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\title{Computing $p$-Adic Cyclotomic Heights}
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\date{{\bf Notes for a Talk} at Harvard on 2004-12-08}
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\author{William Stein}
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\renewcommand{\E}{\mathbb{E}}
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%\usepackage[hyperindex,pdfmark]{hyperref}
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\usepackage[hypertex]{hyperref}
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\begin{document}
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\maketitle
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\tableofcontents
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\section{Introduction}
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Let $E$ be an elliptic curve over~$\Q$ given by a
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minimal Weierstrass equation and suppose
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$$P=(x,y)=\left(\frac{a}{d^2},\frac{b}{d^3}\right)\in E(\Q),$$ with $a,b,d\in\Z$ and
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$\gcd(a,d)=\gcd(b,d)=1$. The
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\defn{naive height} of $P$ is
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$$\tilde{h}(P) = \log\max\{|a|,d^2\},$$
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and the \defn{canonical height} of $P$ is
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$$
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h(P) = \lim_{n\to\infty} \frac{h(2^n P)}{4^n}.
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$$
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This definition is not good for computation, because
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$2^n P$ gets huge very quickly, and computing
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$2^n P$ exactly, for~$n$ large, is not reasonable.
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%The canonical height is quadratic, in the sense that
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%$h(mP) = m^2 h(P)$ for all integer $m$.
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In \cite[\S3.4]{cremona:algs}, Cremona describes an efficient method
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(due mostly to Silverman) for computing $h(P)$. One defines
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\defn{local heights} $\hat{h}_p:E(\Q)\to\R$, for all primes~$p$, and
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$\hat{h}_\infty:E(\Q)\to\R$ such that $$h(P) = \hat{h}_\infty(P) +
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\sum \hat{h}_p(P).$$
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The local heights $\hat{h}_p(P)$ are easy to
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compute explicitly. For example, when $p$ is a prime of good
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reduction, $\hat{h}_p(P) = \max\{0,-\ord_p(x)\}\cdot \log(p)$.
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{\em This talk is {\bf NOT} about local heights $\hat{h}_p$, and we will
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not mention them any further.} Instead, this talk is about a canonical global
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$p$-adic height function
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$$h_p : E(\Q)\to\Q_p.$$
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These height functions are genuine height functions; e.g., $h_p$
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is a quadratic function, i.e, $h_p(mP) = m^2 h(P)$ for all~$m$.
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They appear when defining the $p$-adic regulators that appear in
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$p$-adic analogues of the Birch and Swinnerton-Dyer
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conjecture, in work of Mazur, Tate, Teitelbaum, Greenberg, Schneider, Perrin-Riou
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and many other people.
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\vspace{3ex}
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\noindent{\bf Acknowledgement:} Discussions
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with Mike Harrison, Nick Katz, and Christian Wuthrich. \\
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This is joint work with Barry Mazur and John Tate.
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\section{The $p$-Adic Cyclotomic Height Pairing}
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Let $E$ be an elliptic curve over~$\Q$ and suppose $p\geq 5$ is a prime
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such that $E$ has good ordinary reduction at $p$. Suppose $P\in E(\Q)$
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is a point that reduces to $0\in E(\F_p)$ and to the connected
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component of $\mathcal{E}_{\F_\ell}$ at all bad primes $\ell$.
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We will define functions $\log_p$, $\sigma$, and $d$ below.
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In terms of these functions, the $p$-adic height of $P$ is
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\begin{equation}\label{eqn:heightdef}
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h_p(P) = \frac{1}{p}\cdot \log_p\left(\frac{\sigma(P)}{d(P)}\right) \in \Q_p.
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\end{equation}
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The function $h_p$ satisfies $h_p(nP) = n^2 h_p(P)$ for all integers~$n$,
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so it extends to a function on the full Mordell-Weil group $E(\Q)$.
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Setting $$\langle P, Q\rangle_p = \frac{1}{2}\cdot (h_p(P+Q)-h_p(P)-h_p(Q)),$$
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we obtain a pairing on
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$E(\Q)_{/\tor}$, and the $p$-adic regulator is the discriminant
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of this pairing (which is well defined up to sign). We have
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the following standard conjecture about this height pairing.
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\begin{conjecture}
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The pairing $\langle -, -\rangle_p$ is nondegenerate.
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\end{conjecture}
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Investigations into $p$-adic analogues of the Birch and
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Swinnerton-Dyer conjecture for curves of positive rank inevitably lead
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to questions about this height pairings, which motivate our interest
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in computing it.
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% \begin{remark}
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% There is also an anticyclotomic height pairing on
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% $E(K)\times E(K)$, where $K$ is a quadratic imaginary field.
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% It has been studied by Perrin-Riou, Bernardi,
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% \end{remark}
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We now define each of the undefined quantities in
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(\ref{eqn:heightdef}). The function $\log_p:\Q_p^* \to \Q_p$ is the
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unique group homomorphism with $\log_p(p)=0$ that extends the homomorphism
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$\log_p:1+p\Z_p \to \Q_p$ defined by the usual power series of $\log(x)$ about $1$. Thus
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if $x\in\Q_p^*$, we have
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$$\log_p(x) = \frac{1}{p-1}\cdot \log_p(u^{p-1}),$$
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where $u =
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p^{-\ord_p(x)} \cdot x$ is the unit part of~$x$, and the usual
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series for $\log$ converges on $u^{p-1}$.
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The denominator $d(P)$ is the positive square root of the denominator of the
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$x$-coordinate of~$P$.
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The $\sigma$ function is the most mysterious quantity in
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(\ref{eqn:heightdef}), and it turns out the mystery is closely related
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to the difficulty of computing the $p$-adic number $\E_2(E,\omega)$,
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where $\E_2$ is the $p$-adic weight $2$ Eisenstein series. There are
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{\em many} ways to define or characterize $\sigma$, e.g.,
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\cite{mazur-tate:sigma} contains $11$ different characterizations!
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Let $$x(t) = \frac{1}{t^2} + \cdots \in \Z((t))$$
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be the formal power
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series that expresses $x$ in terms of $t=-x/y$ locally near $0\in E$.
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Then Mazur and Tate prove there is exactly one function $\sigma(t)\in
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t\Z_p[[t]]$ and constant $c\in \Z_p$ that satisfy the equation
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\begin{equation}\label{eqn:sigmadef}
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x(t)
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+ c = -\frac{d}{\omega}\left( \frac{1}{\sigma}
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\frac{d\sigma}{\omega}\right).
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\end{equation}
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This defines $\sigma$, and,
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unwinding the meaning of the expression on the right, it leads to an
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algorithm to compute $\sigma(t)$ to any desired precision,
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which we now sketch.
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If we expand (\ref{eqn:sigmadef}), we can view $c$ as a formal
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variable and solve for $\sigma(t)$ as a power series with coefficients
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that are polynomials in $c$. Each coefficient of $\sigma(t)$ must be
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in $\Z_p$, so when there are denominators in the polynomials in $c$,
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we obtain conditions on $c$ modulo powers of $p$. Taking these
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together for {\em many} coefficients eventually yields enough information to
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get $c\pmod{p^n}$, for a given $n$, hence $\sigma(t) \pmod{p^n}$.
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However, this algorithm is {\em extremely inefficient} and its
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complexity is unclear. Cristian Wuthrich, who has probably done
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more computations with this method than anyone else (and has a nice
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PARI implementation), told me the
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following in email (Oct 2004):
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\begin{quote}
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``I believe that in the integrality algorithm, approximately $p^n$
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coefficients of the sigma function have to be computed to get $c$ up to
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$p^n$ (which gives the height up to $p^{n+1}$). i.e. it is hopelessly
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ineffective for $p>100$.''
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\end{quote}
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For the last 15 or 20 years, the above unsatisifactory algorithm has
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been the standard one for computing $p$-adic heights, e.g., when
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investigating $p$-adic analogues of the BSD conjecture.
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\begin{center}
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{\em Due to a fortuitous combination of events, the
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situation recently improved...}
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\end{center}
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\section{Using Cohomology to Compute $\sigma$}
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Suppose that $E$ is an elliptic curve over $\Q$ given by a Weierstrass equation
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$$
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y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.
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$$
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Let $x(t)$ be the formal series as before, and set
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$$\wp(t) = x(t) + \frac{a_1^2 + 4a_2}{12}\in\Q((t)).$$
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One can show that the function $\wp$ satisfies $(\wp')^2 = 4\wp^3 - g_2 \wp - g_3$, etc.; it's
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the analogue of the usual complex $\wp$-function.
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In \cite{mazur-tate:sigma}, Mazur and Tate prove that
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$$
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x(t) + c = \wp(t) + \frac{1}{12}\cdot \E_2(E,\omega),
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$$
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where $\E_2(E,\omega)$ is the value of the Katz $p$-adic
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weight~$2$ Eisenstein series at $(E,\omega)$, and the equality is of
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elements of $\Q_p((t))$. Thus computing the mysterious $c$ is equivalent
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to computing the $p$-adic number $\E_2(E,\omega)\in\Z_p$.
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``The'' weight~$2$ Eisenstein appears in many ways.
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In the context of clssical modular forms, the function
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$$
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E_2(z) = 1 - 24\sum_{n=1}^{\infty} \sigma_1(n) q^n
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$$
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is holomorphic on $\h$, but is not a modular form of
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level~$1$. There exists a nonzero constant~$A$ such that
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$$
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F_2(z) = E_2(z) + \frac{A}{\pi y} \qquad\qquad y = \Im(z)
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$$
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is not holomorphic, but one can show that it
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transforms like a modular form of level~$1$ and weight~$2$.
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Thus for any integer $N>1$, the difference
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$$F_2(z) - N F_2(Nz) = E_2(z) - N E_2(Nz)$$
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is a modular form for~$\Gamma_0(N)$.
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However, in the context of Katz's $p$-adic modular forms (i.e.,
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functions on pairs $(E,\omega)$),
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there is a $p$-adic Eisenstein series
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$\E_2$ of level~$1$. It's $q$-expansion is
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$$
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\E_2(\Tate(q),\omega_{\can}) = 1 - 24\sum_{n=1}^{\infty} \sigma_1(n)q^n,
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$$
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where $\Tate(q)$ is the Tate curve over $\Q_p$
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with parameter $q$ and $\omega_{\can}$ is the canonical
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nonvanishing differential on the Tate curve.
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This summer, Mazur, Tate, and I explored many ideas for computing
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$\E_2(E,\omega)$ explicitly, where $E$ is a curve with good ordinary
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reduction at~$p$.
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Perhaps the difficulty of computing $\E_2(E,\omega)$ is somehow intrinsic
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to the theory?
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%There are many strategies for trying to compute
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%$\E_2(E,\omega)$, and many lead to frustratingly slow algorithms.
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\subsection{Katz's Email}
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This section contains an email that Nick Katz sent out in
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response to a query from Barry Mazur. It is the basis of the algorithm
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we will describe later.
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\begin{verbatim}
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Date: Thu, 8 Jul 2004 13:53:13 -0400
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From: Nick Katz <nmk@Math.Princeton.EDU>
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Subject: Re: convergence of the Eisenstein series of weight two
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To: mazur@math.harvard.edu, nmkatz@Math.Princeton.EDU
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Cc: tate@math.utexas.edu, was@math.harvard.edu
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\end{verbatim}
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(I have edited the email below, to better fit the style of these
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notes.)
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It seems to me you want to use the interpretation of $P=\E_2$ as the
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``{\em direction of the unit root subspace}''; that should make it fast to
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compute. Concretely, suppose we have a pair $(E, \omega)$ over $\Z_p$, and
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to fix ideas $p$ is not $2$ or $3$. Then we write a Weierstrass
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equation for $E$,
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$$
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y^2 = 4x^3 - g_2x - g_3,
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$$
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so that $\omega=dx/y$, and we denote by
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$\eta$ the differential $xdx/y$.
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Then $\omega$ and $\eta$ form a $\Z_p$ basis of
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$$\H^1 = \H^1_{\dR},$$ and
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the key step is to compute the matrix of
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absolute Frobenius. Here this map is $\Z_p$-linear, since
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we are working over $\Z_p$; otherwise, if we were working
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over the Witt vectors of an $\F_q$, the map would only
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be $\sigma$-linear.
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This calculation goes fast, because the matrix of Frobenius lives
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over the entire $p$-adic moduli space, and we are back in the {\bf glory days
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of Washnitzer-Monsky cohomology} (of the open curve $E - {\cO}$).
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Okay, now suppose we have computed the matrix of $\Frob$ in the
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basis $\omega, \eta$. The unit root subspace is a direct factor, call
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it $U$, of the $\H^1$, and we know that a complimentary direct factor is
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the $\Z_p$ span of $\omega$.
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We also know that $\Frob(\omega)$ lies
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in $p\H^1$, and this tells us that, $\!\!\!\!\mod{p^n}$,
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the subspace~$U$ is the span of
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$\Frob^n(\eta)$.
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What this means concretely is that if we write,
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for each $n$,
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$$
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\Frob^n(\eta) = a_n\omega + b_n\eta,
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$$
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then $b_n$ is a unit (congruent modulo $p$ to the $n$th power of the
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Hasse invariant) and that $P$ is $-12a_n/b_n$.
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See my Antwerp appendix and also my paper {\em $p$-adic
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interpolation of real analytic Eisenstein series}.
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So in terms of speed of convergence, {\em once} you have $\Frob$, you
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have to iterate it $n$ times to calculate $P \pmod{p^n}$.
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\subsection{The Algorithms}
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The following algorithms culminate in an algorithm for computing
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$h_p(P)$ that incorporates Katz's ideas with the discussion elsewhere
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in this talk. I have computed $\sigma$ and $h_p$ in numerous
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cases using the algorithm described below, and using my
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implementations of the ``integrality'' algorithm described above
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and also Wuthrich's algorithm, and the results match.
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Tate has also done several computations of $h_p$ using other
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methods, and again the results match.
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Note: The analysis
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of some of the necessary precision is not complete below.
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Kedlaya's algorithm is an algorithm for computing zeta functions
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of hyperelliptic curves over finite fields. An intermediate step
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in his algorithm is computation of the matrix of absolute Frobenius
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on $p$-adic de Rham cohomology. In Kedlaya's papers, he determines
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the precision of various objects needed to compute this matrix
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to a given precision.
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The first algorithm computes the value $\E_2(E,\omega)$ using
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Kedlaya's algorithm and the method suggested by Katz in the email
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above.
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\begin{algorithm}{Evaluation of $\E_2(E,\omega)$}\label{alg:e2}
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Given an elliptic curve over~$\Q$ and prime~$p$, this algorithm
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computes $\E_2(E,\omega)\in \Q_p$. We
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assume that Kedlaya's algorithm is available for computing a
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presentation of the $p$-adic Monsky-Washnitzer cohomology of
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$E-\{\cO\}$ with Frobenius action.
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\begin{steps}
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\item Let $c_4$ and $c_6$ be the $c$-invariants of a minimal model
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of~$E$. Set
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$$a_4\set -\frac{c_4}{2^4\cdot 3}\qquad\text{and}\qquad
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a_6 \set -\frac{c_6}{2^5\cdot 3^3}.$$
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\item Apply Kedlaya's algorithm to the hyperelliptic curve
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$y^2=x^3 + a_4x + a_6$ (which is isomorphic to $E$) to obtain the matrix
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$M$ of the action of absolute Frobenius
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on the basis
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$$\omega=\frac{dx}{y}, \qquad \eta=\frac{xdx}{y}$$
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to precision $O(p^n)$. We view $M$ as acting
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from the left.
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\item
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We know $M$ to precision $O(p^n)$.
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Compute the $n$th power of $M$ and let
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$\vtwo{a}{b}$ be the second column of $M^n$.
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Then $\Frob^n(\eta) = a\omega + b\eta$.
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\item Output $M$ and $-12a/b$ (which is $\E_2(E,\omega)$), then terminate.
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\end{steps}
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\end{algorithm}
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The next algorithm uses Algorithm~\ref{alg:e2} to compute $\sigma(t)$.
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\begin{algorithm}{The Canonical $p$-adic Sigma Function}\label{alg:sigma}
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Given an elliptic curve~$E$ and a good ordinary prime~$p$, this
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algorithm computes $\sigma(t)\in\Z_p[[t]]$ modulo $(p^n, t^m)$ for
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any given positive integers $n,m$.
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%(I have {\em not} figured out
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% exactly what precision each object below must be computed to.)
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\begin{steps}
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\item Using Algorithm~\ref{alg:e2}, compute $e_2 = \E_2(E,\omega)\in
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\Z_p$ to precision $O(p^n)$.
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\item Compute the formal expansion of $x = x(t) \in \Q[[t]]$
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in terms of the local parameter $t=-x/y$ at infinity
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to precision $O(t^m)$.
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\item Compute the formal logarithm $z(t)=t + \cdots \in \Q((t))$ to precision
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$O(t^m)$ using that $$\ds z(t) = \int \frac{dx/dt}{(2y(t)+a_1x(t) + a_3)},$$
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where $x(t)=t/w(t)$ and $y(t)=-1/w(t)$ are the formal $x$
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and $y$ functions, and $w(t)$ is given by the explicit inductive
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formula in \cite[Ch.~7]{silverman:aec}. (Here $t=-x/y$ and $w=-1/y$ and
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we can write $w$ as a series in $t$.)
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\item Using a power series ``reversion'' (functional inverse)
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algorithm, find the unique power series $F(z)\in\Q[[z]]$ such that
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$t=F(z)$. Here $F$ is the reversion of $z$, which exists because
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$z(t) = t + \cdots$.
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\item Set $\wp(t) \set x(t) + (a_1^2 + 4a_2)/12 \in \Q[[t]]$ (to precision
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$O(t^m)$), where the
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$a_i$ are the coefficients of the Weierstrass equation of $E$.
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Then compute the series $\wp(z) = \wp(F(z))\in \Q((z))$.
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\item Set $\ds g(z)\set \frac{1}{z^2} - \wp(z) + \frac{e_2}{12}\in\Q_p((z))$.
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[Warning:
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The theory suggests the last term should be $-e_2/12$ but the calculations do not
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work unless I use the above formula. There are probably two
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normalizations of $E_2$ in the references.]
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\item Set
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$\ds \sigma(z) \set z\cdot \exp\left(\int \int g(z) \dz \dz\right)
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\in \Q_p[[z]]$.
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\item Set $\sigma(t) \set \sigma(z(t))\in t\cdot \Z_p[[t]]$, where $z(t)$
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is the formal logarithm computed above. Output $\sigma(t)$
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and terminate.
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\end{steps}
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\end{algorithm}
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\begin{remark}
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The trick of changing from $\wp(t)$ to $\wp(z)$ is essential so that
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we can solve a certain differential equation using just operations
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with power series.
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\end{remark}
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The final algorithm uses $\sigma(t)$ to compute the $p$-adic height.
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\begin{algorithm}{$p$-adic Height}
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Given an elliptic curve~$E$ over $\Q$, a good ordinary prime~$p$,
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and an element $P\in E(\Q)$, this algorithm computes the
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$p$-adic height $h_p(P) \in \Q_p$ to precision $O(p^n)$.
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%(I will ignore the precision below.)
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\begin{steps}
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\item{}[Prepare Point] Compute an integer $m$ such that
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$mP$ reduces to $\cO\in E(\F_p)$ and to the connected
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component of $\mathcal{E}_{\F_\ell}$ at all bad primes $\ell$.
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For example,~$m$ could be the least common multiple of the Tamagawa numbers
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of $E$ and $\#E(\F_p)$. Set $Q\set mP$ and write $Q=(x,y)$.
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\item{}[Denominator] Let $d$ be the positive integer square root of the
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denominator of $x$.
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\item{}[Compute $\sigma$] Compute $\sigma(t)$ using
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Algorithm~\ref{alg:sigma}, and set $s \set \sigma(-x/y) \in \Q_p$.
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\item{}[Logs] Compute
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$\ds h_p(Q) \set \frac{1}{p}\log_p\left(\frac{s}{d}\right)$, and
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$\ds h_p(P) \set \frac{1}{m^2} \cdot h_p(Q)$. Output $h_p(P)$ and terminate.
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\end{steps}
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\end{algorithm}
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\section{Future Directions}
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In this section we discuss various directions for future
387
investigation.
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\subsection{Log Convergence}
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Suppose $E_t$ is an elliptic curves over $\Q(t)$. It might be
390
interesting to obtain formula for $\E_2(E_t)$ as
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an element of $\Q_p((t))$. This might shed light on the
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analytic behavior of the $p$-adic modular form $\E_2$, and on Tate's
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recent experimental observations about the behavior of the
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$(1/j)$-expansion of the weight~$0$ modular function $\E_2 E_4/E_6$.
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More precisely, Tate computed the expansion of $\E_2 E_4/E_6$
396
in powers of $1/j$ for $p=2,3,5$, and observed very
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slow convergence. The rest of this
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section is very closely based on an email from Tate about
399
his observation.
400
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Here's a very small result concerning the $p$-adic nature of $\E_2$
402
for $p=2,3,5$. For the primes $p\leq 5$ we can test the convergence of
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a weight $0$ level $1$ $p$-adic modular function~$f$ (with poles only
404
at 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 $a_n =
407
nb_n$, so for example $a_{p^m}=p^mb_{p^m}$ is divisible by $p^m$,
408
which is a tiny hint of~$f$ having ``logarithmic'' $p$-adic
409
convergence.
410
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\begin{theorem}
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The form
413
$$
414
f=\frac{E_2 E_4}{E_6} - 1
415
$$
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has this property, with $g=3\log(E_4)$
417
divisible by $720$ in $\Z_2$, $\Z_3$ and $\Z_5$.
418
\end{theorem}
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I leave the proof as an exercise. The idea is that by well-known
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formulas, if $P=E_2$, $Q=E_4$, and $R=E_6$, then
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$$
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q\frac{dg}{dq} = 3q\frac{d\log(Q)}{dq} = 3q\frac{dQ}{Qdq} = P - \frac{R}{Q}
423
$$
424
and
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$$
426
q\frac{dz}{zdq} = \frac{R}{Q}.
427
$$
428
Now divide the first equality by the second to get the result. Note
429
that for $p=2$ and~$3$, the result for $n=p^m$ seems just right. For
430
$f=PQ/R$,
431
it gives
432
$$
433
v_2(a_{2^m}) \geq m+v_2(720)=m+4,
434
$$
435
and similarly
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$$
437
v_3(a_{3^m}) \geq m+2,
438
$$
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and those inequalities are equalities for $2^m$ and $3^m <200$.
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For the record, in case it might give a clue to what is going on,
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experimentally we have, for $n<200$:
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$$
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v_2(a_n) = l_2(n) + 3s_2(n),
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$$
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where $l_2(n)=1+\lfloor \log_2(n) \rfloor$
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and $s(n)$ is the sum of the digits of $n$ written
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in base $2$. Similarly for $n<200$,
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$$
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v_3(a_n) = l_3(n) + s_3(n).
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$$
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For $p=5$ it seems that at least $v_5(a_n)\geq l_5(n)$; in fact, even
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$$
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v_5(a_n) \geq l_5(2n),
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$$
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with likely equality for $2n=5^m-1$ and $5^m+1$.
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\subsection{Connections with $p$-adic Birch and Swinnerton-Dyer}
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It would also be interesting to do many more computations in support of
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$p$-adic analogues of the BSD conjectures of \cite{mtt}, especially
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when $E/\Q$ has large rank. Substantial theoretical work has been
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done toward these $p$-adic conjectures, and this work may be useful to
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algorithms for computing information about Shafarevich-Tate and Selmer
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groups of elliptic curves. For example, in \cite{pr:exp}, Perrin-Riou
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uses her results about the $p$-adic BSD conjecture in the
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supersingular case to prove that $\Sha(E/\Q)[p]=0$ for certain~$p$ and
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elliptic curves~$E$ of rank $>1$, for which the work of Kolyvagin and
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Kato does not apply. Mazur and Rubin (with my computational input)
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are also obtaining results that could be viewed as fitting into this
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program.
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I have been involved with Andrei Jorza and Stephen Patrikas on a
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project to verify the full Birch and Swinnerton-Dyer conjecture for
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all elliptic curves of conductor $\leq 1000$ and rank $\leq 1$. There
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are many examples in which the rank is $1$ and the upper bound coming
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from Kolyvagin's Euler system is divisible by a prime $p\geq 7$, which
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also divides a Tamagawa number. The results of Kolyvagin and Kato do
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not give a sufficiently tight upper bound on $\Sha(E/\Q)$. However,
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discussions with Greenberg, Pollack, Grigorov,
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and Perrin-Riou have convinced me that it might be possible in many cases to do
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appropriate computations of $p$-adic heights and derivatives of
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$p$-adic $L$-functions, combined with results of Kato and Schneider,
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and obtain a sufficiently strong upper bounds on $\#\Sha(E/\Q)$.
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\subsection{Optimization}
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I would like to optimize the implementation of the algorithm.
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Probably the most time-consuming step is computation of
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$\E_2(E,\omega)$ using Kedlaya's algorithm. My current implementation
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uses Michael Harrison's implementation of Kedlaya's algorithm for
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$y^2=f(x)$, with $f(x)$ of arbitrary degree. (Michael Harrison was a
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Coates student who was in industry for many years, and is now back.)
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Perhaps implementing just what is needed for elliptic curves from
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Kedlaya's algorithm would be more efficient. Also, Harrison tells me
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his implementation isn't nearly as optimized as it might be.
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\subsection{Natural Generalizations}
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\begin{enumerate}
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\item It might be possible to compute $p$-adic heights on Jacobians of
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hyperelliptic curves.
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\item Formulate everything above over number fields, and extend to the case
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of additive reduction.
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\item What about when $p$ is a prime of supersingular reduction?
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\end{enumerate}
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\section{Examples}
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In this section I show you examples of how to use the MAGMA package
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I wrote for computing with $p$-adic heights, and give you a sense
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for how efficient it is.
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\begin{verbatim}
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> function EC(s) return EllipticCurve(CremonaDatabase(),s); end function;
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> E := EC("37A");
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> Attach("kedlaya.m"); // get this from me
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> Attach("padic_height.m"); // get this from me
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> P := good_ordinary_primes(E,100); P;
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[ 5, 7, 11, 13, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73,
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79, 83, 89, 97 ]
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> for p in P do time print p, regulator(E,p,10); end for;
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5 22229672 + O(5^11)
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Time: 0.040
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7 317628041 + O(7^11)
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...
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89 15480467821870438719 + O(89^10)
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Time: 1.190
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97 -11195795337175141289 + O(97^10)
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Time: 1.490
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> E := EC("389A");
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> P := good_ordinary_primes(E,100); P;
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[ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
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67, 71, 73, 79, 83, 89, 97 ]
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> for p in P do time print p, regulator(E,p,10); end for;
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5 -3871266 + O(5^11)
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Time: 0.260
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7 483898350 + O(7^11)
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...
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89 9775723521676164462 + O(89^10)
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Time: 1.330
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97 -13688331881071698338 + O(97^10)
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Time: 1.820
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> E := EC("5077A");
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> P := good_ordinary_primes(E,100); P;
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[ 5, 7, 11, 13, 17, 19, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71,
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73, 79, 83, 89, 97 ]
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> for p in P do time print p, regulator(E,p,10); end for;
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5 655268*5^-2 + O(5^7)
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Time: 0.800
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7 -933185758 + O(7^11)
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...
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89 -3325438607428779200 + O(89^10)
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Time: 1.910
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97 -5353586908063282167 + O(97^10)
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Time: 2.010
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--------
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> E := EC("37A");
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> time regulator(E,5,50);
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115299522541340178416234094637464047 + O(5^51)
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Time: 1.860
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> Valuation(115299522541340178416234094637464047 - 22229672,5);
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9
566
> time regulator(E,97,50);
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-5019271523950156862996295340254565181870308222348277984940964806\
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97957622583267105973403430183075091 + O(97^50)
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Time: 31.7
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\end{verbatim}
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\bibliography{biblio}
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\end{document}
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