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10\title{Computing $p$-Adic Cyclotomic Heights}
11\date{{\bf Notes for a Talk} at Harvard on 2004-12-08}
12\author{William Stein}
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18\begin{document}
19\maketitle
20\tableofcontents
21
22\section{Introduction}
23
24Let $E$ be an elliptic curve over~$\Q$ given by a
25minimal Weierstrass equation and suppose
26$$P=(x,y)=\left(\frac{a}{d^2},\frac{b}{d^3}\right)\in E(\Q),$$ with $a,b,d\in\Z$ and
27$\gcd(a,d)=\gcd(b,d)=1$.  The
28\defn{naive height} of $P$ is
29$$\tilde{h}(P) = \log\max\{|a|,d^2\},$$
30and the \defn{canonical height} of $P$ is
31$$32 h(P) = \lim_{n\to\infty} \frac{h(2^n P)}{4^n}. 33$$
34This definition is not good for computation, because
35$2^n P$ gets huge very quickly, and computing
36$2^n P$ exactly, for~$n$ large, is not reasonable.
37%The canonical height is quadratic, in the sense that
38%$h(mP) = m^2 h(P)$ for all integer $m$.
39
40In \cite[\S3.4]{cremona:algs}, Cremona describes an efficient method
41(due mostly to Silverman) for computing $h(P)$.  One defines
42\defn{local heights} $\hat{h}_p:E(\Q)\to\R$, for all primes~$p$, and
43$\hat{h}_\infty:E(\Q)\to\R$ such that $$h(P) = \hat{h}_\infty(P) + 44\sum \hat{h}_p(P).$$
45The local heights $\hat{h}_p(P)$ are easy to
46compute explicitly.  For example, when $p$ is a prime of good
47reduction, $\hat{h}_p(P) = \max\{0,-\ord_p(x)\}\cdot \log(p)$.
48
49{\em This talk is {\bf NOT} about local heights $\hat{h}_p$, and we will
50not mention them any further.}  Instead, this talk is about a canonical global
51$p$-adic height function
52$$h_p : E(\Q)\to\Q_p.$$
53These height functions are genuine height functions; e.g., $h_p$
54is a quadratic function, i.e, $h_p(mP) = m^2 h(P)$ for all~$m$.
55They appear when defining the $p$-adic regulators that appear in
56$p$-adic analogues of the Birch and Swinnerton-Dyer
57conjecture, in work of Mazur, Tate, Teitelbaum, Greenberg, Schneider, Perrin-Riou
58and many other people.
59
60\vspace{3ex}
61\noindent{\bf Acknowledgement:} Discussions
62with Mike Harrison, Nick Katz, and Christian Wuthrich. \\
63This is joint work with Barry Mazur and John Tate.
64
65\section{The $p$-Adic Cyclotomic Height Pairing}
66Let $E$ be an elliptic curve over~$\Q$ and suppose $p\geq 5$ is a prime
67such that $E$ has good ordinary reduction at $p$.  Suppose $P\in E(\Q)$
68is a point that reduces to $0\in E(\F_p)$ and to the connected
69component of $\mathcal{E}_{\F_\ell}$ at all bad primes $\ell$.
70We will define functions $\log_p$, $\sigma$, and $d$ below.
71In terms of these functions, the $p$-adic height of $P$ is
72\begin{equation}\label{eqn:heightdef}
73  h_p(P) = \frac{1}{p}\cdot \log_p\left(\frac{\sigma(P)}{d(P)}\right) \in \Q_p.
74\end{equation}
75The function $h_p$ satisfies $h_p(nP) = n^2 h_p(P)$ for all integers~$n$,
76so it extends to a function on the full Mordell-Weil group $E(\Q)$.
77Setting $$\langle P, Q\rangle_p = \frac{1}{2}\cdot (h_p(P+Q)-h_p(P)-h_p(Q)),$$
78we obtain a pairing on
79$E(\Q)_{/\tor}$, and the $p$-adic regulator is the discriminant
80of this pairing (which is well defined up to sign).  We have
82\begin{conjecture}
83The pairing $\langle -, -\rangle_p$ is nondegenerate.
84\end{conjecture}
85
86Investigations into $p$-adic analogues of the Birch and
87Swinnerton-Dyer conjecture for curves of positive rank inevitably lead
88to questions about this height pairings, which motivate our interest
89in computing it.
90% \begin{remark}
91% There is also an anticyclotomic height pairing on
92% $E(K)\times E(K)$, where $K$ is a quadratic imaginary field.
93% It has been studied by Perrin-Riou, Bernardi,
94% \end{remark}
95
96We now define each of the undefined quantities in
97(\ref{eqn:heightdef}).  The function $\log_p:\Q_p^* \to \Q_p$ is the
98unique group homomorphism with $\log_p(p)=0$ that extends the homomorphism
99$\log_p:1+p\Z_p \to \Q_p$ defined by the usual power series of $\log(x)$ about $1$.  Thus
100if $x\in\Q_p^*$, we have
101$$\log_p(x) = \frac{1}{p-1}\cdot \log_p(u^{p-1}),$$
102where $u = 103p^{-\ord_p(x)} \cdot x$ is the unit part of~$x$, and the usual
104series for $\log$  converges on $u^{p-1}$.
105
106The denominator $d(P)$ is the positive square root of the denominator of the
107$x$-coordinate of~$P$.
108
109The $\sigma$ function is the most mysterious quantity in
110(\ref{eqn:heightdef}), and it turns out the mystery is closely related
111to the difficulty of computing the $p$-adic number $\E_2(E,\omega)$,
112where $\E_2$ is the $p$-adic weight $2$ Eisenstein series.  There are
113{\em many} ways to define or characterize $\sigma$, e.g.,
114\cite{mazur-tate:sigma} contains $11$ different characterizations!
115Let $$x(t) = \frac{1}{t^2} + \cdots \in \Z((t))$$
116be the formal power
117series that expresses $x$ in terms of $t=-x/y$ locally near $0\in E$.
118Then Mazur and Tate prove there is exactly one function $\sigma(t)\in 119t\Z_p[[t]]$ and constant $c\in \Z_p$ that satisfy the equation
121x(t)
122+ c = -\frac{d}{\omega}\left( \frac{1}{\sigma}
123  \frac{d\sigma}{\omega}\right).
124\end{equation}
125This defines $\sigma$, and,
126unwinding the meaning of the expression on the right, it leads to an
127algorithm to compute $\sigma(t)$ to any desired precision,
128which we now sketch.
129
130If we expand (\ref{eqn:sigmadef}), we can view $c$ as a formal
131variable and solve for $\sigma(t)$ as a power series with coefficients
132that are polynomials in $c$.  Each coefficient of $\sigma(t)$ must be
133in $\Z_p$, so when there are denominators in the polynomials in $c$,
134we obtain conditions on $c$ modulo powers of $p$.  Taking these
135together for {\em many} coefficients eventually yields enough information to
136get $c\pmod{p^n}$, for a given $n$, hence $\sigma(t) \pmod{p^n}$.
137However, this algorithm is {\em extremely inefficient} and its
138complexity is unclear.  Cristian Wuthrich, who has probably done
139more computations with this method than anyone else (and has a nice
140PARI implementation), told me the
141following in email (Oct 2004):
142\begin{quote}
143I believe that in the integrality algorithm, approximately $p^n$
144coefficients of the sigma function have to be computed to get $c$ up to
145$p^n$ (which gives the height up to $p^{n+1}$). i.e. it is hopelessly
146ineffective for $p>100$.''
147\end{quote}
148
149For the last 15 or 20 years, the above unsatisifactory algorithm has
150been the standard one for computing $p$-adic heights, e.g., when
151investigating $p$-adic analogues of the BSD conjecture.
152\begin{center}
153{\em Due to a fortuitous combination of events, the
154    situation recently improved...}
155\end{center}
156\section{Using Cohomology to Compute $\sigma$}
157Suppose that $E$ is an elliptic curve over $\Q$ given by a Weierstrass equation
158$$159y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6. 160$$
161Let $x(t)$ be the formal series as before, and set
162$$\wp(t) = x(t) + \frac{a_1^2 + 4a_2}{12}\in\Q((t)).$$
163One can show that the function $\wp$ satisfies $(\wp')^2 = 4\wp^3 - g_2 \wp - g_3$, etc.; it's
164the analogue of the usual complex $\wp$-function.
165In \cite{mazur-tate:sigma}, Mazur and Tate prove that
166$$167 x(t) + c = \wp(t) + \frac{1}{12}\cdot \E_2(E,\omega), 168$$
169where $\E_2(E,\omega)$ is the value of the Katz $p$-adic
170weight~$2$ Eisenstein series at $(E,\omega)$, and the equality is of
171elements of $\Q_p((t))$.  Thus computing the mysterious $c$ is equivalent
172to computing the $p$-adic number $\E_2(E,\omega)\in\Z_p$.
173
174The'' weight~$2$ Eisenstein appears in many ways.
175In the context of clssical modular forms, the function
176$$177 E_2(z) = 1 - 24\sum_{n=1}^{\infty} \sigma_1(n) q^n 178$$
179is holomorphic on $\h$, but is not a modular form of
180level~$1$.  There exists a nonzero constant~$A$ such that
181$$182 F_2(z) = E_2(z) + \frac{A}{\pi y} \qquad\qquad y = \Im(z) 183$$
184is not holomorphic, but one can show that it
185transforms like a modular form of level~$1$ and weight~$2$.
186Thus for any integer $N>1$, the difference
187$$F_2(z) - N F_2(Nz) = E_2(z) - N E_2(Nz)$$
188is a modular form for~$\Gamma_0(N)$.
189However, in the context of Katz's $p$-adic modular forms (i.e.,
190functions on pairs $(E,\omega)$),
191there is a $p$-adic Eisenstein series
192$\E_2$ of level~$1$.  It's $q$-expansion is
193$$194 \E_2(\Tate(q),\omega_{\can}) = 1 - 24\sum_{n=1}^{\infty} \sigma_1(n)q^n, 195$$
196where $\Tate(q)$ is the Tate curve over $\Q_p$
197with parameter $q$ and $\omega_{\can}$ is the canonical
198nonvanishing differential on the Tate curve.
199
200
201This summer, Mazur, Tate, and I explored many ideas for computing
202$\E_2(E,\omega)$ explicitly, where $E$ is a curve with good ordinary
203reduction at~$p$.
204Perhaps the difficulty of computing $\E_2(E,\omega)$ is somehow intrinsic
205to the theory?
206%There are many strategies for trying to compute
207%$\E_2(E,\omega)$, and many lead to frustratingly slow algorithms.
208
209\subsection{Katz's Email}
210This section contains an email that Nick Katz sent out in
211response to a query from Barry Mazur.  It is the basis of the algorithm
212we will describe later.
213\begin{verbatim}
214Date: Thu, 8 Jul 2004 13:53:13 -0400
215From: Nick Katz <nmk@Math.Princeton.EDU>
216Subject: Re: convergence of the Eisenstein series of weight two
217To: mazur@math.harvard.edu, nmkatz@Math.Princeton.EDU
218Cc: tate@math.utexas.edu, was@math.harvard.edu
219\end{verbatim}
220(I have edited the email below, to better fit the style of these
221notes.)
222
223It seems to me you want to use the interpretation of $P=\E_2$ as the
224{\em direction of the unit root subspace}''; that should make it fast to
225compute. Concretely, suppose we have a pair $(E, \omega)$ over $\Z_p$, and
226to fix ideas $p$ is not $2$ or $3$.  Then we write a Weierstrass
227equation for $E$,
228$$229 y^2 = 4x^3 - g_2x - g_3, 230$$
231so that $\omega=dx/y$, and we denote by
232$\eta$ the differential $xdx/y$.
233Then $\omega$ and $\eta$ form a $\Z_p$ basis of
234$$\H^1 = \H^1_{\dR},$$ and
235the key step is to compute the matrix of
236absolute Frobenius. Here this map is $\Z_p$-linear, since
237we are working over $\Z_p$; otherwise, if we were working
238over the Witt vectors of an $\F_q$, the map would only
239be $\sigma$-linear.
240This calculation goes fast, because the matrix of Frobenius lives
241over the entire $p$-adic moduli space, and we are back in the {\bf glory days
242of Washnitzer-Monsky cohomology} (of the open curve $E - {\cO}$).
243
244        Okay, now suppose we have computed the matrix of $\Frob$ in the
245basis $\omega, \eta$. The unit root subspace is a direct factor, call
246it $U$, of the $\H^1$, and we know that a complimentary direct factor is
247the $\Z_p$ span of $\omega$.
248We also know that $\Frob(\omega)$ lies
249in $p\H^1$, and this tells us that, $\!\!\!\!\mod{p^n}$,
250the subspace~$U$ is the span of
251$\Frob^n(\eta)$.
252What this means concretely is that if we write,
253for each $n$,
254$$255 \Frob^n(\eta) = a_n\omega + b_n\eta, 256$$
257then $b_n$ is a unit (congruent modulo $p$ to the $n$th power of the
258Hasse invariant) and that $P$ is $-12a_n/b_n$.
259See my Antwerp appendix and also my paper {\em $p$-adic
260interpolation of real analytic Eisenstein series}.
261
262        So in terms of speed of convergence, {\em once} you have $\Frob$, you
263have to iterate it $n$ times to calculate $P \pmod{p^n}$.
264
265
266\subsection{The Algorithms}
267The following algorithms culminate in an algorithm for computing
268$h_p(P)$ that incorporates Katz's ideas with the discussion elsewhere
269in this talk.  I have computed $\sigma$ and $h_p$ in numerous
270cases using the algorithm described below, and using my
271implementations of the integrality'' algorithm described above
272and also Wuthrich's algorithm, and the results match.
273Tate has also done several computations of $h_p$ using other
274methods, and again the results match.
275Note: The analysis
276of some of the necessary precision is not complete below.
277
278Kedlaya's algorithm is an algorithm for computing zeta functions
279of hyperelliptic curves over finite fields.  An intermediate step
280in his algorithm is computation of the matrix of absolute Frobenius
281on $p$-adic de Rham cohomology.  In Kedlaya's papers, he determines
282the precision of various objects needed to compute this matrix
283to a given precision.
284
285The first algorithm computes the value $\E_2(E,\omega)$ using
286Kedlaya's algorithm and the method suggested by Katz in the email
287above.
288\begin{algorithm}{Evaluation of $\E_2(E,\omega)$}\label{alg:e2}
289  Given an elliptic curve over~$\Q$ and prime~$p$, this algorithm
290  computes $\E_2(E,\omega)\in \Q_p$.  We
291  assume that Kedlaya's algorithm is available for computing a
292  presentation of the $p$-adic Monsky-Washnitzer cohomology of
293  $E-\{\cO\}$ with Frobenius action.
294\begin{steps}
295\item  Let $c_4$ and $c_6$ be the $c$-invariants of a minimal model
296of~$E$.  Set
297$$a_4\set -\frac{c_4}{2^4\cdot 3}\qquad\text{and}\qquad 298a_6 \set -\frac{c_6}{2^5\cdot 3^3}.$$
299\item Apply Kedlaya's algorithm to the hyperelliptic curve
300$y^2=x^3 + a_4x + a_6$ (which is isomorphic to $E$) to obtain the matrix
301$M$ of the action of absolute Frobenius
302on the basis
303$$\omega=\frac{dx}{y}, \qquad \eta=\frac{xdx}{y}$$
304to precision $O(p^n)$.   We view $M$ as acting
305from the left.
306\item
307We know $M$ to precision $O(p^n)$.
308Compute the $n$th power of $M$ and let
309$\vtwo{a}{b}$ be the second column of $M^n$.
310Then $\Frob^n(\eta) = a\omega + b\eta$.
311
312\item Output $M$ and $-12a/b$ (which is $\E_2(E,\omega)$), then terminate.
313\end{steps}
314\end{algorithm}
315
316The next algorithm uses Algorithm~\ref{alg:e2} to compute $\sigma(t)$.
317\begin{algorithm}{The Canonical $p$-adic Sigma Function}\label{alg:sigma}
318  Given an elliptic curve~$E$ and a good ordinary prime~$p$, this
319  algorithm computes $\sigma(t)\in\Z_p[[t]]$ modulo $(p^n, t^m)$ for
320  any given positive integers $n,m$.
321%(I have {\em not} figured out
322%  exactly what precision each object below must be computed to.)
323\begin{steps}
324\item Using Algorithm~\ref{alg:e2}, compute $e_2 = \E_2(E,\omega)\in 325 \Z_p$ to precision $O(p^n)$.
326\item Compute the formal expansion of $x = x(t) \in \Q[[t]]$
327in terms of the local parameter $t=-x/y$ at infinity
328to precision $O(t^m)$.
329\item Compute the formal logarithm $z(t)=t + \cdots \in \Q((t))$ to precision
330$O(t^m)$ using that $$\ds z(t) = \int \frac{dx/dt}{(2y(t)+a_1x(t) + a_3)},$$
331where $x(t)=t/w(t)$ and $y(t)=-1/w(t)$ are the formal $x$
332and $y$ functions, and $w(t)$ is given by the explicit inductive
333formula in \cite[Ch.~7]{silverman:aec}. (Here $t=-x/y$ and $w=-1/y$ and
334we can write $w$ as a series in $t$.)
335\item Using a power series reversion'' (functional inverse)
336  algorithm, find the unique power series $F(z)\in\Q[[z]]$ such that
337  $t=F(z)$.  Here $F$ is the reversion of $z$, which exists because
338  $z(t) = t + \cdots$.
339\item Set $\wp(t) \set x(t) + (a_1^2 + 4a_2)/12 \in \Q[[t]]$ (to precision
340$O(t^m)$), where the
341  $a_i$ are the coefficients of the Weierstrass equation of $E$.
342Then compute the series $\wp(z) = \wp(F(z))\in \Q((z))$.
343\item Set $\ds g(z)\set \frac{1}{z^2} - \wp(z) + \frac{e_2}{12}\in\Q_p((z))$.
344  [Warning:
345  The theory suggests the last term should be $-e_2/12$ but the calculations do not
346  work unless I use the above formula. There are probably two
347  normalizations of $E_2$ in the references.]
348\item Set
349$\ds \sigma(z) \set z\cdot \exp\left(\int \int g(z) \dz \dz\right) 350\in \Q_p[[z]]$.
351\item Set $\sigma(t) \set \sigma(z(t))\in t\cdot \Z_p[[t]]$, where $z(t)$
352is the formal logarithm computed above.  Output $\sigma(t)$
353and terminate.
354\end{steps}
355\end{algorithm}
356
357\begin{remark}
358  The trick of changing from $\wp(t)$ to $\wp(z)$ is essential so that
359  we can solve a certain differential equation using just operations
360  with power series.
361\end{remark}
362
363The final algorithm uses $\sigma(t)$ to compute the $p$-adic height.
364\begin{algorithm}{$p$-adic Height}
365Given an elliptic curve~$E$ over $\Q$, a good ordinary prime~$p$,
366and an element $P\in E(\Q)$, this algorithm computes the
367$p$-adic height $h_p(P) \in \Q_p$ to precision $O(p^n)$.
368%(I will ignore the precision below.)
369\begin{steps}
370\item{}[Prepare Point] Compute an integer $m$ such that
371$mP$ reduces to $\cO\in E(\F_p)$ and to the connected
372component of $\mathcal{E}_{\F_\ell}$ at all bad primes $\ell$.
373For example,~$m$ could be the least common multiple of the Tamagawa numbers
374of $E$ and $\#E(\F_p)$.  Set $Q\set mP$ and write $Q=(x,y)$.
375\item{}[Denominator] Let $d$ be the positive integer square root of the
376denominator of $x$.
377\item{}[Compute $\sigma$] Compute $\sigma(t)$ using
378  Algorithm~\ref{alg:sigma}, and set $s \set \sigma(-x/y) \in \Q_p$.
379\item{}[Logs] Compute
380$\ds h_p(Q) \set \frac{1}{p}\log_p\left(\frac{s}{d}\right)$, and
381$\ds h_p(P) \set \frac{1}{m^2} \cdot h_p(Q)$.  Output $h_p(P)$ and terminate.
382\end{steps}
383\end{algorithm}
384
385\section{Future Directions}
386In this section we discuss various directions for future
387investigation.
388\subsection{Log Convergence}
389Suppose $E_t$ is an elliptic curves over $\Q(t)$.  It might be
390interesting to obtain formula for $\E_2(E_t)$ as
391an element of $\Q_p((t))$.  This might shed light on the
392analytic behavior of the $p$-adic modular form $\E_2$, and on Tate's
393recent experimental observations about the behavior of the
394$(1/j)$-expansion of the weight~$0$ modular function $\E_2 E_4/E_6$.
395More precisely, Tate computed the expansion of $\E_2 E_4/E_6$
396in powers of $1/j$ for $p=2,3,5$, and observed very
397slow  convergence.  The rest of this
398section is very closely based on an email from Tate about
399his observation.
400
401Here's a very small result concerning the $p$-adic nature of $\E_2$
402for $p=2,3,5$. For the primes $p\leq 5$ we can test the convergence of
403a weight $0$ level $1$ $p$-adic modular function~$f$ (with poles only
404at infinity) by expanding in powers of $z=1/j$. Say
405$f=\sum_{n=1}^{\infty} a_nz^n$.  If $f=zdg/dz$ for some formal series
406$g=\sum b_nz^n$ with $p$-integral coefficients $b_n$, then $a_n = 407nb_n$, so for example $a_{p^m}=p^mb_{p^m}$ is divisible by $p^m$,
408which is a tiny hint of~$f$ having logarithmic'' $p$-adic
409convergence.
410
411\begin{theorem}
412The form
413$$414f=\frac{E_2 E_4}{E_6} - 1 415$$
416has this property, with $g=3\log(E_4)$
417divisible by $720$ in $\Z_2$, $\Z_3$ and $\Z_5$.
418\end{theorem}
419I leave the proof as an exercise. The idea is that by well-known
420formulas, if $P=E_2$, $Q=E_4$, and $R=E_6$, then
421$$422 q\frac{dg}{dq} = 3q\frac{d\log(Q)}{dq} = 3q\frac{dQ}{Qdq} = P - \frac{R}{Q} 423$$
424and
425$$426 q\frac{dz}{zdq} = \frac{R}{Q}. 427$$
428Now divide the first equality by the second to get the result. Note
429that for $p=2$ and~$3$, the result for $n=p^m$ seems just right. For
430$f=PQ/R$,
431it gives
432$$433v_2(a_{2^m}) \geq m+v_2(720)=m+4, 434$$
435and similarly
436$$437v_3(a_{3^m}) \geq m+2, 438$$
439and those inequalities are equalities for $2^m$ and $3^m <200$.
440
441For the record, in case it might give a clue to what is going on,
442experimentally we have, for $n<200$:
443$$444 v_2(a_n) = l_2(n) + 3s_2(n), 445$$
446where $l_2(n)=1+\lfloor \log_2(n) \rfloor$
447and $s(n)$ is the sum of the digits of $n$ written
448in base $2$. Similarly for $n<200$,
449$$450 v_3(a_n) = l_3(n) + s_3(n). 451$$
452For $p=5$ it seems that at least $v_5(a_n)\geq l_5(n)$; in fact, even
453$$454 v_5(a_n) \geq l_5(2n), 455$$
456with likely equality for $2n=5^m-1$ and $5^m+1$.
457
458
459
460\subsection{Connections with $p$-adic Birch and Swinnerton-Dyer}
461 It would also be interesting to do many more computations in support of
462$p$-adic analogues of the BSD conjectures of \cite{mtt}, especially
463when $E/\Q$ has large rank.  Substantial theoretical work has been
464done toward these $p$-adic conjectures, and this work may be useful to
465algorithms for computing information about Shafarevich-Tate and Selmer
466groups of elliptic curves.  For example, in \cite{pr:exp}, Perrin-Riou
467uses her results about the $p$-adic BSD conjecture in the
468supersingular case to prove that $\Sha(E/\Q)[p]=0$ for certain~$p$ and
469elliptic curves~$E$ of rank $>1$, for which the work of Kolyvagin and
470Kato does not apply.  Mazur and Rubin (with my computational input)
471are also obtaining results that could be viewed as fitting into this
472program.
473
474I have been involved with Andrei Jorza and Stephen Patrikas on a
475project to verify the full Birch and Swinnerton-Dyer conjecture for
476all elliptic curves of conductor $\leq 1000$ and rank $\leq 1$.  There
477are many examples in which the rank is $1$ and the upper bound coming
478from Kolyvagin's Euler system is divisible by a prime $p\geq 7$, which
479also divides a Tamagawa number.  The results of Kolyvagin and Kato do
480not give a sufficiently tight upper bound on $\Sha(E/\Q)$.  However,
481discussions with Greenberg,  Pollack, Grigorov,
482and Perrin-Riou have convinced me that it might be possible in many cases to do
483appropriate computations of $p$-adic heights and derivatives of
484$p$-adic $L$-functions, combined with results of Kato and Schneider,
485and obtain a sufficiently strong upper bounds on $\#\Sha(E/\Q)$.
486
487\subsection{Optimization}
488
489I would like to optimize the implementation of the algorithm.
490Probably the most time-consuming step is computation of
491$\E_2(E,\omega)$ using Kedlaya's algorithm.  My current implementation
492uses Michael Harrison's implementation of Kedlaya's algorithm for
493$y^2=f(x)$, with $f(x)$ of arbitrary degree.  (Michael Harrison was a
494Coates student who was in industry for many years, and is now back.)
495
496Perhaps implementing just what is needed for elliptic curves from
497Kedlaya's algorithm would be more efficient.  Also, Harrison tells me
498his implementation isn't nearly as optimized as it might be.
499
500\subsection{Natural Generalizations}
501\begin{enumerate}
502\item It might be possible to compute $p$-adic heights on Jacobians of
503hyperelliptic curves.
504
505\item Formulate everything above over number fields, and extend to the case
507
508\item What about when $p$ is a prime of supersingular reduction?
509\end{enumerate}
510
511\section{Examples}
512In this section I show you examples of how to use the MAGMA package
513I wrote for computing with $p$-adic heights, and give you a sense
514for how efficient it is.
515
516\begin{verbatim}
517> function EC(s) return EllipticCurve(CremonaDatabase(),s); end function;
518> E := EC("37A");
519> Attach("kedlaya.m");        // get this from me
520> Attach("padic_height.m");   // get this from me
521> P := good_ordinary_primes(E,100); P;
522[ 5, 7, 11, 13, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73,
52379, 83, 89, 97 ]
524> for p in P do time print p, regulator(E,p,10); end for;
5255 22229672 + O(5^11)
526Time: 0.040
5277 317628041 + O(7^11)
528...
52989 15480467821870438719 + O(89^10)
530Time: 1.190
53197 -11195795337175141289 + O(97^10)
532Time: 1.490
533> E := EC("389A");
534> P := good_ordinary_primes(E,100); P;
535[ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
53667, 71, 73, 79, 83, 89, 97 ]
537> for p in P do time print p, regulator(E,p,10); end for;
5385 -3871266 + O(5^11)
539Time: 0.260
5407 483898350 + O(7^11)
541...
54289 9775723521676164462 + O(89^10)
543Time: 1.330
54497 -13688331881071698338 + O(97^10)
545Time: 1.820
546> E := EC("5077A");
547> P := good_ordinary_primes(E,100); P;
548[ 5, 7, 11, 13, 17, 19, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71,
54973, 79, 83, 89, 97 ]
550> for p in P do time print p, regulator(E,p,10); end for;
5515 655268*5^-2 + O(5^7)
552Time: 0.800
5537 -933185758 + O(7^11)
554...
55589 -3325438607428779200 + O(89^10)
556Time: 1.910
55797 -5353586908063282167 + O(97^10)
558Time: 2.010
559--------
560> E := EC("37A");
561> time regulator(E,5,50);
562115299522541340178416234094637464047 + O(5^51)
563Time: 1.860
564> Valuation(115299522541340178416234094637464047 - 22229672,5);
5659
566> time regulator(E,97,50);
567-5019271523950156862996295340254565181870308222348277984940964806\
568     97957622583267105973403430183075091 + O(97^50)
569Time: 31.7
570\end{verbatim}
571
572
573\bibliography{biblio}
574\end{document}
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