\documentclass[11pt]{article}1\hoffset=-0.05\textwidth2\textwidth=1.10\textwidth3\voffset=-0.05\textheight4\textheight=1.10\textheight5\bibliographystyle{amsalpha}6\include{macros}7%\renewcommand{\set}{\leftarrow}8\renewcommand{\set}{=}9\title{Computing $p$-Adic Cyclotomic Heights}10\date{{\bf Notes for a Talk} at Harvard on 2004-12-08}11\author{William Stein}12\renewcommand{\E}{\mathbb{E}}13%\usepackage[hyperindex,pdfmark]{hyperref}14\usepackage[hypertex]{hyperref}151617\begin{document}18\maketitle19\tableofcontents2021\section{Introduction}2223Let $E$ be an elliptic curve over~$\Q$ given by a24minimal Weierstrass equation and suppose25$$P=(x,y)=\left(\frac{a}{d^2},\frac{b}{d^3}\right)\in E(\Q),$$ with $a,b,d\in\Z$ and26$\gcd(a,d)=\gcd(b,d)=1$. The27\defn{naive height} of $P$ is28$$\tilde{h}(P) = \log\max\{|a|,d^2\},$$29and the \defn{canonical height} of $P$ is30$$31h(P) = \lim_{n\to\infty} \frac{h(2^n P)}{4^n}.32$$33This definition is not good for computation, because34$2^n P$ gets huge very quickly, and computing35$2^n P$ exactly, for~$n$ large, is not reasonable.36%The canonical height is quadratic, in the sense that37%$h(mP) = m^2 h(P)$ for all integer $m$.3839In \cite[\S3.4]{cremona:algs}, Cremona describes an efficient method40(due mostly to Silverman) for computing $h(P)$. One defines41\defn{local heights} $\hat{h}_p:E(\Q)\to\R$, for all primes~$p$, and42$\hat{h}_\infty:E(\Q)\to\R$ such that $$h(P) = \hat{h}_\infty(P) +43\sum \hat{h}_p(P).$$44The local heights $\hat{h}_p(P)$ are easy to45compute explicitly. For example, when $p$ is a prime of good46reduction, $\hat{h}_p(P) = \max\{0,-\ord_p(x)\}\cdot \log(p)$.4748{\em This talk is {\bf NOT} about local heights $\hat{h}_p$, and we will49not mention them any further.} Instead, this talk is about a canonical global50$p$-adic height function51$$h_p : E(\Q)\to\Q_p.$$52These height functions are genuine height functions; e.g., $h_p$53is a quadratic function, i.e, $h_p(mP) = m^2 h(P)$ for all~$m$.54They appear when defining the $p$-adic regulators that appear in55$p$-adic analogues of the Birch and Swinnerton-Dyer56conjecture, in work of Mazur, Tate, Teitelbaum, Greenberg, Schneider, Perrin-Riou57and many other people.5859\vspace{3ex}60\noindent{\bf Acknowledgement:} Discussions61with Mike Harrison, Nick Katz, and Christian Wuthrich. \\62This is joint work with Barry Mazur and John Tate.6364\section{The $p$-Adic Cyclotomic Height Pairing}65Let $E$ be an elliptic curve over~$\Q$ and suppose $p\geq 5$ is a prime66such that $E$ has good ordinary reduction at $p$. Suppose $P\in E(\Q)$67is a point that reduces to $0\in E(\F_p)$ and to the connected68component of $\mathcal{E}_{\F_\ell}$ at all bad primes $\ell$.69We will define functions $\log_p$, $\sigma$, and $d$ below.70In terms of these functions, the $p$-adic height of $P$ is71\begin{equation}\label{eqn:heightdef}72h_p(P) = \frac{1}{p}\cdot \log_p\left(\frac{\sigma(P)}{d(P)}\right) \in \Q_p.73\end{equation}74The function $h_p$ satisfies $h_p(nP) = n^2 h_p(P)$ for all integers~$n$,75so it extends to a function on the full Mordell-Weil group $E(\Q)$.76Setting $$\langle P, Q\rangle_p = \frac{1}{2}\cdot (h_p(P+Q)-h_p(P)-h_p(Q)),$$77we obtain a pairing on78$E(\Q)_{/\tor}$, and the $p$-adic regulator is the discriminant79of this pairing (which is well defined up to sign). We have80the following standard conjecture about this height pairing.81\begin{conjecture}82The pairing $\langle -, -\rangle_p$ is nondegenerate.83\end{conjecture}8485Investigations into $p$-adic analogues of the Birch and86Swinnerton-Dyer conjecture for curves of positive rank inevitably lead87to questions about this height pairings, which motivate our interest88in computing it.89% \begin{remark}90% There is also an anticyclotomic height pairing on91% $E(K)\times E(K)$, where $K$ is a quadratic imaginary field.92% It has been studied by Perrin-Riou, Bernardi,93% \end{remark}9495We now define each of the undefined quantities in96(\ref{eqn:heightdef}). The function $\log_p:\Q_p^* \to \Q_p$ is the97unique group homomorphism with $\log_p(p)=0$ that extends the homomorphism98$\log_p:1+p\Z_p \to \Q_p$ defined by the usual power series of $\log(x)$ about $1$. Thus99if $x\in\Q_p^*$, we have100$$\log_p(x) = \frac{1}{p-1}\cdot \log_p(u^{p-1}),$$101where $u =102p^{-\ord_p(x)} \cdot x$ is the unit part of~$x$, and the usual103series for $\log$ converges on $u^{p-1}$.104105The denominator $d(P)$ is the positive square root of the denominator of the106$x$-coordinate of~$P$.107108The $\sigma$ function is the most mysterious quantity in109(\ref{eqn:heightdef}), and it turns out the mystery is closely related110to the difficulty of computing the $p$-adic number $\E_2(E,\omega)$,111where $\E_2$ is the $p$-adic weight $2$ Eisenstein series. There are112{\em many} ways to define or characterize $\sigma$, e.g.,113\cite{mazur-tate:sigma} contains $11$ different characterizations!114Let $$x(t) = \frac{1}{t^2} + \cdots \in \Z((t))$$115be the formal power116series that expresses $x$ in terms of $t=-x/y$ locally near $0\in E$.117Then Mazur and Tate prove there is exactly one function $\sigma(t)\in118t\Z_p[[t]]$ and constant $c\in \Z_p$ that satisfy the equation119\begin{equation}\label{eqn:sigmadef}120x(t)121+ c = -\frac{d}{\omega}\left( \frac{1}{\sigma}122\frac{d\sigma}{\omega}\right).123\end{equation}124This defines $\sigma$, and,125unwinding the meaning of the expression on the right, it leads to an126algorithm to compute $\sigma(t)$ to any desired precision,127which we now sketch.128129If we expand (\ref{eqn:sigmadef}), we can view $c$ as a formal130variable and solve for $\sigma(t)$ as a power series with coefficients131that are polynomials in $c$. Each coefficient of $\sigma(t)$ must be132in $\Z_p$, so when there are denominators in the polynomials in $c$,133we obtain conditions on $c$ modulo powers of $p$. Taking these134together for {\em many} coefficients eventually yields enough information to135get $c\pmod{p^n}$, for a given $n$, hence $\sigma(t) \pmod{p^n}$.136However, this algorithm is {\em extremely inefficient} and its137complexity is unclear. Cristian Wuthrich, who has probably done138more computations with this method than anyone else (and has a nice139PARI implementation), told me the140following in email (Oct 2004):141\begin{quote}142``I believe that in the integrality algorithm, approximately $p^n$143coefficients of the sigma function have to be computed to get $c$ up to144$p^n$ (which gives the height up to $p^{n+1}$). i.e. it is hopelessly145ineffective for $p>100$.''146\end{quote}147148For the last 15 or 20 years, the above unsatisifactory algorithm has149been the standard one for computing $p$-adic heights, e.g., when150investigating $p$-adic analogues of the BSD conjecture.151\begin{center}152{\em Due to a fortuitous combination of events, the153situation recently improved...}154\end{center}155\section{Using Cohomology to Compute $\sigma$}156Suppose that $E$ is an elliptic curve over $\Q$ given by a Weierstrass equation157$$158y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.159$$160Let $x(t)$ be the formal series as before, and set161$$\wp(t) = x(t) + \frac{a_1^2 + 4a_2}{12}\in\Q((t)).$$162One can show that the function $\wp$ satisfies $(\wp')^2 = 4\wp^3 - g_2 \wp - g_3$, etc.; it's163the analogue of the usual complex $\wp$-function.164In \cite{mazur-tate:sigma}, Mazur and Tate prove that165$$166x(t) + c = \wp(t) + \frac{1}{12}\cdot \E_2(E,\omega),167$$168where $\E_2(E,\omega)$ is the value of the Katz $p$-adic169weight~$2$ Eisenstein series at $(E,\omega)$, and the equality is of170elements of $\Q_p((t))$. Thus computing the mysterious $c$ is equivalent171to computing the $p$-adic number $\E_2(E,\omega)\in\Z_p$.172173``The'' weight~$2$ Eisenstein appears in many ways.174In the context of clssical modular forms, the function175$$176E_2(z) = 1 - 24\sum_{n=1}^{\infty} \sigma_1(n) q^n177$$178is holomorphic on $\h$, but is not a modular form of179level~$1$. There exists a nonzero constant~$A$ such that180$$181F_2(z) = E_2(z) + \frac{A}{\pi y} \qquad\qquad y = \Im(z)182$$183is not holomorphic, but one can show that it184transforms like a modular form of level~$1$ and weight~$2$.185Thus for any integer $N>1$, the difference186$$F_2(z) - N F_2(Nz) = E_2(z) - N E_2(Nz)$$187is a modular form for~$\Gamma_0(N)$.188However, in the context of Katz's $p$-adic modular forms (i.e.,189functions on pairs $(E,\omega)$),190there is a $p$-adic Eisenstein series191$\E_2$ of level~$1$. It's $q$-expansion is192$$193\E_2(\Tate(q),\omega_{\can}) = 1 - 24\sum_{n=1}^{\infty} \sigma_1(n)q^n,194$$195where $\Tate(q)$ is the Tate curve over $\Q_p$196with parameter $q$ and $\omega_{\can}$ is the canonical197nonvanishing differential on the Tate curve.198199200This summer, Mazur, Tate, and I explored many ideas for computing201$\E_2(E,\omega)$ explicitly, where $E$ is a curve with good ordinary202reduction at~$p$.203Perhaps the difficulty of computing $\E_2(E,\omega)$ is somehow intrinsic204to the theory?205%There are many strategies for trying to compute206%$\E_2(E,\omega)$, and many lead to frustratingly slow algorithms.207208\subsection{Katz's Email}209This section contains an email that Nick Katz sent out in210response to a query from Barry Mazur. It is the basis of the algorithm211we will describe later.212\begin{verbatim}213Date: Thu, 8 Jul 2004 13:53:13 -0400214From: Nick Katz <nmk@Math.Princeton.EDU>215Subject: Re: convergence of the Eisenstein series of weight two216To: mazur@math.harvard.edu, nmkatz@Math.Princeton.EDU217Cc: tate@math.utexas.edu, was@math.harvard.edu218\end{verbatim}219(I have edited the email below, to better fit the style of these220notes.)221222It seems to me you want to use the interpretation of $P=\E_2$ as the223``{\em direction of the unit root subspace}''; that should make it fast to224compute. Concretely, suppose we have a pair $(E, \omega)$ over $\Z_p$, and225to fix ideas $p$ is not $2$ or $3$. Then we write a Weierstrass226equation for $E$,227$$228y^2 = 4x^3 - g_2x - g_3,229$$230so that $\omega=dx/y$, and we denote by231$\eta$ the differential $xdx/y$.232Then $\omega$ and $\eta$ form a $\Z_p$ basis of233$$\H^1 = \H^1_{\dR},$$ and234the key step is to compute the matrix of235absolute Frobenius. Here this map is $\Z_p$-linear, since236we are working over $\Z_p$; otherwise, if we were working237over the Witt vectors of an $\F_q$, the map would only238be $\sigma$-linear.239This calculation goes fast, because the matrix of Frobenius lives240over the entire $p$-adic moduli space, and we are back in the {\bf glory days241of Washnitzer-Monsky cohomology} (of the open curve $E - {\cO}$).242243Okay, now suppose we have computed the matrix of $\Frob$ in the244basis $\omega, \eta$. The unit root subspace is a direct factor, call245it $U$, of the $\H^1$, and we know that a complimentary direct factor is246the $\Z_p$ span of $\omega$.247We also know that $\Frob(\omega)$ lies248in $p\H^1$, and this tells us that, $\!\!\!\!\mod{p^n}$,249the subspace~$U$ is the span of250$\Frob^n(\eta)$.251What this means concretely is that if we write,252for each $n$,253$$254\Frob^n(\eta) = a_n\omega + b_n\eta,255$$256then $b_n$ is a unit (congruent modulo $p$ to the $n$th power of the257Hasse invariant) and that $P$ is $-12a_n/b_n$.258See my Antwerp appendix and also my paper {\em $p$-adic259interpolation of real analytic Eisenstein series}.260261So in terms of speed of convergence, {\em once} you have $\Frob$, you262have to iterate it $n$ times to calculate $P \pmod{p^n}$.263264265\subsection{The Algorithms}266The following algorithms culminate in an algorithm for computing267$h_p(P)$ that incorporates Katz's ideas with the discussion elsewhere268in this talk. I have computed $\sigma$ and $h_p$ in numerous269cases using the algorithm described below, and using my270implementations of the ``integrality'' algorithm described above271and also Wuthrich's algorithm, and the results match.272Tate has also done several computations of $h_p$ using other273methods, and again the results match.274Note: The analysis275of some of the necessary precision is not complete below.276277Kedlaya's algorithm is an algorithm for computing zeta functions278of hyperelliptic curves over finite fields. An intermediate step279in his algorithm is computation of the matrix of absolute Frobenius280on $p$-adic de Rham cohomology. In Kedlaya's papers, he determines281the precision of various objects needed to compute this matrix282to a given precision.283284The first algorithm computes the value $\E_2(E,\omega)$ using285Kedlaya's algorithm and the method suggested by Katz in the email286above.287\begin{algorithm}{Evaluation of $\E_2(E,\omega)$}\label{alg:e2}288Given an elliptic curve over~$\Q$ and prime~$p$, this algorithm289computes $\E_2(E,\omega)\in \Q_p$. We290assume that Kedlaya's algorithm is available for computing a291presentation of the $p$-adic Monsky-Washnitzer cohomology of292$E-\{\cO\}$ with Frobenius action.293\begin{steps}294\item Let $c_4$ and $c_6$ be the $c$-invariants of a minimal model295of~$E$. Set296$$a_4\set -\frac{c_4}{2^4\cdot 3}\qquad\text{and}\qquad297a_6 \set -\frac{c_6}{2^5\cdot 3^3}.$$298\item Apply Kedlaya's algorithm to the hyperelliptic curve299$y^2=x^3 + a_4x + a_6$ (which is isomorphic to $E$) to obtain the matrix300$M$ of the action of absolute Frobenius301on the basis302$$\omega=\frac{dx}{y}, \qquad \eta=\frac{xdx}{y}$$303to precision $O(p^n)$. We view $M$ as acting304from the left.305\item306We know $M$ to precision $O(p^n)$.307Compute the $n$th power of $M$ and let308$\vtwo{a}{b}$ be the second column of $M^n$.309Then $\Frob^n(\eta) = a\omega + b\eta$.310311\item Output $M$ and $-12a/b$ (which is $\E_2(E,\omega)$), then terminate.312\end{steps}313\end{algorithm}314315The next algorithm uses Algorithm~\ref{alg:e2} to compute $\sigma(t)$.316\begin{algorithm}{The Canonical $p$-adic Sigma Function}\label{alg:sigma}317Given an elliptic curve~$E$ and a good ordinary prime~$p$, this318algorithm computes $\sigma(t)\in\Z_p[[t]]$ modulo $(p^n, t^m)$ for319any given positive integers $n,m$.320%(I have {\em not} figured out321% exactly what precision each object below must be computed to.)322\begin{steps}323\item Using Algorithm~\ref{alg:e2}, compute $e_2 = \E_2(E,\omega)\in324\Z_p$ to precision $O(p^n)$.325\item Compute the formal expansion of $x = x(t) \in \Q[[t]]$326in terms of the local parameter $t=-x/y$ at infinity327to precision $O(t^m)$.328\item Compute the formal logarithm $z(t)=t + \cdots \in \Q((t))$ to precision329$O(t^m)$ using that $$\ds z(t) = \int \frac{dx/dt}{(2y(t)+a_1x(t) + a_3)},$$330where $x(t)=t/w(t)$ and $y(t)=-1/w(t)$ are the formal $x$331and $y$ functions, and $w(t)$ is given by the explicit inductive332formula in \cite[Ch.~7]{silverman:aec}. (Here $t=-x/y$ and $w=-1/y$ and333we can write $w$ as a series in $t$.)334\item Using a power series ``reversion'' (functional inverse)335algorithm, find the unique power series $F(z)\in\Q[[z]]$ such that336$t=F(z)$. Here $F$ is the reversion of $z$, which exists because337$z(t) = t + \cdots$.338\item Set $\wp(t) \set x(t) + (a_1^2 + 4a_2)/12 \in \Q[[t]]$ (to precision339$O(t^m)$), where the340$a_i$ are the coefficients of the Weierstrass equation of $E$.341Then compute the series $\wp(z) = \wp(F(z))\in \Q((z))$.342\item Set $\ds g(z)\set \frac{1}{z^2} - \wp(z) + \frac{e_2}{12}\in\Q_p((z))$.343[Warning:344The theory suggests the last term should be $-e_2/12$ but the calculations do not345work unless I use the above formula. There are probably two346normalizations of $E_2$ in the references.]347\item Set348$\ds \sigma(z) \set z\cdot \exp\left(\int \int g(z) \dz \dz\right)349\in \Q_p[[z]]$.350\item Set $\sigma(t) \set \sigma(z(t))\in t\cdot \Z_p[[t]]$, where $z(t)$351is the formal logarithm computed above. Output $\sigma(t)$352and terminate.353\end{steps}354\end{algorithm}355356\begin{remark}357The trick of changing from $\wp(t)$ to $\wp(z)$ is essential so that358we can solve a certain differential equation using just operations359with power series.360\end{remark}361362The final algorithm uses $\sigma(t)$ to compute the $p$-adic height.363\begin{algorithm}{$p$-adic Height}364Given an elliptic curve~$E$ over $\Q$, a good ordinary prime~$p$,365and an element $P\in E(\Q)$, this algorithm computes the366$p$-adic height $h_p(P) \in \Q_p$ to precision $O(p^n)$.367%(I will ignore the precision below.)368\begin{steps}369\item{}[Prepare Point] Compute an integer $m$ such that370$mP$ reduces to $\cO\in E(\F_p)$ and to the connected371component of $\mathcal{E}_{\F_\ell}$ at all bad primes $\ell$.372For example,~$m$ could be the least common multiple of the Tamagawa numbers373of $E$ and $\#E(\F_p)$. Set $Q\set mP$ and write $Q=(x,y)$.374\item{}[Denominator] Let $d$ be the positive integer square root of the375denominator of $x$.376\item{}[Compute $\sigma$] Compute $\sigma(t)$ using377Algorithm~\ref{alg:sigma}, and set $s \set \sigma(-x/y) \in \Q_p$.378\item{}[Logs] Compute379$\ds h_p(Q) \set \frac{1}{p}\log_p\left(\frac{s}{d}\right)$, and380$\ds h_p(P) \set \frac{1}{m^2} \cdot h_p(Q)$. Output $h_p(P)$ and terminate.381\end{steps}382\end{algorithm}383384\section{Future Directions}385In this section we discuss various directions for future386investigation.387\subsection{Log Convergence}388Suppose $E_t$ is an elliptic curves over $\Q(t)$. It might be389interesting to obtain formula for $\E_2(E_t)$ as390an element of $\Q_p((t))$. This might shed light on the391analytic behavior of the $p$-adic modular form $\E_2$, and on Tate's392recent experimental observations about the behavior of the393$(1/j)$-expansion of the weight~$0$ modular function $\E_2 E_4/E_6$.394More precisely, Tate computed the expansion of $\E_2 E_4/E_6$395in powers of $1/j$ for $p=2,3,5$, and observed very396slow convergence. The rest of this397section is very closely based on an email from Tate about398his observation.399400Here's a very small result concerning the $p$-adic nature of $\E_2$401for $p=2,3,5$. For the primes $p\leq 5$ we can test the convergence of402a weight $0$ level $1$ $p$-adic modular function~$f$ (with poles only403at infinity) by expanding in powers of $z=1/j$. Say404$f=\sum_{n=1}^{\infty} a_nz^n$. If $f=zdg/dz$ for some formal series405$g=\sum b_nz^n$ with $p$-integral coefficients $b_n$, then $a_n =406nb_n$, so for example $a_{p^m}=p^mb_{p^m}$ is divisible by $p^m$,407which is a tiny hint of~$f$ having ``logarithmic'' $p$-adic408convergence.409410\begin{theorem}411The form412$$413f=\frac{E_2 E_4}{E_6} - 1414$$415has this property, with $g=3\log(E_4)$416divisible by $720$ in $\Z_2$, $\Z_3$ and $\Z_5$.417\end{theorem}418I leave the proof as an exercise. The idea is that by well-known419formulas, if $P=E_2$, $Q=E_4$, and $R=E_6$, then420$$421q\frac{dg}{dq} = 3q\frac{d\log(Q)}{dq} = 3q\frac{dQ}{Qdq} = P - \frac{R}{Q}422$$423and424$$425q\frac{dz}{zdq} = \frac{R}{Q}.426$$427Now divide the first equality by the second to get the result. Note428that for $p=2$ and~$3$, the result for $n=p^m$ seems just right. For429$f=PQ/R$,430it gives431$$432v_2(a_{2^m}) \geq m+v_2(720)=m+4,433$$434and similarly435$$436v_3(a_{3^m}) \geq m+2,437$$438and those inequalities are equalities for $2^m$ and $3^m <200$.439440For the record, in case it might give a clue to what is going on,441experimentally we have, for $n<200$:442$$443v_2(a_n) = l_2(n) + 3s_2(n),444$$445where $l_2(n)=1+\lfloor \log_2(n) \rfloor$446and $s(n)$ is the sum of the digits of $n$ written447in base $2$. Similarly for $n<200$,448$$449v_3(a_n) = l_3(n) + s_3(n).450$$451For $p=5$ it seems that at least $v_5(a_n)\geq l_5(n)$; in fact, even452$$453v_5(a_n) \geq l_5(2n),454$$455with likely equality for $2n=5^m-1$ and $5^m+1$.456457458459\subsection{Connections with $p$-adic Birch and Swinnerton-Dyer}460It would also be interesting to do many more computations in support of461$p$-adic analogues of the BSD conjectures of \cite{mtt}, especially462when $E/\Q$ has large rank. Substantial theoretical work has been463done toward these $p$-adic conjectures, and this work may be useful to464algorithms for computing information about Shafarevich-Tate and Selmer465groups of elliptic curves. For example, in \cite{pr:exp}, Perrin-Riou466uses her results about the $p$-adic BSD conjecture in the467supersingular case to prove that $\Sha(E/\Q)[p]=0$ for certain~$p$ and468elliptic curves~$E$ of rank $>1$, for which the work of Kolyvagin and469Kato does not apply. Mazur and Rubin (with my computational input)470are also obtaining results that could be viewed as fitting into this471program.472473I have been involved with Andrei Jorza and Stephen Patrikas on a474project to verify the full Birch and Swinnerton-Dyer conjecture for475all elliptic curves of conductor $\leq 1000$ and rank $\leq 1$. There476are many examples in which the rank is $1$ and the upper bound coming477from Kolyvagin's Euler system is divisible by a prime $p\geq 7$, which478also divides a Tamagawa number. The results of Kolyvagin and Kato do479not give a sufficiently tight upper bound on $\Sha(E/\Q)$. However,480discussions with Greenberg, Pollack, Grigorov,481and Perrin-Riou have convinced me that it might be possible in many cases to do482appropriate computations of $p$-adic heights and derivatives of483$p$-adic $L$-functions, combined with results of Kato and Schneider,484and obtain a sufficiently strong upper bounds on $\#\Sha(E/\Q)$.485486\subsection{Optimization}487488I would like to optimize the implementation of the algorithm.489Probably the most time-consuming step is computation of490$\E_2(E,\omega)$ using Kedlaya's algorithm. My current implementation491uses Michael Harrison's implementation of Kedlaya's algorithm for492$y^2=f(x)$, with $f(x)$ of arbitrary degree. (Michael Harrison was a493Coates student who was in industry for many years, and is now back.)494495Perhaps implementing just what is needed for elliptic curves from496Kedlaya's algorithm would be more efficient. Also, Harrison tells me497his implementation isn't nearly as optimized as it might be.498499\subsection{Natural Generalizations}500\begin{enumerate}501\item It might be possible to compute $p$-adic heights on Jacobians of502hyperelliptic curves.503504\item Formulate everything above over number fields, and extend to the case505of additive reduction.506507\item What about when $p$ is a prime of supersingular reduction?508\end{enumerate}509510\section{Examples}511In this section I show you examples of how to use the MAGMA package512I wrote for computing with $p$-adic heights, and give you a sense513for how efficient it is.514515\begin{verbatim}516> function EC(s) return EllipticCurve(CremonaDatabase(),s); end function;517> E := EC("37A");518> Attach("kedlaya.m"); // get this from me519> Attach("padic_height.m"); // get this from me520> P := good_ordinary_primes(E,100); P;521[ 5, 7, 11, 13, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73,52279, 83, 89, 97 ]523> for p in P do time print p, regulator(E,p,10); end for;5245 22229672 + O(5^11)525Time: 0.0405267 317628041 + O(7^11)527...52889 15480467821870438719 + O(89^10)529Time: 1.19053097 -11195795337175141289 + O(97^10)531Time: 1.490532> E := EC("389A");533> P := good_ordinary_primes(E,100); P;534[ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,53567, 71, 73, 79, 83, 89, 97 ]536> for p in P do time print p, regulator(E,p,10); end for;5375 -3871266 + O(5^11)538Time: 0.2605397 483898350 + O(7^11)540...54189 9775723521676164462 + O(89^10)542Time: 1.33054397 -13688331881071698338 + O(97^10)544Time: 1.820545> E := EC("5077A");546> P := good_ordinary_primes(E,100); P;547[ 5, 7, 11, 13, 17, 19, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71,54873, 79, 83, 89, 97 ]549> for p in P do time print p, regulator(E,p,10); end for;5505 655268*5^-2 + O(5^7)551Time: 0.8005527 -933185758 + O(7^11)553...55489 -3325438607428779200 + O(89^10)555Time: 1.91055697 -5353586908063282167 + O(97^10)557Time: 2.010558--------559> E := EC("37A");560> time regulator(E,5,50);561115299522541340178416234094637464047 + O(5^51)562Time: 1.860563> Valuation(115299522541340178416234094637464047 - 22229672,5);5649565> time regulator(E,97,50);566-5019271523950156862996295340254565181870308222348277984940964806\56797957622583267105973403430183075091 + O(97^50)568Time: 31.7569\end{verbatim}570571572\bibliography{biblio}573\end{document}574575576577578579580