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Author: William A. Stein
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Dear William,
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Thanks loads for the Sha computations, and I think I have an idea about why
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the bizarre
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behavior, but let me ponder a bit more this weekend. I also promised some
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precision regarding
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the other project I was hoping to interest you in; that is, the search for
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evidence
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for, or against, what I call the "sign conjecture". Here are some words
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about that. This sign
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conjecture is, in a particular instance, equivalent to the assertion that the
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$p$-adic anti-cyclotomic height pairing is {\it not} identically zero which
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brings me to the
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situation I suggest working in.
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Fix $E$ an elliptic curve over ${\bf Q}$ of conductor $N$, and of
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Mordell-Weil rank $2$ (or any
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even positive number). Fix $p$ a prime number with respect to which $E$
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has good ordinary
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reduction. Let $K$ range through quadratic imaginary fields with the
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following properties:
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{\bf (a)} The discriminant of $K$ is prime to $Np$ (this can be
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significantly weakened).
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{\bf (b)}The ideal $p{\cal O}_K$ splits to the product of two ideals which
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I will call $\pi$ and
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${\bar \pi}$.
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{\bf (c)} The field $K$ satisfies the ``Heegner condition" with respect to
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$N$, which means
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that there is an ideal ${\cal N} \subset {\cal O}_K$ such that ${\cal
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O}_K/{\cal N} $ is a cyclic
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group of order $N$.
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{\bf (d)} The Mordell-Weil group of $E$ over $K$ (which is necessarily of
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odd rank) is of rank
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one more than the rank of $E({\bf Q})$. Equivalently, the Mordell-Weil
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rank of the twist of $E$
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by the quadratic character determining $K$ has Mordell-Weil rank one over
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${\bf Q}$.
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Now what I would want to know is whether or not the $p$-adic
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anti-cyclotomic height pairing
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$$E(K) \times E(K) \to {\bf Q}_p$$
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is identically trivial, or not. If $P,Q$ are points in $E(K)$, let this
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pairing be denoted $<P,Q>$. I
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should also say that this pairing is equivariant for complex conjugation,
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where the action of
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complex conjugation on the ``
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${\bf Q}_p$" is multiplication by $-1$; so $$<{\bar P},{\bar Q}> = -
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<P,Q>.$$ In particular, the
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only possibly nontrivial piece of this pairing comes from taking the height
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of some point that is
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not an eigen-point for complex conjugation, and, in particular, not defined
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over ${\bf Q}$.
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Let me give you the recipe for computing $<P,Q>$. Put
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$S =\{\pi, {\bar
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\pi}
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\}$ and for some of the notation I will use, let us just consult page 14
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(and surround) of
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Perrin-Riou's notes. In particular, she defines a subgroup of finite index
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$E_S^{(1)}(K)\subset E(K)$ (this is defined by imposing conditions on the
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reduction of points to
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$S$) and gives a somewhat naive definition for the computation of the
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pairing $<P,P>$ restricted
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to points $P\in E_S^{(1)}(K)$. First consider the anti-cylotomic mapping
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$${\tilde \rho}: {\rm ideals \ rel \ prime\ to\ } S \longrightarrow
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{\bf Z}_p$$
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defined as follows. Note that if we are given a principal ideal
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relatively prime to $S$, $(\alpha)
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\subset {\cal O}_K$, then $\alpha$ is well-defined modulo an $e$-th root of
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$1$ where $e$ is
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usually $2$, and in exceptional situations is $4$ or $6$. Therefore,
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taking an appropriate power
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of any ideal $I$ relatively prime to $S$ to make it principal, and then to
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obliterate this finite
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amibiguity, we render that power $I^{\nu}$ generated by a canonical
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element, call it
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$\alpha_\nu$, which is a unit mod $\pi$ and mod ${\bar \pi}$. Fix $$\lambda:
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{\cal O}_\pi^* \to {\bf Z_p}$$ a ``logarithm" (we can assume that it is an
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isomorphism between
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${\cal O}_\pi^*$ mod its torsion with $ {\bf Z_p}$) and $${\bar \lambda}:
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{\cal O}_{\bar\pi}^* \to {\bf Z_p}$$ its complex conjugate. Now put
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$${\tilde \rho}(I): = {1\over \nu}\cdot (\lambda(\alpha_\nu) - {\bar
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\lambda}(\alpha_\nu))
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\in {1\over \nu}\cdot {\bf Z_p} \subset {\bf Q_p}.$$
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OK, now write $P$ in ``coordinates", as in Perrin-Riou's manuscript.
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Specifically, we write
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$$P = ({x(p)\over e(P)^2}, {y(p)\over e(P)^3})$$ where one has a bit of
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head-ache if these
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``numerators and denominators" are really ideals rather than numbers, but
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in any event, we
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then put (as P.-R. does on page 14):
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$$H_2(P) = {1\over 2}{\tilde \rho}(e(P)^2\cdot x(p)),$$ noting that our
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definition of the
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subgroup $E_S^{(1)}(K)\subset E(K)$ is fashioned to guarantee that the
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ideal $e(P)^2\cdot x(p)$ is
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prime to $S$.
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Now comes the standard ``Tate trick"; that is, we average, to define:
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$$<P,P>:= {\rm lim}_{n \to {\cal 1}}{ H_2(p^n\cdot P)\over p^{2n}}.$$
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Is it feasible to make some computations along the above lines?
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Best,
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Barry
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