it is nondegenerate
on one side (the side, of course having smaller rank); for
more details see, e.g., \cite[Conj.~11]{mazur-rubin:pairings_arith}.
\end{remark}
%\edit{Mazur: It is possible that in the quadratic imaginary situation the terms
%(in the rho-height formula) involving the $d_w$'s can be more neatly
% packaged? I haven't tried, but since it is SO neat in the cyclotomic
% case, it may be worth a look here.}
\section{The Algorithms}\label{sec:cohom}
Fix an elliptic curve $E$ over $\Q$ and a good ordinary prime $p\geq
5$. In this section we discuss algorithms for computing the
cyclotomic $p$-adic height of elements of $E(\Q)$.
\subsection{Computing the $p$-adic $\sigma$-function}
First we explicitly solve the differential equation
(\ref{eqn:sigmadef}). Let $z(t)$ be the formal logarithm on $E$,
which is given by $z(t) = \int \frac{\omega}{dt} = t + \cdots$ (here
the symbol $\int$ means formal integration with $0$ constant term).
There is a unique function $F(z)\in\Q((z))$ such that $t = F(z(t))$.
Set $x(z) = x(F(z))$. Rewrite (\ref{eqn:sigmadef}) as
\begin{equation}\label{eqn:sigmaz}
x(z) + c = -\frac{d}{\omega}\left(\frac{d\log(\sigma)}{\omega}\right).
\end{equation}
A crucial observation is that
$$
x(z)+c = \frac{1}{z^2} - \frac{a_1^2+4a_2}{12} + c + \cdots;
$$
in particular, the coefficient of $1/z$ in the expansion of $g(z)=x(z)+c$
is~$0$.
Since $z = \int (\omega/dt)$ we have $dz = (\omega/dt)dt = \omega$,
hence $dz/\omega=1$,
so
\begin{equation}\label{eqn:inz}
-\frac{d}{\omega}\left(\frac{d\log(\sigma)}{\omega}\right)
= -\frac{dz}{\omega}\frac{d}{dz} \left( \frac{d \log(\sigma)}{\omega} \right)
= -\frac{d}{dz} \left( \frac{d\log(\sigma)}{dz}\right).
\end{equation}
Write $\sigma(z) = z\sigma_0(z)$ where $\sigma_0(z)$ has nonzero constant
term. Then
\begin{equation}\label{eqn:inz2}
-\frac{d}{dz} \left( \frac{d\log(\sigma)}{dz}\right)
= \frac{1}{z^2} - \frac{d}{dz} \left( \frac{d\log(\sigma_0)}{dz}\right).
\end{equation}
Thus combining (\ref{eqn:sigmaz})--(\ref{eqn:inz2}) and changing sign gives
$$
\frac{1}{z^2} - x(z) - c =
\frac{d}{dz} \left( \frac{d\log(\sigma_0)}{dz}\right).
$$
This is particularly nice, since $g(z) = \frac{1}{z^2} - x(z) - c \in
\Q[[z]]$. We can thus solve for $\sigma_0(z)$ by formally integrating
twice and exponentiating:
$$
\sigma_0(z) = \exp\left(\int\int g(z) dz dz\right),
$$
where we choose the constants in the double integral
to be $0$, so $\int\int g = 0 + 0z + \cdots $.
Using (\ref{eqn:wpe2})
we can rewrite $g(z)$ in terms of $e_2 = \E_2(E,\omega)$
and $\wp(z)$ as
$$
g(z) = \frac{1}{z^2} -(x(z) + c) = \frac{1}{z^2} - \wp(z)+ \frac{e_2}{12}.
$$
Combining everything and using that $\sigma(z) =
z\sigma_0(z)$ yields
$$
\sigma(z) = z \cdot \exp\left(\int\int
\left(\frac{1}{z^2} - \wp(z) + \frac{e_2}{12} \right)dz dz\right),
$$
Finally, to compute $\sigma(t)$ we compute $\sigma(z)$ and obtain
$\sigma(t)$ as $\sigma(z(t))$.
We formalize the resulting algorithm below.
\begin{algorithm}{The Canonical $p$-adic Sigma Function}\label{alg:sigma}
Given an elliptic curve~$E$ over~$\Q$, a good ordinary prime~$p$
for~$E$, and an approximation $e_2$ for $\E_2(E,\omega)$,
this algorithm computes an approximation to $\sigma(t)\in\Z_p[[t]]$.
\begin{steps}
\item{}[Compute Formal Log]\label{alg:sigma1} Compute the formal
logarithm $z(t)=t + \cdots \in \Q((t))$ using that
\begin{equation}\label{eqn:formal_log}
\ds z(t) = \int
\frac{dx/dt}{2y(t)+a_1x(t) + a_3}, \qquad\text{(0 constant term)}
\end{equation}
where $x(t)=t/w(t)$ and
$y(t)=-1/w(t)$ are the local expansions of~$x$ and~$y$ in terms of
$t=-x/y$, and $w(t) = \sum_{n\geq 0} s_n t^n$ is given by the
following explicit inductive formula (see, e.g.,
\cite[pg.~18]{bluher:formal}):
$$
s_0 = s_1 = s_2 = 0, \qquad s_3 = 1, \qquad \text{and for $n\geq 4$,}
$$
$$
s_n = a_1 s_{n-1} + a_2 s_{n-2} + a_3\sum_{i+j=n} s_i s_j
+ a_4 \sum_{i+j = n-1} s_i s_j + a_6 \sum_{i+j+k=n} s_i s_j s_k.
$$
% in
% \cite[Ch.~IV.1]{silverman:aec}.\footnote{Note that the formula for the
% formal group addition on page 114 of \cite{silverman:aec} is
% incorrect. In the numerator, the summand $-2a_4\lambda\nu -
% 3a_6\lambda^2\nu$ should be replaced by $+2a_4\lambda\nu +
% 3a_6\lambda^2\nu$.}
\item{}[Reversion] 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{}[Compute $\wp$]
Compute $\alpha(t) \set x(t) + (a_1^2 + 4a_2)/12 \in \Q[[t]]$, where the
$a_i$ are as in (\ref{eqn:weq}).
Then compute the series $\wp(z) = \alpha(F(z))\in \Q((z))$.
\item{}[Compute $\sigma(z)$]
Set $\ds g(z)\set \frac{1}{z^2} - \wp(z) + \frac{e_2}{12}\in\Q_p((z))$,
% \edit{The theory suggests the last term should be $-e_2/12$ but the calculations do not
% work unless I use $+e_2/12$. There are probably two
% normalizations of $E_2$ in the references. GULP -- here we
%need to be extra careful. William: I more convinced by Katz's
%argument that his $\E_2$ is right than the one in Mazur-Tate.}
and compute
$$\ds \sigma(z) \set z\cdot \exp\left(\int \int g(z) dz dz\right)
\in \Q_p[[z]].$$
\item{}[Compute $\sigma(t)$] Set $\sigma(t) \set \sigma(z(t))\in
t\cdot \Z_p[[t]]$, where $z(t)$ is the formal logarithm computed in
Step~\ref{alg:sigma1}. Output $\sigma(t)$ and terminate.
\end{steps}
\end{algorithm}
%\begin{remark}
% The trick of changing from $\wp(t)$ to $\wp(z)$ allows us to solve a
% certain differential equation using just simple operations with
% power series.
%\end{remark}
%\edit{Insert explanation of exactly how we used the formal
%logarithm to transform a difficult differential equation
%in terms of $t$ into an easy one. I learned this from
%a paper or thesis of Wuthrich.}
\subsection{Computing $\E_2(E,\omega)$ using cohomology}
\label{sec:e2cohom}
This section is about a fast method of computation of $\E_2(E,\omega)$
for individual ordinary elliptic curves, ``one at a time''. The key
input is \cite[App.~2]{katz:padicprop} (see also
\cite{katz:padic_interp}), which gives an interpretation of
$\E_2(E,\omega)$ as the ``direction'' of the unit root eigenspace (cf.
formula A.2.4.1 of \cite[App.~2]{katz:padicprop}) of Frobenius acting
on the one-dimensional de Rham cohomology of $E$.
Concretely, consider an elliptic curve~$E$ over $\Z_p$ with good
ordinary reduction. Assume that $p\geq 5$. Fix a Weierstrass
equation for~$E$ of the form $ y^2 = 4x^3 - g_2x - g_3, $ The
differentials $\omega=dx/y$ and $\eta=xdx/y$ form a $\Z_p$-basis for
the first $p$-adic de Rham cohomology group~$\H^1$ of~$E$, and we wish
to compute the matrix~$F$ of absolute Frobenius with respect to this
basis. Frobenius is $\Z_p$-linear, since we are working over $\Z_p$;
if we were working over the Witt vectors of $\F_q$, then Frobenius
would only be semi-linear.
We explicitly calculate~$F$ (to a specified precision) using Kedlaya's
algorithm, which makes use of Monsky-Washnitzer cohomology of the
affine curve $E-\cO$. Kedlaya designed his algorithm for computation
of zeta functions of hyperelliptic curves over finite fields. An
intermediate step in Kedlaya's algorithm is computation of the matrix
of absolute Frobenius on $p$-adic de Rham cohomology, via
Monsky-Washnitzer cohomology. For more details see
\cite{kedlaya:counting_mw} and \cite{kedlaya:counting-errata}. For
recent formulations and applications of fast algorithms to compute
Frobenius eigenvalues, see \cite{lauder-wan}.
Now that we have computed $F$, we deduce $\E_2(E,\omega)$ as follows.
The unit root subspace is a direct factor, call it~$U$, of $\H^1$, and
we know that a complementary direct factor is the $\Z_p$ span
of~$\omega$. We also know that $F(\omega)$ lies in $p\H^1$, and this
tells us that, $\!\!\!\!\mod{p^n}$, the subspace~$U$ is the span of
$F^n(\eta)$. Thus if for each~$n$, we write $ F^n(\eta) = a_n\omega +
b_n\eta, $ then $b_n$ is a unit (congruent $\!\!\!\pmod{p}$ to the $n$th
power of the Hasse invariant) and $\E_2(E,\omega) \con -12a_n/b_n
\pmod{p^n}.$ Note that $a_n$ and $b_n$ are the entries of the second
column of the matrix $F^n$.
%In terms of speed of convergence, once we have $F$, we iterate
%it~$n$ times to calculate $P \pmod{p^n}$.
\begin{algorithm}{Evaluation of $\E_2(E,\omega)$}\label{alg:e2}
Given an elliptic curve over~$\Q$ and a good ordinary prime~$p\geq
5$, this algorithm approximates $\E_2(E,\omega)\in \Z_p$ modulo $p^n$.
\begin{steps}
\item{}[Invariants] 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{}[Kedlaya] 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
$F$ (modulo $p^n$) of the action of absolute Frobenius on the basis
$$
\omega=\frac{dx}{y}, \qquad \eta=\frac{xdx}{y}.
$$
We view $F$ as acting from the left.
\item{}[Iterate Frobenius]
Compute the second column $\vtwo{a}{b}$ of $F^n$,
so $\Frob^n(\eta) = a\omega + b\eta$.
\item{}[Finished]
Output $-12a/b$ (which is a number modulo $p^n$, since~$b$
is a unit).
\end{steps}
\end{algorithm}
\subsection{Computing $\E_2(E,\omega)$ using integrality}
\label{sec:integrality}
The algorithm in this section is more elementary than the one in
Section~\ref{sec:e2cohom}, and is directly motivated by
Theorem~\ref{thm:uniqde}. In practice it is very slow, except if~$p$
is small (e.g., $p=5$) and we only require $\E_2(E,\omega)$ to very
low precision. Our guess is that it should be exponentially hard to
compute a quantity using a log convergent series for it, and that this
``integrality'' method is essentially the same as using log convergent
expansions.
Let~$c$ be an indeterminate and in view of (\ref{eqn:ce2}), write $e_2
= -12c + a_1^2 + 4a_2 \in \Q[c]$. If we run Algorithm~\ref{alg:sigma}
with this (formal) value of $e_2$, we obtain a series $\sigma(t,c) \in
\Q[c][[t]]$. For each prime $p\geq 5$, Theorem~\ref{thm:uniqde}
implies that there is a unique choice of $c_p\in \Z_p$ such that
$\sigma(t,c_p) = t + \cdots \in t\Z_p[[t]]$ is odd. Upon fixing a
prime~$p$, we compute the coefficients of $\sigma(t,c)$, which are
polynomials in~$\Q[c]$; integrality of $\sigma(t,c_p)$ then imposes
conditions that together must determine~$c_p$ up to some precision,
which depends on the number of coefficients that we consider. Having
computed~$c_p$ to some precision, we recover $\E_2(E,\omega)$ as
$-12c_p + a_1^2 + 4a_2$. We formalize the above as an algorithm.
\begin{algorithm}{Integrality}\label{alg:int}
Given an elliptic curve over~$\Q$ and a good ordinary prime~$p\geq
5$, this algorithm approximates the associated $p$-adic
$\sigma$-function.
\begin{steps}
\item{}[Formal Series] Use Algorithm~\ref{alg:sigma} with
$e_2 = -12c + a_1^2 + 4a_2$ to compute $\sigma(t) \in \Q[c][[t]]$
to some precision.
\item{}[Approximate $c_p$] Obtain constraints on $c$ using
that the coefficients of $\sigma$ must be in $\Z_p$. These determine
$c$ to some precision. (For more details see the
example in Section~\ref{sec:ex-integrality}).
\end{steps}
\end{algorithm}
\subsection{Computing cyclotomic $p$-adic heights}
Finally we give an algorithm for computing the cyclotomic $p$-adic
height $h_p(P)$ that combines Algorithm~\ref{alg:e2} with the
discussion elsewhere in this paper. We have computed $\sigma$ and
$h_p$ in numerous cases using the algorithm described below, and
implementations of the ``integrality'' algorithm described above, and
the results match.
% We have also done
%several computations of $h_p$ using other methods \edit{Tate's --
% explain}, and again the results match.
\begin{algorithm}{The $p$-adic Height}\label{alg:padic_height}
Given an elliptic curve~$E$ over $\Q$, a good ordinary prime~$p$,
and a non-torsion element $P\in E(\Q)$, this algorithm approximates
the $p$-adic height $h_p(P) \in \Q_p$.
\begin{steps}
\item{}[Prepare Point] Compute a positive 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 $\sigma$] Approximate $\sigma(t)$ using
Algorithm~\ref{alg:sigma} together with either
Algorithm~\ref{alg:e2} or Algorithm~\ref{alg:int},
and set $s \set \sigma(-x/y) \in
\Q_p$.
\item{}[Height] Compute
$\ds h_p(Q) \set \frac{1}{p}\log_p\left(\frac{s}{d}\right)$, then
$\ds h_p(P) \set \frac{1}{m^2} \cdot h_p(Q)$.
Output $h_p(P)$ and terminate.
\end{steps}
\end{algorithm}
\section{Sample Computations}\label{sec:sample}
We did the calculations in this section using SAGE \cite{sage} and
Magma \cite{magma}. In particular, SAGE includes an optimized
implementation due to J. Balakrishnan, R. Bradshaw, D. Harvey,
Y. Qiang, and W. Stein of our algorithm for computing $p$-adic
heights for elliptic curves over $\Q$. This implementation
includes further tricks, e.g., for series manipulation, which
are not described in this paper.
\subsection{The rank one curve of conductor $37$}\label{sec:ex-integrality}
Let $E$ be the rank $1$ curve $y^2+y=x^3-x$
of conductor $37$. The point $P=(0,0)$ is
a generator for $E(\Q)$.
We illustrate the above algorithms in detail
by computing the $p$-adic height of~$P$ for the good
ordinary prime~$p=5$.
The steps of
Algorithm~\ref{alg:padic_height} are as follows:
\begin{enumerate}
\item{}[Prepare Point] The component group of $\cE_{\F_{37}}$ is trivial.
The group $E(\F_{5})$ has order $8$ and the reduction of $P$ to $E(\F_{5})$
also has order~$8$, so let
$$
Q=8P = \left(\frac{21}{25},\,\, -\frac{69}{125}\right).
$$
\item{}[Denominator] We have $d=5$.
\item{}[Compute $\sigma$] We illustrate computation of $\sigma(t)$
using both Algorithm~\ref{alg:e2} and Algorithm~\ref{alg:int}.
\begin{enumerate}
\item{}[Compute $\sigma(t,c)$]
We use Algorithm~\ref{alg:sigma}
with $e_2 = 12c - a_1^2 - 4a_2$ to compute~$\sigma$ as a series
in~$t$ with coefficients polynomials in~$c$, as follows:
\begin{enumerate}
\item{}[Compute Formal Log]
Using the recurrence, we find that
\begin{align*}
w(t) &=t^{3} + t^{6} - t^{7} + 2t^{9} - 4t^{10} + 2t^{11} + 5t^{12} - 5t^{13} + 5t^{14} + \cdots
\end{align*}
Thus
\begin{align*}
x(t) &= t^{-2} - t + t^{2} - t^{4} + 2t^{5} - t^{6} - 2t^{7}
+ 6t^{8} - 6t^{9} - 3t^{10} + \cdots \\
y(t) &= -t^{-3} + 1 - t + t^{3} - 2t^{4} + t^{5} + 2t^{6} - 6t^{7} + 6t^{8} + 3t^{9} + \cdots
\end{align*}
so integrating (\ref{eqn:formal_log}) we see that the formal logarithm is
$$
z(t) =
t + \frac{1}{2}t^{4} - \frac{2}{5}t^{5} + \frac{6}{7}t^{7} - \frac{3}{2}t^{8} + \frac{2}{3}t^{9} + 2t^{10} - \frac{60}{11}t^{11} + 5t^{12} + \cdots
$$
\item{}[Reversion]
Using reversion, we find $F$
with $F(z(t)) = t$:
$$
F(z) = z - \frac{1}{2}z^{4} + \frac{2}{5}z^{5} + \frac{1}{7}z^{7} - \frac{3}{10}z^{8} + \frac{2}{15}z^{9} - \frac{1}{28}z^{10} +
\frac{54}{385}z^{11} + \cdots
$$
\item{}[Compute $\wp$]
We have $a_1=a_2=0$, so
$$
\alpha(t) = x(t) + (a_1^2 + 4a_2)/12 = x(t),
$$
so
$$
\wp(z) = x(F(z)) = z^{-2} + \frac{1}{5}z^{2} - \frac{1}{28}z^{4}
+ \frac{1}{75}z^{6} - \frac{3}{1540}z^{8} + \cdots %\frac{1943}{3822000}z^{10} + \cdots
$$
Note that the coefficient of $z^{-1}$ is $0$ and all exponents
are even.
\item{}[Compute $\sigma(t,c)$]
Noting again that $a_1=a_2 = 0$, we have
\begin{align*}
g(z,c) &= \frac{1}{z^2} - \wp(z) + \frac{12c-a_1^2 - 4a_2}{12}\\
&= c -\frac{1}{5}z^{2} + \frac{1}{28}z^{4} - \frac{1}{75}z^{6} + \frac{3}{1540}z^{8} - \frac{1943}{3822000}z^{10} + \cdots
\end{align*}
Formally integrating twice and exponentiating, we obtain
\begin{align*}
\sigma(z,c) &= z\cdot \exp\left(\int \int g(z,c) dz dz\right) \\
&= z\cdot \exp\Bigl(
\frac{c}{2} \cdot z^2 -\frac{1}{60}z^{4} + \frac{1}{840}z^{6} - \frac{1}{4200}z^{8} + \frac{1}{46200}z^{10} \\
& \qquad - \frac{1943}{504504000}z^{12} + \cdots
\Bigr)\\
&=
z + \frac{1}{2}cz^{3} + \left(\frac{1}{8}c^2 - \frac{1}{60}\right)z^{5} +
\left(\frac{1}{48}c^3 - \frac{1}{120}c + \frac{1}{840}\right)z^{7} +\\
& \qquad \left(\frac{1}{384}c^4 - \frac{1}{480}c^2 + \frac{1}{1680}c - \frac{1}{10080}\right)z^{9} + \cdots
\end{align*}
Finally,
\begin{align*}
\sigma(t) = \sigma(z(t)) &=
t + \frac{1}{2}ct^{3} + \frac{1}{2}t^{4} + \left(\frac{1}{8}c^2 - \frac{5}{12}\right)t^{5} + \frac{3}{4}ct^{6} + \\
& \qquad \left(\frac{1}{48}c^3 - \frac{73}{120}c + \frac{103}{120}\right)t^{7} + \cdots
\\
\end{align*}
\end{enumerate}
\item{}[Approximate] The first coefficient of $\sigma(t)$
that gives integrality information is the coefficient of $t^7$.
Since
$$
\frac{1}{48}c^3 - \frac{73}{120}c + \frac{103}{120} \in \Z_5,
$$
multiplying by $5$ we see that
$$
\frac{5}{48}c^3 - \frac{73}{24}c + \frac{103}{24} \con 0 \pmod{5}.
$$
Thus
$$
c \con \frac{103}{24} \cdot \frac{24}{73} \con 1 \pmod{5}.
$$
The next useful coefficient is the coefficient of $t^{11}$, which is
$$
\frac{1}{3840}c^5 - \frac{169}{2880}c^3 + \frac{5701}{6720}c^2 + \frac{127339}{100800}c - \frac{40111}{7200}
$$
Multiplying by $25$, reducing coefficients, and using integrality yields the congruence
$$
10c^5 + 5c^3 + 20c^2 + 2c + 3\con 0 \pmod{25}.
$$
Writing $c=1+5d$ and substituting gives the equation $10d+15 \con 0
\pmod{25}$, so $2d + 3 \con 0 \pmod{5}$. Thus $d \con 1 \pmod{5}$,
hence $c = 1 + 5 + O(5^2)$. Repeating the procedure above with
more terms, we next get new information from the coefficient of
$t^{31}$, where we deduce that $c = 1 + 5 + 4\cdot 5^2 + O(5^3)$.
\end{enumerate}
\noindent{\bf Using Algorithm~\ref{alg:e2}:}
Using Kedlaya's algorithm (as implemented in \cite{magma}) we
find almost instantly that
$$
\E_2(E,\omega) = 2 + 4\cdot5 + 2\cdot5^3 + 5^4 + 3\cdot5^5 + 2\cdot5^6 + 5^8 + 3\cdot5^9 + 4\cdot5^{10}
+ \cdots.
$$
Thus
$$
c = \frac{1}{12} \E_2(E,\omega) = 1 + 5 + 4\cdot5^2 + 5^3 + 5^4 + 5^6 + 4\cdot5^7 + 3\cdot5^8 + 2\cdot5^9 + 4\cdot 5^{10} + \cdots,
$$
which is consistent with what we found above using integrality.
\item{}[Height]
For $Q=(x,y)=8(0,0)$ as above, we have
$$
s = \sigma\left(-\frac{x}{y}\right) = \sigma\left(\frac{35}{23}\right) = 4\cdot 5 + 5^2 + 5^3 + 5^4 + \cdots,
$$
so
\begin{align*}
h_5(Q) &= \frac{1}{5}\cdot \log_5\left(\frac{s}{5}\right) =
\frac{1}{5}\cdot \log_5(4 + 5 + 5^2 + 5^3 + 2\cdot 5^5 + \cdots)\\
&= 3 + 5 + 2\cdot 5^3 + 3\cdot 5^4 + \cdots.
\end{align*}
Finally,
$$
h_5(P) = \frac{1}{8^2} \cdot h_5(Q) = 2 + 4\cdot 5 + 5^2 + 2\cdot 5^3 + 2\cdot 5^4 + \cdots.
$$
\end{enumerate}
% Table~\ref{tab:37} lists the good ordinary primes with $5 < p<100$
% and the corresponding $p$-adic height (mod $p^3$) of $P$, where in each case
% we used Algorithm~\ref{alg:e2} to compute~$\sigma$.
% \begin{figure}[H]
% \caption{The Rank $1$ Curve of Conductor $37$\label{tab:37}}
% \begin{center}
% \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
% $p$ &
% 5& 7& 11& 13& 23& 29& 31& 41& 43& 47 & 53 \\\hline
% $h_p(P)$ & 37 & 605& -218 & $-47\cdot 13^2$ & -3069& -9942&-333 & -12807 &-2982 &3684 &
% $20869\cdot 53^{-2}$ \\\hline
% \end{tabular}
% \vspace{2ex}
% \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\hline
% $p$ &
% 59& 61& 67& 71& 73& 79& 83& 89& 97 \\\hline
% $h_p(P)$ & -92910 & 27875 &$-1215\cdot 67$ &-30692 &-133761 & 42634 &2618 &-145804
% & -45277\\\hline
% \end{tabular}
% \end{center}
% \end{figure}
\begin{remark}
A {\em very} good check to see whether or not any implementation of
the algorithms in this paper is really correct, is just to make
control experiments every once in a while, by computing $h(P)$ and
then comparing it with $h(2P)/4$, $h(3P)/9$, etc. In particular,
compute $h(P) - h(nP)/n^2 $ for several~$n$ and check that the
result is $p$-adically small. We have done this in many cases for
the implementation used to compute the tables in this section.
\end{remark}
\subsection{Curves of ranks $1$, $2$, $3$, $4$, and $5$}
\subsubsection{Rank $1$}
The first (ordered by conductor) curve of rank $1$ is the curve with
Cremona label 37A, which we considered in
Section~\ref{sec:ex-integrality} above.
\begin{center}
\begin{tabular}{|c|l|}\hline
$p$ & $p$-adic regulator of {\bf 37A}\\ \hline
5 & $1 + 5 + 5^2 + 3\cdot5^5 + 4\cdot5^6 + O(5^7)$ \\\hline
7 & $1 + 7 + 3\cdot7^2 + 7^3 + 6\cdot7^4 + 2\cdot7^5 + 4\cdot7^6 + O(7^7)$ \\\hline
11 & $7 + 9\cdot11 + 7\cdot11^2 + 8\cdot11^3 + 9\cdot11^4 + 2\cdot11^5 + 7\cdot11^6 + O(11^7)$ \\\hline
13 & $12\cdot13 + 5\cdot13^2 + 9\cdot13^3 + 10\cdot13^4 + 4\cdot13^5 + 2\cdot13^6 + O(13^7)$ \\\hline
23 & $20 + 10\cdot23 + 18\cdot23^2 + 16\cdot23^3 + 13\cdot23^4 + 4\cdot23^5 + 15\cdot23^6 + O(23^7)$ \\\hline
29 & $19 + 4\cdot29 + 26\cdot29^2 + 2\cdot29^3 + 26\cdot29^4 + 26\cdot29^5 + 17\cdot29^6 + O(29^7)$ \\\hline
31 & $15 + 10\cdot31 + 13\cdot31^2 + 2\cdot31^3 + 24\cdot31^4 + 9\cdot31^5 + 8\cdot31^6 + O(31^7)$ \\\hline
41 & $30 + 2\cdot41 + 23\cdot41^2 + 15\cdot41^3 + 27\cdot41^4 + 8\cdot41^5 + 17\cdot41^6 + O(41^7)$ \\\hline
43 & $30 + 30\cdot43 + 22\cdot43^2 + 38\cdot43^3 + 11\cdot43^4 + 29\cdot43^5 + O(43^6)$ \\\hline
47 & $11 + 37\cdot47 + 27\cdot47^2 + 23\cdot47^3 + 22\cdot47^4 + 34\cdot47^5 + 3\cdot47^6 + O(47^7)$ \\\hline
53 & $26\cdot53^{-2} + 30\cdot53^{-1} + 20 + 47\cdot53 + 10\cdot53^2 + 32\cdot53^3 + O(53^4)$ \\\hline
\end{tabular}
\end{center}
Note that when $p=53$ we have $\#E(\F_p)=p$,
i.e., $p$ is anomalous.
\subsection{Rank $2$}
The first curve of rank $2$ is the curve 389A of conductor 389.
The $p$-adic regulators of this curve are as follows:
\begin{center}
\begin{tabular}{|c|l|}\hline
$p$ & $p$-adic regulator of {\bf 389A}\\ \hline
5 & $1 + 2\cdot5 + 2\cdot5^2 + 4\cdot5^3 + 3\cdot5^4 + 4\cdot5^5 + 3\cdot5^6 + O(5^7)$ \\\hline
7 & $6 + 3\cdot7^2 + 2\cdot7^3 + 6\cdot7^4 + 7^5 + 2\cdot7^6 + O(7^7)$ \\\hline
11 & $4 + 7\cdot11 + 6\cdot11^2 + 11^3 + 9\cdot11^4 + 10\cdot11^5 + 3\cdot11^6 + O(11^7)$ \\\hline
13 & $9 + 12\cdot13 + 10\cdot13^2 + 5\cdot13^3 + 5\cdot13^4 + 13^5 + 9\cdot13^6 + O(13^7)$ \\\hline
17 & $4 + 8\cdot17 + 15\cdot17^2 + 11\cdot17^3 + 13\cdot17^4 + 16\cdot17^5 + 6\cdot17^6 + O(17^7)$ \\\hline
19 & $3 + 5\cdot19 + 8\cdot19^2 + 16\cdot19^3 + 13\cdot19^4 + 14\cdot19^5 + 11\cdot19^6 + O(19^7)$ \\\hline
23 & $17 + 23 + 22\cdot23^2 + 16\cdot23^3 + 3\cdot23^4 + 15\cdot23^5 + O(23^7)$ \\\hline
29 & $9 + 14\cdot29 + 22\cdot29^2 + 29^3 + 22\cdot29^4 + 29^5 + 20\cdot29^6 + O(29^7)$ \\\hline
31 & $1 + 17\cdot31 + 4\cdot31^2 + 16\cdot31^3 + 18\cdot31^4 + 21\cdot31^5 + 8\cdot31^6 + O(31^7)$ \\\hline
37 & $28 + 37 + 11\cdot37^2 + 7\cdot37^3 + 3\cdot37^4 + 24\cdot37^5 + 17\cdot37^6 + O(37^7)$ \\\hline
41 & $20 + 26\cdot41 + 41^2 + 29\cdot41^3 + 38\cdot41^4 + 31\cdot41^5 + 23\cdot41^6 + O(41^7)$ \\\hline
43 & $40 + 25\cdot43 + 15\cdot43^2 + 18\cdot43^3 + 36\cdot43^4 + 35\cdot43^5 + O(43^6)$ \\\hline
47 & $25 + 24\cdot47 + 7\cdot47^2 + 11\cdot47^3 + 35\cdot47^4 + 3\cdot47^5 + 9\cdot47^6 + O(47^7)$ \\\hline
\end{tabular}
\end{center}
\subsection{Rank $3$}
The first curve of rank $3$ is the curve 5077A of conductor 5077.
The $p$-adic regulators of this curve are as follows:
\begin{center}
\begin{tabular}{|c|l|}\hline
$p$ & $p$-adic regulator of {\bf 5077A}\\ \hline
5 & $5^{-2} + 5^{-1} + 4 + 2\cdot5 + 2\cdot5^2 + 2\cdot5^3 + 4\cdot5^4 + 2\cdot5^5 + 5^6 + O(5^7)$ \\\hline
7 & $1 + 3\cdot7 + 3\cdot7^2 + 4\cdot7^3 + 4\cdot7^5 + O(7^7)$ \\\hline
11 & $6 + 11 + 5\cdot11^2 + 11^3 + 11^4 + 8\cdot11^5 + 3\cdot11^6 + O(11^7)$ \\\hline
13 & $2 + 6\cdot13 + 13^3 + 6\cdot13^4 + 13^5 + 4\cdot13^6 + O(13^7)$ \\\hline
17 & $11 + 15\cdot17 + 8\cdot17^2 + 16\cdot17^3 + 9\cdot17^4 + 5\cdot17^5 + 11\cdot17^6 + O(17^7)$ \\\hline
19 & $17 + 9\cdot19 + 10\cdot19^2 + 15\cdot19^3 + 6\cdot19^4 + 13\cdot19^5 + 17\cdot19^6 + O(19^7)$ \\\hline
23 & $7 + 17\cdot23 + 19\cdot23^3 + 21\cdot23^4 + 19\cdot23^5 + 22\cdot23^6 + O(23^7)$ \\\hline
29 & $8 + 16\cdot29 + 11\cdot29^2 + 20\cdot29^3 + 9\cdot29^4 + 8\cdot29^5 + 24\cdot29^6 + O(29^7)$ \\\hline
31 & $17 + 11\cdot31 + 28\cdot31^2 + 3\cdot31^3 + 17\cdot31^5 + 29\cdot31^6 + O(31^7)$ \\\hline
43 & $9 + 13\cdot43 + 15\cdot43^2 + 32\cdot43^3 + 28\cdot43^4 + 18\cdot43^5 + 3\cdot43^6 + O(43^7)$ \\\hline
47 & $29 + 3\cdot47 + 46\cdot47^2 + 4\cdot47^3 + 23\cdot47^4 + 25\cdot47^5 + 37\cdot47^6 + O(47^7)$ \\\hline
\end{tabular}
\end{center}
For $p=5$ and $E$ the curve 5077A, we have $\#E(\F_5) = 10$,
so $a_p\con 1\pmod{5}$, hence $p$ is anamolous.
%\edit{William: Barry, is 5077A still considered anamolous at $5$
%even though we don't have $a_p=1$?}
\subsection{Rank $4$}
Next we consider the curve of rank $4$ with smallest
known conductor ($234446 = 2\cdot 117223$):
$$
y^2 + xy = x^3 - x^2 - 79x + 289.
$$
Note that computation of the $p$-adic heights is just as fast for this
curve as the above curves, i.e., our algorithm for computing heights
is insensitive to the conductor, only the prime $p$ (of course,
computing the Mordell-Weil group could take much longer if the
conductor is large).
\begin{center}
\begin{tabular}{|c|l|}\hline
$p$ & $p$-adic regulator of rank $4$ curve\\ \hline
5 & $2\cdot5^{-2} + 2\cdot5^{-1} + 3\cdot5 + 5^2 + 4\cdot5^3 + 4\cdot5^4 + 3\cdot5^5 + 3\cdot5^6 + O(5^7)$ \\\hline
7 & $6\cdot7 + 4\cdot7^2 + 5\cdot7^3 + 5\cdot7^5 + 3\cdot7^6 + O(7^7)$ \\\hline
11 & $5 + 10\cdot11 + 5\cdot11^2 + 11^3 + 3\cdot11^5 + 11^6 + O(11^7)$ \\\hline
13 & $12 + 2\cdot13 + 4\cdot13^2 + 10\cdot13^3 + 3\cdot13^4 + 5\cdot13^5 + 7\cdot13^6 + O(13^7)$ \\\hline
17 & $15 + 8\cdot17 + 13\cdot17^2 + 5\cdot17^3 + 13\cdot17^4 + 7\cdot17^5 + 14\cdot17^6 + O(17^7)$ \\\hline
19 & $14 + 16\cdot19 + 15\cdot19^2 + 6\cdot19^3 + 10\cdot19^4 + 7\cdot19^5 + 13\cdot19^6 + O(19^7)$ \\\hline
23 & $3 + 15\cdot23 + 15\cdot23^2 + 12\cdot23^4 + 20\cdot23^5 + 7\cdot23^6 + O(23^7)$ \\\hline
29 & $25 + 4\cdot29 + 18\cdot29^2 + 5\cdot29^3 + 27\cdot29^4 + 23\cdot29^5 + 27\cdot29^6 + O(29^7)$ \\\hline
31 & $21 + 26\cdot31 + 22\cdot31^2 + 25\cdot31^3 + 31^4 + 3\cdot31^5 + 14\cdot31^6 + O(31^7)$ \\\hline
37 & $34 + 14\cdot37 + 32\cdot37^2 + 25\cdot37^3 + 28\cdot37^4 + 36\cdot37^5 + O(37^6)$ \\\hline
41 & $33 + 38\cdot41 + 9\cdot41^2 + 35\cdot41^3 + 25\cdot41^4 + 15\cdot41^5 + 30\cdot41^6 + O(41^7)$ \\\hline
43 & $14 + 34\cdot43 + 12\cdot43^2 + 26\cdot43^3 + 32\cdot43^4 + 26\cdot43^5 + O(43^6)$ \\\hline
47 & $43 + 47 + 17\cdot47^2 + 28\cdot47^3 + 40\cdot47^4 + 6\cdot47^5 + 7\cdot47^6 + O(47^7)$ \\\hline
\end{tabular}
\end{center}
\subsection{Rank $5$}
Next we consider the curve of rank $5$ with smallest
known conductor, which is the prime 19047851.
The curve is
$$
y^2 + y = x^3 - 79x + 342
$$
\begin{center}
\begin{tabular}{|c|l|}\hline
$p$ & $p$-adic regulator of rank $5$ curve\\ \hline
5 & $2\cdot5 + 5^2 + 5^3 + 2\cdot5^4 + 5^5 + 5^6 + O(5^7)$ \\\hline
7 & $2 + 6\cdot7 + 4\cdot7^2 + 3\cdot7^3 + 6\cdot7^4 + 2\cdot7^5 + 4\cdot7^6 + O(7^7)$ \\\hline
11 & $10 + 11 + 6\cdot11^2 + 2\cdot11^3 + 6\cdot11^4 + 7\cdot11^5 + 5\cdot11^6 + O(11^7)$ \\\hline
13 & $11 + 8\cdot13 + 3\cdot13^2 + 4\cdot13^3 + 10\cdot13^4 + 5\cdot13^5 + 6\cdot13^6 + O(13^7)$ \\\hline
17 & $4 + 11\cdot17 + 4\cdot17^2 + 5\cdot17^3 + 13\cdot17^4 + 5\cdot17^5 + 2\cdot17^6 + O(17^7)$ \\\hline
19 & $11 + 7\cdot19 + 11\cdot19^2 + 7\cdot19^3 + 9\cdot19^4 + 6\cdot19^5 + 10\cdot19^6 + O(19^7)$ \\\hline
23 & $14 + 14\cdot23 + 20\cdot23^2 + 6\cdot23^3 + 19\cdot23^4 + 9\cdot23^5 + 15\cdot23^6 + O(23^7)$ \\\hline
29 & $3 + 5\cdot29 + 20\cdot29^3 + 21\cdot29^4 + 18\cdot29^5 + 11\cdot29^6 + O(29^7)$ \\\hline
31 & $4 + 26\cdot31 + 11\cdot31^2 + 12\cdot31^3 + 3\cdot31^4 + 15\cdot31^5 + 22\cdot31^6 + O(31^7)$ \\\hline
37 & $3 + 20\cdot37 + 11\cdot37^2 + 17\cdot37^3 + 33\cdot37^4 + 5\cdot37^5 + O(37^7)$ \\\hline
41 & $3 + 41 + 35\cdot41^2 + 29\cdot41^3 + 22\cdot41^4 + 27\cdot41^5 + 25\cdot41^6 + O(41^7)$ \\\hline
43 & $35 + 41\cdot43 + 43^2 + 11\cdot43^3 + 32\cdot43^4 + 11\cdot43^5 + 18\cdot43^6 + O(43^7)$ \\\hline
47 & $25 + 39\cdot47 + 45\cdot47^2 + 25\cdot47^3 + 42\cdot47^4 + 13\cdot47^5 + O(47^6)$ \\\hline
\end{tabular}
\end{center}
Note that the regulator for $p=5$ is not a unit, and $\#E(F_5)=9$.
This is the only example of a regulator in our tables with
positive valuation.
%\begin{figure}[H]\caption{Twists of $X_0(11)$}\label{tbl:twists}
%\begin{center}
%\begin{tabular}{|c||c|c|c|c|c|c|} \hline
% & 5 & 7 & 13 & 17 & 23 \\ \hline\hline
%$-47$ & 241 & 48 & 8448 & 71282 & $23^{-2}\cdot 125405$ \\\hline
%232 & 522 & 239 & 8966 & 57386 & 16874 \\ \hline
%\end{tabular}
%\end{center}
%\end{figure}
\part{Computing expansions for $\E_2$ in terms of classical modular forms}\label{part2}
We next study convergence of $\E_2$ in the general context of $p$-adic
and overconvergent modular forms. Coleman, Gouvea, and Jochnowitz
prove in \cite{MR1317641} that $\E_2$ is {\em transcendental} over the
ring of overconvergent modular forms, so $\E_2$ is certainly
non-overconvergent. However, $\E_2$ is {\em log convergent} in a
sense that we make precise in this part of the paper.
%The questions of this section were
%motivated by searching for a (slow) method of computation of $\E_2$,
%which, however, works for every ordinary elliptic curve ``once and for
%all.''
%More precisely, suppose $E_t$ is an elliptic curves over $\Q(t)$ with
%invariant differential $\omega_t$. It would be interesting to obtain
%a formula for $\E_2(E_t,\omega_t)$ as an element of $\Q_p((t))$. This
%might shed light on the analytic behavior of the $p$-adic modular
%form~$\E_2$. Note that Wuthrich \cite{wuthrich:heightfamily} has
%studied the related problem of finding the height of a point $P_t$ on
%$E_t$.
\section{Questions about rates of convergence}\label{sec:logconvgen}
Fix $p$ a prime number, which, in this section, we will assume is $\ge
5$. We only consider modular forms of positive even integral weight,
on $\Gamma_0(M)$ for some~$M$, and with Fourier coefficients in
$\C_p$. By a {\it classical modular form} we will mean one with these
properties, and by a {\it Katz modular form} we mean a $p$-adic
modular form in the sense of Katz (\cite{katz:padicprop}), again with
these properties, i.e., of integral weight $k \ge 0$, of tame
level~$N$ for a positive integer~$N$ prime to~$p$, and with Fourier
coefficients in $\C_p$. A {\em $p$-integral modular form} is a
modular form with Fourier coefficients in $\Z_p$.
Note that throughout Sections~\ref{sec:logconvgen} and
\ref{sec:preciselogconv}, all our modular forms can be taken to be
with coefficients in~$\Q_p$.
If~$f$ is a classical, or Katz, modular form, we will often simply
identify the form~$f$ with its Fourier expansion, $f = \sum_{n \ge 0}
c_f(n)q^n$. By $\ord_p(f)$ we mean the greatest lower bound of the
non-negative integers $ {\rm ord}_p(c_f(n))$ for $n \ge 0$. The
valuation $\ord_p$ on $\C_p$ here is given its natural normalization,
i.e., $\ord_p(p) =1$.
We say two $p$-integral modular forms are {\em congruent}
modulo~$p^n$, denoted $$f \equiv g\ \pmod{p^n},$$ if their corresponding
Fourier coefficients are congruent modulo $p^n$. Equivalently, $f
\equiv g \pmod{p^n}$ if $\ord_p(f-g) \ge n$.
Recall the traditional notation,
$$
\sigma_{k-1}(n)=\sum_{0\,\, <\,\, d \ | \ n}d^{k-1},
$$
and put $\sigma(n)=\sigma_1(n)$.
Let $E_k= -b_k/2k + \sum_{n=0}^{\infty}\sigma_{k-1}(n) q^n$ be the
Eisenstein series of even weight $k \ge 2$, and denote by ${\cal E}_k$
the ``other natural normalization" of the Eisenstein series,
$$
{\cal E}_k= 1 -\frac{2k}{b_k} \cdot
\sum_{n=0}^{\infty}\sigma_{k-1}(n) q^n,$$ for $k \ge 2$. We have
$${\cal E}_{p-1} \equiv 1 \pmod{p}.
$$
(Note that ${\cal E}_k$ is the $q$-expansion of the Katz
modular form that we denote by ${\bf E}_k$ elsewhere in
this paper.)
For $k > 2$ these are classical modular forms of level $1$, while the
Fourier series $E_2= -1/24 + \sum_{n=0}^{\infty}\sigma(n) q^n$, and
the corresponding ${\cal E}_2$, are not; nevertheless, they may all be
viewed as Katz modular forms of tame level $1$.
Put
$$
\sigma^{(p)}(n)=\sum_{0 \,\,<\,\, d \ | \ n; \ (p,d)=1}d,
$$
so that we have:
\begin{equation}\label{eqn:logconv1}
\sigma(n)= \sigma^{(p)}(n) + p\sigma^{(p)}(n/p) + p^2\sigma^{(p)}(n/p^2)+\cdots
\end{equation}
where the convention is that $\sigma^{(p)}(r)=0$ if $r$ is not an
integer.
Let $V = V_p$ be the operator on power series given by the rule:
$$V\left(\sum _{n\ge 0}c_nq^n\right) = \ \sum _{n\ge 0}c_nq^{pn}.$$
If $F=\sum _{n\ge 0}c_nq^n$ is a classical modular form of weight $k$
on $\Gamma_0(M)$, then $V(F)$ is (the Fourier expansion of) a
classical modular form of weight $k$ on $\Gamma_0(Mp)$
(cf. \cite[Ch. VIII]{lang:modular}).
The Fourier series
$$
E_2^{(p)}= (1-pV)E_2 = \frac{p-1}{24} + \sum \sigma_{1}^{(p)}(n)q^n
$$
is, in contrast to $E_2$, a classical modular form (of weight~$2$ on
$\Gamma_0(p)$) and we can invert the formula of its definition to
give the following equality of Fourier series:
\begin{equation}\label{eqn:logconv2}
E_2 = \sum_{\nu \ge 0} p^{\nu}V^{\nu} E_2^{(p)},
\end{equation}
this equality being, for the corresponding Fourier coefficients other
than the constant terms, another way of phrasing (\ref{eqn:logconv1}).
\begin{definition}[Convergence Rate]\label{defn:convrate} We
call a function $\alpha(\nu)$
taking values that are either positive integers or $+\infty$ on
integers $\nu = 0, \pm 1, \pm 2, \dots$ a {\em convergence rate} if
$\alpha(\nu)$ is a non-decreasing function such that $\alpha(\nu) =
0$ for $ \nu \le 0$, $\alpha(\nu+\mu) \le \alpha(\nu) +
\alpha(\mu)$, and $\alpha(\nu)$ tends to $+\infty$ as $\nu$ does.
\end{definition}
A simple nontrivial example of a convergence rate is
$$
\alpha(\nu) = \begin{cases} 0 & \text{ for } \nu \leq 0,\\
\nu & \text{ for } \nu \geq 0.
\end{cases}
$$
If $\alpha(\nu)$ is a convergence rate, put $T\alpha(\nu) =
\alpha(\nu-1)$; note that $T\alpha(\nu)$ is also a convergence rate
($T$ translates the graph of $\alpha$ one to the right).
Given a collection $\{\alpha_j\}_{j \in J}$ of convergence rates, the
``max" function $\alpha(\nu)= {\rm max}_{j \in J}\ \alpha_j(\nu)$ is
again a convergence rate.
\begin{definition}[$\alpha$-Convergent] Let $\alpha$ be a
convergence rate. A Katz modular form $f$ is $ \alpha$-{\em
convergent} if there is a function $a:\Z_{\geq
0} \to \Z_{\geq 0}$ such that
\begin{equation}\label{eqn:logconv3}
f = \sum_{\nu=0}^{\infty} p^{a(\nu)}f_{\nu}{\cal
E}_{p-1}^{-\nu}
\end{equation}
with $f_\nu$ a classical $p$-integral modular form (of weight
$k+\nu(p-1)$ and level $N$) and $a(\nu) \ge \alpha(\nu)$ for
all $\nu \ge 0$.
\end{definition}
If $ \alpha' \le \alpha$ are convergence rates and a modular form~$f$
is $\alpha$-{\em convergent} then it is also
$\alpha'$-{\em convergent}. As formulated,
an expansion of the shape of (\ref{eqn:logconv3}) for a given $f$ is
not unique but \cite{katz:padicprop} and \cite{gouvea:slnm} make a
certain sequence of choices that enable them to get canonical
expansions of the type (\ref{eqn:logconv3}), dependent on those
initial choices. Specifically, let $M_{\rm classical} (N, k, \Z_p)$
denote the $\Z_p$-module of classical modular forms on $\Gamma_0(N)$
of weight $k$ and with Fourier coefficients in $\Z_p$. Multiplication
by ${\cal E}_{p-1}$ allows one to identify $M_{\rm classical} (N, k,
\Z_p)$ with a saturated ${\Z}_p$-lattice in $M_{\rm classical} (N,
k+p-1, \Z_p)$. (The lattice is saturated because multiplication by
$E_{p-1} \!\!\!\pmod{p}$ is injective, since it is the identity map on
$q$-expansions.) {\em Fix}, for each $k$, a $\Z_p$-module,
$$
C(N, k+p-1, \Z_p)\subset M_{\rm classical} (N, k+p-1, \Z_p)
$$
that is complementary to ${\cal E}_{p-1}\cdot M_{\rm classical} (N, k,
\Z_p)\subset M_{\rm classical} (N, k+p-1, \Z_p)$. Requiring the
classical modular forms $f_\nu$ of the expansion (\ref{eqn:logconv3})
to lie in these complementary submodules, i.e., $f_\nu \in C (N,
k+\nu(p-1), \Z_p)$ for all~$\nu$, pins down the expansion uniquely.
Let us call an expansion of the form $$f = \sum_{\nu=0}^{\infty}
p^{a(\nu)}f_\nu{\cal E}_{p-1}^{-\nu}$$ pinned down by a choice of
complementary submodules as described above a {\em Katz expansion} of
$f$.
A {\em classical} $p$-integral modular form is, of course, $
\alpha$-convergent for every $\alpha$. For any given convergence
rate~$\alpha$, the $ \alpha$-convergent Katz modular forms of tame
level $N$ are closed under multiplication, and the collection of them
forms an algebra over the ring of classical modular forms of level $N$
(with Fourier coefficients in $\Z_p$). Any Katz $p$-integral modular
form is $\alpha$-convergent, for some convergence rate~$\alpha$
(see~\cite{gouvea:slnm}).
% .\edit{Why; I couldn't easily see this -- William. THis is given
% very explicitly here, as I remember. But maybe we want a more
% specific citation in this book?}
\begin{proposition}\label{prop:convcong}
A Katz $p$-integral modular form $f$ of weight $k$ and tame level
$N$ as above is $\alpha$-convergent if and only if the Fourier
series of $f{\cal E}_{p-1}^{\nu}$ is congruent to the Fourier series
of a classical $p$-integral modular form (of weight $k + \nu(p-1)$
and level $N$) modulo $p^{\alpha(\nu+1)}$ for every integer $\nu \ge
0$.
\end{proposition}
\begin{proof}
We use the $q$-expansion principle. Specifically, if $G_{\nu}$ is
a classical modular form such that $f{\cal E}_{p-1}^{\nu}\equiv
G_{\nu}\pmod{p^{\alpha(\nu+1)}}$ then $g_{\nu}=
p^{-\alpha(\nu+1)}(f{\cal E}_{p-1}^{\nu}-G_{\nu})$ is again a Katz
modular form, and we can produce the requisite $\alpha$-convergent
Katz expansion by inductive consideration of these $g_{\nu}$'s.
(Note that the other implication is trivial. Also note our running
hypothesis that $p\geq 5$.)
%\edit{*** We only proved <==. What about ==>. I think this other
%direction is trivial by reducing both sides modulo $p^*$, {\em except}
%possibly if $p=3$, since then we get $\cE_2$ appearing in first
%of the sum. I guess we excluded $p\leq 3$ for our running hypothesis.
%We might mention that this is a place where we use this exclusion. -- William}
\end{proof}
In view of this, we may define, for any $f$ as in
Proposition~\ref{prop:convcong}, the function $a_f(\nu)$ (for $\nu \ge
0$) as follows: $a_f(0)=0$, and for $\nu \ge 1$, $a_f(\nu)$ is the
largest integer $a$ such that $f{\cal E}_{p-1}^{\nu-1}$ is congruent
to a classical $p$-integral modular form (of weight $k + (\nu-1)(p-1)$ and level
$N$) modulo $p^a$.
\begin{corollary}
The Katz $p$-integral modular form $f$ is $\alpha$-convergent for
any convergence rate~$\alpha$ that is majorized by the function
$a_f$. (I.e., for which $\alpha(\nu) \leq a_f(\nu)$ for all
$\nu\geq 0$.)
\end{corollary}
\begin{definition}[Overconvergent of Radius $r$] Let $r \in \Q$ be a
positive rational number. A Katz $p$-integral modular form $f$ of
tame level $N$ is {\em overconvergent of radius $r$} if and only if
it is $\alpha$-convergent for some function $\alpha$ such that
$\alpha(\nu) \ge r\cdot \nu $ for all $\nu$, and $\alpha(\nu)-
r\cdot \nu$ tends to infinity with $\nu$.
\end{definition}
\begin{remarks} It is convenient to say, for two function
$\alpha(\nu)$ and $\alpha'(\nu)$, that
$$\alpha(\nu) \ggeq \alpha'(\nu)$$
if $\alpha(\nu) \ge \alpha'(\nu)$ and $\alpha(\nu) - \alpha'(\nu)$
tends to infinity with $\nu$. So, we may rephrase the above definition
as saying that $f$ is overconvergent with radius $r$ if it is
$\alpha$-convergent with $\alpha(\nu) \ggeq r\cdot \nu $. The above
definition is equivalent to the definition of \cite{katz:padicprop,
gouvea:slnm} except for the fact that the word {\em radius} in these
references does not denote the rational number $r$ above, but rather a
choice of $p$-adic number whose $\ord_p$ is $r$. We may think of
our manner of phrasing the definition as being a {\em definition by
Katz expansion convergence rate} as opposed to what one might call
the {\em definition by rigid analytic geometric behavior}, meaning the
equivalent, and standard, formulation (cf. \cite{katz:padicprop}) given
by considering $f$ as a rigid analytic function on an appropriate
extension of the Hasse domain in the (rigid analytic space
associated to) $X_0(N)$.
\end{remarks}
\begin{definition}[(Precisely) Log Convergent]\label{defn:logconv}
A Katz $p$-integral modular form $f$ is {\em log-convergent} if
$c\cdot \log(\nu) \le a_f(\nu)$ for some positive constant $c$ and
all but finitely many~$\nu$ (equivalently: if it is
$\alpha$-convergent for $\alpha(\nu) = c\cdot \log(\nu)$ for some
positive constant~$c$). We will say that $f$ is {\em precisely
log-convergent} if there are positive constants $c, C$ such that
$c\cdot \log(\nu) \le a_f(\nu) \le C\cdot \log(\nu)$ for all but
finitely many~$\nu$.
\end{definition}
\begin{remark}
As in Definition~\ref{defn:convrate} above, we may think of this
manner of phrasing the definition as being a {\em definition by Katz
expansion convergence rate}. This seems to us to be of some
specific interest in connection with the algorithms that we present
in this article for the computation of $\E_2$. For more theoretical
concerns, however, we think it would be interesting to give, if
possible, an equivalent {\em definition by rigid analytic geometric
behavior}: is there some explicit behavior at the ``rim" of the
Hasse domain that characterizes log-convergence?
\end{remark}
\begin{proposition}\label{prop:katzadmit}
Let $p\geq 5$.
Let~$f$ be a Katz $p$-integral modular form of
weight~$k$ and tame level~$N$ that admits an expansion of the type
$$f = \sum_{\nu=0}^{\infty} p^{\nu}{\cal F}_{\nu}{\cal
E}_{p-1}^{-\nu}$$ where, for all $\nu \ge 0$, ${\cal F}_\nu$ is a
classical $p$-integral modular form (of weight $k + \nu(p-1)$) on
$\Gamma_0(p^{\nu+1})$. Then $f$ is log-convergent
and $$\liminf_{n\to\infty} \frac{a_f(n)}{\log(n)} \geq \frac{1}{\log(p)}.$$
\end{proposition}
\begin{proof}
The classical modular form ${\cal F}_\nu$ on $\Gamma_0(p^{\nu+1})$
is an overconvergent Katz modular form of radius $r$ for any $r$
such that $r < {\frac{1}{p^{\nu-1}(p+1)}}$ (cf. \cite{katz:padicprop},
\cite[Cor.~II.2.8]{gouvea:slnm}). Let
$${\cal F}_\nu = \sum_{\mu=0}^{\infty} f_{\mu}^{(\nu)}{\cal E}_{p-1}^{-\mu}$$ be its Katz expansion. So,
$$\ord_p( f_{\mu}^{(\nu)}) \ggeq \left(\frac{1}{p^{\nu-1}(p+1)}
-\epsilon_{\mu, \nu}\right)\cdot \mu$$ for any choice of positive
$\epsilon_{\mu, \nu}$. We have $$f = \sum_{\nu=0}^{\infty}
p^{\nu}\sum_{\mu=0}^{\infty} f_{\mu}^{(\nu)}{\cal
E}_{p-1}^{-(\mu+\nu)},$$ or (substituting $\gamma = \mu + \nu$)
$$
f = \sum_{\gamma =0}^{\infty} \left\{\sum_{\nu=0}^{\gamma} p^{\nu}\ f_{\gamma-\nu}^{(\nu)}\right\}{\cal E}_{p-1}^{-\gamma}.
$$
Putting $G_{\gamma} = \sum_{\nu=0}^{\gamma} p^{\nu}\ f_{\gamma-\nu}^{(\nu)}$ we may write the above expansion as
$$f = \sum_{\gamma =0}^{\infty} G_{\gamma}{\cal E}_{p-1}^{-\gamma},$$ and we must show that
$$\ord_p (G_{\gamma}) \ge c\cdot {\rm log}(\gamma)$$ for some positive constant $c$.
For any $\nu \le \gamma$ we have
$$
\ord_p \left( p^{\nu}\ f_{\gamma-\nu}^{(\nu)}\right) \ggeq \nu +
\left(\frac{1}{p^{\nu-1}(p+1)}
-\epsilon_{\gamma-\nu, \nu}\right)(\gamma-\nu).
$$
We need to find a lower bound for the minimum value achieved by the
right-hand side of this equation. To prepare for this, first note
that at the extreme value $\nu =0$ we compute $\ord_p (\
f_{\gamma}^{(0)}) \ge \left(\frac{p}{(p+1)} -\epsilon_{\gamma, 0}\right)\cdot \gamma$,
and to study the remaining cases, $\nu =1,\dots, \gamma$, we look at
the function
$$R(t)= t + \left(\frac{1}{p^{t-1}(p+1)}\right)(\gamma-t)$$
in the range $1\le t \le \gamma$. This, by calculus, has a unique
minimum at $t = t_\gamma \in (1, \gamma)$ given by the equation
\begin{equation}\label{eqn:logconv4}
\frac{p+1}{p}\cdot
p^{t_\gamma} = \log(p)\cdot (\gamma - t_\gamma) +1.
\end{equation}
Define $e_{\gamma} = t_{\gamma}- \log_p (\gamma)$ and substituting, we get:
\begin{equation}
p^{e_\gamma} = \frac{p\log(p)}{p+1} - \frac{p\log(p)}{p+1} \frac{e_{\gamma}}{\gamma} +A_{\gamma}
\end{equation}
where $A_{\gamma}$ goes to zero, as $\gamma$ goes to $\infty$.
If $e_\gamma$ is positive we get that $$ p^{e_\gamma} \le \frac{p\log(p)}{p+1} +A_{\gamma}$$ and so $e_\gamma$ is bounded from above, independent of $\gamma$, while if $e_{\gamma} = -d_{\gamma} $ with $d_\gamma$ positive, we have
$$ \frac {1}{p^{d_\gamma}} = \frac{p\log(p)}{p+1} + \frac{p\log(p)}{p+1} \frac{d_{\gamma}}{\gamma} +A_{\gamma}.$$
Recall that since $t_\gamma > 0$ we also have $d_\gamma <
\log_p(\gamma)$, so that the right hand side of the displayed equation
tends to $\frac{p\log(p)}{p+1}$ as $\gamma$ goes to~$\infty$, so the
equation forces~$d_\gamma$ to be bounded from above, as $\gamma$ tends
to~$\infty$.
This discussion gives:
\begin{lemma} \label{tate:lem6.4}
The quantity $|t_\gamma-\log_p(\gamma)|$ is bounded independent of $\gamma$.
\end{lemma}
Substituting $t_{\gamma} = \log_p(\gamma) + e_{\gamma}$ in the
defining equation for $R(t)$ and noting the boundedness of
$|e_{\gamma}|$, we get that $|R(t_{\gamma})-\log_p(\gamma)|$ is bounded as
$\gamma$ goes to~$\infty$, thereby establishing our proposition.
\end{proof}
%\begin{remark}
% The ``$c$" of Definition~\ref{defn:logconv} can be taken to be any
% positive number $< {\rm log}(p)$.
%\end{remark}
\begin{corollary}\label{cor:katzlc}
For all $p\geq 5$, the Katz modular form $f=E_2$ is log-convergent
and
$$
\liminf_{n\to\infty} \frac{a_f(n)}{\log(n)} \ge \frac{1}{\log(p)}.
$$
\end{corollary}
\begin{proof}
The modular forms $V^{\nu}E_2^{(p)}$ are classical modular forms on
$\Gamma_0(p^{\nu+1})$ and therefore formula (\ref{eqn:logconv1}) exhibits $E_2$
as having a Katz expansion of the shape of (\ref{eqn:logconv3}).
Proposition~\ref{prop:katzadmit} then implies the corollary.
\end{proof}
\begin{remark}\label{rem:precise}
Is $E_2$ {\em precisely} log-convergent? The minimal $c$ (cf.
Definition~\ref{defn:logconv}) that can be taken in the $\log$-convergence
rate for $f=E_2$ is $\limsup_{n\to \infty}(a_f(n)/\log(n))$.
Is this minimal $c$ equal to $1/\log(p)$? It is for $p=5$,
as we will show in Section~\ref{sec:preciselogconv}.
The
previous discussion tells us that, as a kind of generalization of
the well-known congruence
$$
E_2{\cal E}_{p-1} \equiv E_{p+1}\pmod{p},
$$
we have that for any $\epsilon > 0$, and all but finitely many $\nu$,
there are classical modular forms ${\cal G}_{\nu}$ of
level $1$ and weight $2+\nu(p-1)$ such that $$E_2{\cal
E}_{p-1}^{\nu} \equiv {\cal G}_{\nu}
\pmod{p^{\lfloor(1-\epsilon){\rm log}_p(\nu)\rfloor}} .$$
\end{remark}
Let $\theta = qd/dq$ denote the standard shift operator; so that if $f
= \sum_{n\ge 0}c_nq^n$, then $\theta(f )= \sum_{n\ge 0}nc_nq^n$. We
have $\ord_p(\theta (f)) \ge \ord_p(f)$. The operator
$\theta$ preserves Katz modular forms, and {\em almost} preserves
classical modular forms in the sense that if $f$ is a classical
modular form of weight $k \ge 2$ then so is $F=\theta(f) -kfE_2/12$
(cf.~\cite{katz:padicprop}). Note, also, that $\ord_p(F) \ge \ord_p(f)$.
\begin{corollary}
The operator $\theta$ preserves log-convergent Katz modular forms.
\end{corollary}
\begin{proof}
Let $f$ be a log-convergent Katz $p$-integral modular form of weight
$k$, of tame conductor~$N$ with a Katz expansion,
\begin{equation}\label{eqn:logconv5}
f = \sum_{\nu=0}^{\infty} p^{a(\nu)}f_{\nu}{\cal E}_{p-1}^{-\nu}
\end{equation}
where $a(\nu) \ge c\cdot {\rm log}(\nu)$ for some positive $c$, and
the $f_{\nu}$'s are classical $p$-integral modular forms on
$\Gamma_0(N)$. Let $F_{\nu}= \theta(f_{\nu}) -(k+
\nu(p-1))f_{\nu}E_2/12$ (which is a classical modular form of weight
$k+2+ \nu(p-1)$ on $\Gamma_0(N)$). Put
$$
G= \theta(E_{p-1}) - \frac{p-1}{12} {\cal E}_{p-1}E_2.
$$
Apply the derivation $\theta$ to (\ref{eqn:logconv5}) to get
\begin{align*}
\theta(f) &= \sum_{\nu=0}^{\infty} p^{a(\nu)}\Bigl\{(F_{\nu}+(k+ \nu(p-1))f_{\nu}E_2/12) {\cal E}_{p-1}^{-\nu} - \\
& \qquad\qquad \qquad\qquad \qquad \nu f_{\nu}{\cal E}_{p-1}^{-\nu-1}\left(G+
\frac{p-1}{12} {\cal E}_{p-1}E_2\right) \Bigr\}.
\end{align*}
or:
$$ \theta(f) = A + BE_2 - C - DE_2,$$ where
\begin{align*}
A&=\sum_{\nu=0}^{\infty} p^{a(\nu)}F_{\nu}{\cal E}_{p-1}^{-\nu},\\
B&= \sum_{\nu=0}^{\infty} p^{a(\nu)}(k+ \nu(p-1))f_{\nu}/12) {\cal E}_{p-1}^{-\nu},\\
C&= \sum_{\nu=0}^{\infty} p^{a(\nu)}\nu f_{\nu}G {\cal E}_{p-1}^{-\nu-1},\\
D&= \sum_{\nu=0}^{\infty} p^{a(\nu)}\frac{p-1}{12}\nu
f_{\nu}{\cal E}_{p-1}.
\end{align*}
Now $A,B, C,D$ are all log-convergent, as is $E_2$ by
Corollary~\ref{cor:katzlc}. Therefore so is $ \theta(f) $.
\end{proof}
\section{Precise log convergence of $E_2$ for $p=2,3,5$}\label{sec:preciselogconv}
In this section we assume $p=2$, $3$ or $5$ and let $P, Q, R$ denote
the Eisenstein series of level 1 of weights $2, 4, 6$, respectively,
normalized so that the constant term in its Fourier expansion is 1.
Let~$f$ be a Katz form of tame level~1 and weight~$k$. Write
$k=4d+6e$, with~$d$ an integer $\ge -1$ and $e=0$ or~$1$. Then
$fQ^{-d}R^{-e}$ is a Katz form of weight~$0$, that is, a Katz
function. Since~$0$ is the only supersingular value of $j$ for
$p=2,3,5$, a Katz function has an expansion in powers of $j^{-1}$
convergent everywhere on the disc $|j^{-1}| \le 1$. Hence, putting
$z=j^{-1}$, we can write
$$
f = Q^dR^e \sum_{n=0}^\infty c_f(n)z^n
= \sum_{n=0}^\infty R^e\Delta^nQ^{-3n+d}.
$$
with $c_f(n) \in \Q_p$ and $c_f(n) \to 0$ as $ n\to \infty$.
Let $$C_{f,p}(N) = \min_{n>N}(\ord_p(c_f(n))).$$
\begin{theorem}\label{tate:thm6.1}
For $p=5$, we have $C_{f,5}(N) = a_f(3N+1-d)$, for
all large~$N$.
\end{theorem}
\begin{proof}
Notice that for $p=5$, ${\cal E}_{p-1}=Q$. Let $\nu=3N+1-d$ for large $N$.
Then
$$
Q^{\nu-1} f = \sum_{n=0}^N c(n) R^e \Delta^n Q^{3(N-n)}
+ R^eQ^d\sum_{n>N}c(n)z^n
=F+G,
$$
say. We have $\ord_5(G)=\min_{n>N} (\ord_5(c(n))=C_{f,5}(N)$.
\footnote{To justify this claim we extend our definition of $\ord_p$
from the ring of Katz forms with Fourier coefficients in~$\Z$ to the
ring $\Z_p[[q]]$ of all formal power series with coefficients
in~$\Z$. Moreover, since $z \in q+q^2\Z_p[[q]]$, we have
$\Z_p[[q]]=\Z_p[[z]]$, and for a formal series $g=\sum a_nq^n = \sum
b^nz^n$, we have $\ord_p(g) =\min(\ord_p(a_n))=\min(\ord_p(b_n))$.
Also (Gauss Lemma) the rule $\ord(g_1g_2)=\ord(g_1)+\ord(g_2)$
holds. Since $\ord_5(R)=\ord_5(Q)=0$, it follows that
$\ord_5(G)=C_{f,5}(N)$ as claimed.}
Since $F$ is a classical modular form of weight $12N+6e$ it follows
from the definition of $a_f$ that $a_f(\nu) \ge C_{f,5}(N)$. On the
other hand, since $\{R^e\Delta^nQ^{3(N-n)} : 0\le n\le N\}$ is a basis
for the space of classical modular forms of weight $12N+6e$, it is
clear that for any such classical form $F'$, the difference
$Q^{\nu-1}f-F'$ is a 5-adic Katz form which can be written as
$R^eQ^{3N}g$ with $g$ a Katz function whose $z$-expansion coefficients
are $c(n)$ for $n>N$. Thus $\ord_5(Q^{\nu-1}f-F')\le C_{f,5}(N)$.
\end{proof}
We have defined $f$ to be log convergent if
$$
\liminf_{n\to\infty}\frac{a_f(n)}{\log(n)}>0,
$$
and to be precisely log convergent if in addition
$$
\limsup_{n\to\infty}\frac{a_f(n)}{\log(n)}< \infty.
$$
\begin{lemma}\label{tate:lem6.2}
Suppose $h(n)$ and $H(n)$ are nondecreasing funcions
defined for all sufficiently large positive integers $n$. If
for some integers $r>0$ and $s$ we have $H(N)=h(rN+s)$ for all
sufficiently large integers N, then
$$
\liminf_{n\to\infty}\frac{h(n)}{\log(n)}
=\liminf_{N\to\infty}\frac{H(N)}{\log(N)},
$$
and
$$
\limsup_{n\to\infty}\frac{h(n)}{\log(n)}
=\limsup_{N\to\infty}\frac{H(N)}{\log(N)}.
$$
\end{lemma}
\begin{proof}
We use the fact that $\frac{\log(rx+s)}{\log(x)} \to 1$ as
$x\to \infty.$ For $n$ and $N$ related by
$$
rN+s \le n \le r(N+1)+s
$$
we have
$$
\frac{h(n)}{\log(n)} \le \frac{h(r(N+1)+s}{\log(rN+s)}
=\frac{H(N+1)}{\log(N+1)} \cdot \frac{\log(N+1)}{\log(rN+s)}.
$$
Similarly,
$$
\frac{h(n)}{\log(n)} \ge \frac{h(rN+s}{\log(r(N+1)+s)}
=\frac{H(N)}{\log(N)} \cdot \frac{\log(N)}{\log(r(N+1)+s)}.
$$
This proves the lemma, because the second factor of the right hand
term in each line approaches 1 as $N$ goes to infinity.
\end{proof}
Theorem~\ref{tate:thm6.1} and Lemma~\ref{tate:lem6.2} show that for
$p=5$ we can replace $a_f$ by $C_f$ in the definition of log
convergent and precisely log convergent. Therefore we define log
convergent and precisely log convergent for $p=2$ and $p=3$ by using
$C_{f,p}$ as a replacement for $a_f$.
\begin{theorem}\label{tate:thm6.3}
For $p=2, 3\text{ or }5$, the weight 2 Eisenstein series $P=\E_2$ is
precisely log convergent. In fact,
$$ \lim_{n\to\infty} \frac{C_{P,p}(n)}{\log(n)} = \frac{1}{\log(p)}.$$
%\edit{BARRY (asked by John):
% Does Remark~\ref{rem:precise} imply
% that for $f=P$ that $\limsup$ is $\ge 1/\log(p)$, so that this
% theorem can be improved and simplified to
% simply: $\lim_{n \to \infty} (C_{P,p}(n)/log(n) = 1/log(p)$ for
% $p= 2,3$, AND 5? In particular, can we replace the
%$\le$ by an $=$ in the final limit of the theorem?}
\end{theorem}
During the proof of this theorem we write $c(n)=c_P(n)$ and
$C_p(n)=C_{P,p}$.
The cases $p=2, 3$ follow immediately from results
of Koblitz (cf. \cite{koblitz}).
Koblitz writes $ P=\sum a_nj^{-n}\frac{qdj}{jdq}$. Since $dj/j=-dz/z$,
and as we will see later in this proof, $qdz/zdq=R/Q$, Koblitz's $a_n$
is the negative of our $c(n)$, hence $\ord_p(c(n))=\ord_p(a_n)$.
Koblitz shows that if we let $l_p(n) = 1 + \lfloor \log(n)/\log(p) \rfloor$, the
number of digits in the expression of $n$ in base $p$, and let $s_p(n)$
denote the sum of those digits, then $\ord_2(c(n))=l_2(n)+3s_2(n)$ and
$\ord_3(c(n))=l_3(n)+s_3(n)$. From this it is an easy exercise to show
$$
C_2(n)=\lfloor \log(n+1)/\log(2)\rfloor +4 \quad\text{ and }\quad
C_3(n)=\lfloor(\log(n+1)/\log(3)\rfloor + 2,
$$
formulas from which cases $p=2$ and $p=3$ of the theorem are evident.
Investigating the case $p=5$ we found experimentally with a PARI program
that the following conjecture holds for $n<1029$.
\begin{conjecture}\label{tate:conj6.4}
We have $\ord_5(c(n))\ge l_5(2n)$, with equality if $n$ written in
base 5 contains only the digits 0,1 or 2, but no 3 or 4.
\end{conjecture}
It is easy to see that Conjecture~\ref{tate:conj6.4} implies that
$$
\limsup_{n\to\infty} \frac{C_5(n)}{\log(n)} = \frac{1}{\log(5)}.
$$
We already know from Corollary~\ref{cor:katzlc} that
$$
\liminf_{n\to\infty} \frac{a_P(n)}{\log(n)} \ge \frac{1}{\log(5)}.
$$
By Lemma~\ref{tate:lem6.2}, this is equivalent to
$$
\liminf_{n\to\infty} \frac{C_{P,5}(n)}{\log(n)} \ge \frac{1}{\log(5)}.
$$
Hence to finish the proof of Theorem~\ref{tate:thm6.3},
we need only prove
\begin{equation}\label{eqn:thm6.3eq}
\limsup_{n\to\infty} \frac{C_{P,5}(n)}{\log(n)} \le \frac{1}{\log(5)}.
\end{equation}
To prove (\ref{eqn:thm6.3eq}) it is enough to prove that
Conjecture~\ref{tate:conj6.4} holds for $n=5^m$, that is,
$\ord_5(c(n))=m+1$. Indeed that equality implies that $C_5(n)\le m+1$
for $n<5^m$ and, choosing $m$ such that $5^{m-1}\le n < 5^m$, shows
that for every $n$ we have $C_5(n) \le m+1 \le \log(n)/\log(5) +2$.
To prove $\ord_5(c(n))=m+1$ we use two lemmas.
\begin{lemma}\label{tate:lem6.5}
We have $\frac{PQ}{R}-1=3\frac{zdQ}{Qdz}.$
\end{lemma}
\begin{proof}
Let $\theta$ denote the classical operator $qd/dq$.
From the formula $\Delta = q\prod_{n\geq 1}(1-q^n)^{24}$ we get
by logarithmic differentiation the classical formula
$$\frac{\theta\Delta}{\Delta} = P.$$
From $z = 1/j = \Delta/Q^3$ we get by logarithmic differentiation that
$$\frac{\theta z}{z} = \frac{\theta\Delta}{\Delta} - 3\frac{\theta Q}{Q}
= P - 3\frac{\theta Q}{Q}.$$
By a formula of Ramanujan (cf. \cite[Thm.~4]{jps}) we have
$$
3\frac{\theta Q}{Q} = P - \frac{R}{Q}.
$$
Substituting gives
$$
\frac{\theta z}{z} = \frac{R}{Q},
$$
and dividing the next to last equation by the last proves the lemma.
\end{proof}
\begin{lemma}\label{tate:lem6.6}
Let $F=\sum_{n\ge 1} \sigma_3(n)q^n$, so that $Q=1+240F$.
Then $F\equiv \sum_{m\ge 0} (z^{5^m}+ z^{2\cdot 5^m}) \pmod 5$.
\end{lemma}
\begin{proof}
Guessing this result by computer experiment, we asked Serre
for a proof. He immediately supplied two, one of which
is the following. During the rest of
this proof all congruences are understood to be modulo 5. Since
$F=z+3z^2+\cdots$, the statement to be proved is equivalent to
$F-F^5 \equiv z+3z^2$. Using the trivial congruence $Q\equiv 1$
and the congruence $P\equiv R$ (the case $p=5$ of a congruence
of Swinnerton-Dyer, (cf. \cite[Thm.~5]{jps}), we note that
$$z=\Delta/Q^3\equiv \Delta = (Q^3-R^2)/1728 \equiv 2-2R^2.$$
The case $p=5, k=4$ of formula (**) in section 2.2 of \cite{jps} reads
$F-F^5 \equiv \theta^3R$. By Ramanujan's formula
$$
\theta R=(PR-Q^2)/2\equiv 3R^2-3,
$$
one finds that indeed
$$
\theta^3R \equiv 2R^4-R^2-1 \equiv z+3z^2,
$$
which proves Lemma~\ref{tate:lem6.6}.
\end{proof}
Let $F=\sum_{n\ge 1}b(n)z^n$. By Lemma~\ref{tate:lem6.6}, $b(5^m)$ and
$b(2\cdot 5^m)$ are not divisible by 5. Therefore the $5^m$th and
$2\cdot 5^m$th coefficients of $zdF/dz=\sum_{n\ge 1}nb(n)z^n$ are
divisible exactly by $5^m$. By Lemma~\ref{tate:lem6.5} we have
$$
\sum_{n\ge 1}c(n)z^n = \frac{PQ}{R}-1=3\frac{zdQ}{Qdz}=3\frac{240zdF}{(1+240F)dz}.
$$
This shows that $\ord_5(c(5^m))=\ord_5(c(2\cdot 5^m))=m+1$ thereby
completing the proof of Theorem~\ref{tate:thm6.3}.
\begin{remark}
For $p=2$ or 3 a simple analogue of Lemma~\ref{tate:lem6.6} holds,
namely $F\equiv \sum_{m\ge 0} z^{p^m} \pmod p$. This can be used to
obtain Koblitz's result for the very special case $n=p^m$.
\end{remark}
\section{Discussion}
\subsection{Log convergence}
The running hypothesis in Section~\ref{sec:logconvgen} is that $p \ge 5$,
but in Section~\ref{sec:preciselogconv} we considered only $p=2,3,5$.
In dealing with the different primes, our discussion changes
strikingly, depending on the three slightly different cases:
\begin{enumerate}
\item[(1)] \label{case1} $p = 2,3$
\item[(2)] \label{case2} $p = 5$
\item[(3)] \label{case3} $p \ge 5$
\end{enumerate}
For (\ref{case1}), in Section~\ref{sec:preciselogconv} we used
expansions in powers of $z=1/j$ to give a careful analysis of
convergence rates, and in contrast, the general discussion of
Section~\ref{sec:logconvgen} {\em must} keep away from those cases
$p=2,3$, in order to maintain the formulation that it currently has.
The prime $p=5$ is in a very fortunate position because it can be
covered by the general discussion a la (\ref{case3}); but we have also
given a precise ``power series in $1/j$'' treatment of $p=5$. These
issues suggest four questions:
\begin{enumerate}
\item Is there any relationship between the convergence rate analysis
we give, and computation-time estimates for the actual algorithms?
\item We have produced an algebra of log-convergent modular forms, and
it has at least one new element that the overconvergent forms do not
have, namely~$\E_2$. Moreover, it is closed under the action of
$\theta$, i.e., ``Tate twist''. Are there other interesting Hecke
eigenforms in this algebra that we should know about? Going the
other way, are there any Hecke eigenforms that are {\em not}
log-convergent? Is there something corresponding to the
``eigencurve'' (it would have to be, at the very least, a surface)
that $p$-adically interpolates log-convergent eigenforms? Is a limit
(in the sense of $\ord_p$'s of Fourier coefficients) of
log-convergent eigenforms again log-convergent? For this last
question to make sense, we probably need to know the following:
\item Is there a rigid-analytic growth type of definition (growth at
the rim of the Hasse domain) that characterizes log-convergence,
just as there is such a definition characterizing overconvergence?
\item Almost certainly one could treat the case $p=7$ by expansions in
powers of $1/(j-1728) = \Delta/R^2$ in the same way that we did
$p=5$ with powers of $1/j = \Delta/Q^3$. The case $p=13$ might be
more interesting.
\end{enumerate}
%\edit{Items 2,3 in section 7 are still too "chatty" for an actual paper.
%So we should remember to "rework" this a bit.}
\subsection{Uniformity in the algorithms}
We are most thankful to Kiran Kedlaya and Alan Lauder for some e-mail
communications regarding an early draft of our article. The topic they
address is the extent to which the algorithms for the computation of
${\bf E}_2$ of an elliptic curve are ``uniform'' in the elliptic
curve, and, in particular, whether one can get fast algorithms for
computing ${\bf E}_2$ of specific families of elliptic curves. In
this section we give a brief synopsis of their comments.
A ``reason'' why ${\bf E}_2$ should turn out not to be overconvergent
is that Katz's formula relates it to the direction of the unit-root
subspace in one-dimensional de Rham cohomology, and that seems only to
make (at least naive) sense in the ordinary case (and not for points
in a supersingular disc, not even ones close to the boundary).
Nevertheless, part of the algorithm has good uniformity properties.
\begin{enumerate}
\item {\em Calculating the matrix of Frobenius:} One can calculate the
matrix of Frobenius for, say, all elliptic curves in the Legendre
family (or any one-parameter family) and the result is
overconvergent everywhere, so this should be relatively efficient.
This can be done either by the algorithm developed by Kedlaya, or
also using the Gauss-Manin connection, as in Lauder's work, which is
probably faster. An approach to computing the ``full'' Frobenius
matrix ``all at once'' for elliptic curves in the Legendre family
has been written up and implemented in Magma by Ralf Gerkmann: See
\cite{gerkmann} for the paper and program. Lauder's paper
\cite{lauder} also discusses Kedlaya's algorithm ``all at once'' for
a one-parameter family of hyperelliptic curves using the Gauss-Manin
connection.
\item {\em Extracting the unit root subspace in de Rham cohomology:}
To compute ${\bf E}_2$ for an individual elliptic curve, one can
specialize the Frobenius matrix and extract the unit root. But
extracting only the unit root part over the entire family at once
would involve non-overconvergent series, and consequently might be
slow. The {\it unit root zeta function}, which encodes the unit
root of Frobenius over a family of ordinary elliptic curves, has
been very well studied by Dwork and Wan (cf. \cite{MR1740990}).
\end{enumerate}
\subsection{Other future projects}
\begin{enumerate}
%\item the question about the value of E_2 on the canonical lifting
\item Explicitly compute anticyclotomic $p$-adic heights,
and apply this to the study of universal norm questions
that arise in \cite{mazur-rubin:large_selmer}.
\item Further investigate Kedlaya's algorithm with a parameter in
connection with log convergence and computation of heights.
\item Determine if the equality
$\lim_{n \to \infty} a_{P}(n)/\log(n) = 1/\log(p)$
holds for all primes~$p$, as it does for $p=5$
by Theorem \ref{tate:thm6.3}.
\end{enumerate}
\newcommand{\etalchar}[1]{$^{#1}$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{GJP{\etalchar{+}}05}
\bibitem[Bes04]{MR2076563}
Amnon Besser, \emph{The {$p$}-adic height pairings of {C}oleman-{G}ross and of
{N}ekov\'a\v r}, Number theory, CRM Proc. Lecture Notes, vol.~36, Amer. Math.
Soc., Providence, RI, 2004, pp.~13--25. \MR{2076563 (2005f:11130)}
\bibitem[Blu]{bluher:formal}
Antonia~W. Bluher, \emph{A {L}eisurely {I}ntroduction to {F}ormal {G}roups and
{E}lliptic {C}urves,\\ {\tt
http://www.math.uiuc.edu/algebraic-number-theory/0076/}}.
\bibitem[BCP97]{magma}
W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
{T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3--4,
235--265, Computational algebra and number theory (London, 1993). \MR{1 484
478}
\bibitem[Col91]{MR1091621}
Robert~F. Coleman, \emph{The universal vectorial bi-extension and {$p$}-adic
heights}, Invent. Math. \textbf{103} (1991), no.~3, 631--650. \MR{1091621
(92k:14021)}
\bibitem[CGJ95]{MR1317641}
Robert~F. Coleman, Fernando~Q. Gouv{\^e}a, and Naomi Jochnowitz, \emph{{$E\sb
2$}, {$\Theta$}, and overconvergence}, Internat. Math. Res. Notices (1995),
no.~1, 23--41 (electronic). \MR{1317641 (96d:11047)}
\bibitem[Ger05]{gerkmann}
Ralf Gerkmann,
{\tt http://www.mathematik.uni-mainz.de/\\
\~{ }gerkmann/ellcurves.html}, (2005).
\bibitem[Gre03]{MR1977007} Ralph Greenberg, \emph{Galois theory for
the {S}elmer group of an abelian variety}, Compositio Math.
\textbf{136} (2003), no.~3, 255--297. \MR{1977007 (2004c:11097)}
\bibitem[GJP{\etalchar{+}}05]{bsdalg1}
G.~Grigorov, A.~Jorza, S.~Patrikis, C.~Patrascu, and W.~Stein,
\emph{Verification of the {B}irch and {S}winnerton-{D}yer {C}onjecture for
{S}pecific {E}lliptic {C}urves}, (Submitted)\\{\tt
http://modular.fas.harvard.edu/papers/bsdalg/} (2005).
\bibitem[Gou88]{gouvea:slnm}
F.\thinspace{}Q. Gouv\^{e}a, \emph{Arithmetic of $p$-adic modular forms},
Springer-Verlag, Berlin, 1988. \MR{91e:11056}
\bibitem[IW03]{MR2021039}
Adrian Iovita and Annette Werner, \emph{{$p$}-adic height pairings on abelian
varieties with semistable ordinary reduction}, J. Reine Angew. Math.
\textbf{564} (2003), 181--203. \MR{2021039 (2004j:11066)}
\bibitem[SJ05]{sage}
William Stein and David Joyner, \emph{Sage: System for algebra and geometry
experimentation}, Communications in Computer Algebra (SIGSAM Bulletin) (July
2005), {\tt http://sage.sourceforge.net/}.
\bibitem[Kat73]{katz:padicprop}
Nicholas~M. Katz, \emph{{$p$}-adic properties of modular schemes and modular
forms}, Modular functions of one variable, III (Proc. Internat. Summer
School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, pp.~69--190.
Lecture Notes in Mathematics, Vol. 350. \MR{0447119 (56 \#5434)}
\bibitem[Kat76]{katz:padic_interp}
\bysame, \emph{{$p$}-adic interpolation of real analytic {E}isenstein series},
Ann. of Math. (2) \textbf{104} (1976), no.~3, 459--571. \MR{0506271 (58
\#22071)}
\bibitem[Ked01]{kedlaya:counting_mw}
Kiran~S. Kedlaya, \emph{Counting points on hyperelliptic curves using
{M}onsky-{W}ashnitzer cohomology}, J. Ramanujan Math. Soc. \textbf{16}
(2001), no.~4, 323--338. \MR{1877805 (2002m:14019)}
\bibitem[Ked03]{kedlaya:counting-errata}
K.\thinspace{}S. Kedlaya, \emph{Errata for: ``{C}ounting points on
hyperelliptic curves using {M}onsky-{W}ashnitzer cohomology'' [{J}.
{R}amanujan {M}ath.\ {S}oc.\ {\bf 16} (2001), no. 4, 323--338}, J. Ramanujan
Math. Soc. \textbf{18} (2003), no.~4, 417--418, Dedicated to Professor K. S.
Padmanabhan. \MR{2 043 934}
\bibitem[Kob77]{koblitz}
Neil Koblitz, {\em $2$-adic and $3$-adic ordinals of the
(1/$j$)-expansion coefficients for the weight 2 Eisenstein series,
Bull. L.M.S. 9 (1977), 188-192.}
\bibitem[Lan95]{lang:modular}
S.~Lang, \emph{Introduction to modular forms}, Springer-Verlag, Berlin, 1995,
With appendixes by D. Zagier and W. Feit, Corrected reprint of the 1976
original.
\bibitem[Lau03]{lauder}
A.\thinspace{}G.\thinspace{}B. Lauder,
\emph{Rigid cohomology and $p$-adic point counting},
to appear in a special issue of J. de Théorie des Nombres de Bordeaux,\\
{\tt http://www.maths.ox.ac.uk/\~{ }lauder/}.
\bibitem[LW02]{lauder-wan} A.\thinspace{}G.\thinspace{}B. Lauder and
D. Wan, \emph{Counting rational points on varieties over finite
fields of small characteristic}, to appear in an MSRI
Computational Number Theory Proceedings (October, 2002).
\bibitem[MR04]{mazur-rubin:pairings_arith}
Barry Mazur and Karl Rubin, \emph{Pairings in the arithmetic of elliptic
curves}, Modular curves and abelian varieties, Progr. Math., vol. 224,
Birkh\"auser, Basel, 2004, pp.~151--163. \MR{MR2058649 (2005g:11095)}
\bibitem[MT83]{mazur-tate:canonical}
B.~Mazur and J.~Tate, \emph{Canonical height pairings via biextensions},
Arithmetic and geometry, Vol. I, Progr. Math., vol.~35, Birkh\"auser Boston,
Boston, MA, 1983, pp.~195--237. \MR{717595 (85j:14081)}
\bibitem[MT87]{mazur-tate:refined}
\bysame, \emph{Refined conjectures of the ``{B}irch and {S}winnerton-{D}yer
type''}, Duke Math. J. \textbf{54} (1987), no.~2, 711--750. \MR{899413
(88k:11039)}
\bibitem[MT91]{mazur-tate:sigma}
\bysame, \emph{The {$p$}-adic sigma function}, Duke Math. J. \textbf{62}
(1991), no.~3, 663--688. \MR{93d:11059}
\bibitem[MTT86]{mtt}
B.~Mazur, J.~Tate, and J.~Teitelbaum, \emph{On {$p$}-adic analogues of the
conjectures of {B}irch and {S}winnerton-{D}yer}, Invent. Math. \textbf{84}
(1986), no.~1, 1--48. \MR{830037 (87e:11076)}
\bibitem[Nek93]{MR1263527}
Jan Nekov{\'a}{\v{r}}, \emph{On {$p$}-adic height pairings}, S\'eminaire de
Th\'eorie des Nombres, Paris, 1990--91, Progr. Math., vol. 108, Birkh\"auser
Boston, Boston, MA, 1993, pp.~127--202. \MR{1263527 (95j:11050)}
\bibitem[Nek03]{nekovar:selmer}
\bysame, \emph{Selmer Complexes}, 2003, see\\
{\tt http://www.math.jussieu.fr/\~{ }nekovar/pu/}.
\bibitem[Pla94]{MR1299736}
Andrew Plater, \emph{Supersingular {$p$}-adic height pairings on elliptic
curves}, Arithmetic geometry (Tempe, AZ, 1993), Contemp. Math., vol. 174,
Amer. Math. Soc., Providence, RI, 1994, pp.~95--105. \MR{1299736
(95h:11056)}
\bibitem[PR03a]{perrin-riou:expmath}
Bernadette Perrin-Riou, \emph{Arithm\'etique des courbes elliptiques \`a
r\'eduction supersinguli\`ere en {$p$}}, Experiment. Math. \textbf{12}
(2003), no.~2, 155--186. \MR{2016704 (2005h:11138)}
\bibitem[PR03b]{pr:exp}
\bysame, \emph{Arithm\'etique des courbes elliptiques \`a r\'eduction
supersinguli\`ere en {$p$}}, Experiment. Math. \textbf{12} (2003), no.~2,
155--186. \MR{2016704}
\bibitem[RM05]{mazur-rubin:large_selmer}
K.~Rubin and B.~Mazur, \emph{Organizing the
arithmetic of elliptic curves}, in preparation.
\bibitem[Sch82]{schneider:height1}
Peter Schneider, \emph{{$p$}-adic height pairings. {I}}, Invent. Math.
\textbf{69} (1982), no.~3, 401--409. \MR{679765 (84e:14034)}
\bibitem[Sch85]{schneider:height2}
\bysame, \emph{{$p$}-adic height pairings. {II}}, Invent. Math. \textbf{79}
(1985), no.~2, 329--374. \MR{778132 (86j:11063)}
\bibitem[Ser73]{jps}
J-P. Serre, \emph{Congruences et formes modulaires [d'apr\`es
{H}.\thinspace{}{P}.\thinspace{}{F}.\ {S}winnerton-{D}yer]}, S\'eminaire
Bourbaki, 24e ann\'ee (1971/1972), Exp. No. 416 (Berlin), Springer, 1973,
pp.~319--338. Lecture Notes in Math., Vol. 317.
\bibitem[Sil92]{silverman:aec}
J.\thinspace{}H. Silverman, \emph{The arithmetic of elliptic curves},
Springer-Verlag, New York, 1992, Corrected reprint of the 1986 original.
\bibitem[Wan99]{MR1740990}
Daqing Wan, \emph{Dwork's conjecture on unit root zeta functions}, Ann. of
Math. (2) \textbf{150} (1999), no.~3, 867--927. \MR{MR1740990 (2001a:11108)}
\bibitem[Wut04]{wuthrich:heightfamily}
Christian Wuthrich, \emph{On {$p$}-adic heights in families of elliptic
curves}, J. London Math. Soc. (2) \textbf{70} (2004), no.~1, 23--40.
\MR{2064750}
\bibitem[Zar90]{MR1042777}
Yuri~G. Zarhin, \emph{{$p$}-adic heights on abelian varieties}, S\'eminaire de
Th\'eorie des Nombres, Paris 1987--88, Progr. Math., vol.~81, Birkh\"auser
Boston, Boston, MA, 1990, pp.~317--341. \MR{1042777 (91f:11043)}
\end{thebibliography}
\Addresses
\end{document}