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\magnification=12001Dear William,2\bigskip3Thanks loads for the Sha computations, and I think I have an idea about why4the bizarre5behavior, but let me ponder a bit more this weekend. I also promised some6precision regarding7the other project I was hoping to interest you in; that is, the search for8evidence9for, or against, what I call the "sign conjecture". Here are some words10about that. This sign11conjecture is, in a particular instance, equivalent to the assertion that the12$p$-adic anti-cyclotomic height pairing is {\it not} identically zero which13brings me to the14situation I suggest working in.15\bigskip16Fix $E$ an elliptic curve over ${\bf Q}$ of conductor $N$, and of17Mordell-Weil rank $2$ (or any18even positive number). Fix $p$ a prime number with respect to which $E$19has good ordinary20reduction. Let $K$ range through quadratic imaginary fields with the21following properties:2223\smallskip24{\bf (a)} The discriminant of $K$ is prime to $Np$ (this can be25significantly weakened).26\smallskip27{\bf (b)}The ideal $p{\cal O}_K$ splits to the product of two ideals which28I will call $\pi$ and29${\bar \pi}$.30\smallskip31{\bf (c)} The field $K$ satisfies the ``Heegner condition" with respect to32$N$, which means33that there is an ideal ${\cal N} \subset {\cal O}_K$ such that ${\cal34O}_K/{\cal N} $ is a cyclic35group of order $N$.36\smallskip37{\bf (d)} The Mordell-Weil group of $E$ over $K$ (which is necessarily of38odd rank) is of rank39one more than the rank of $E({\bf Q})$. Equivalently, the Mordell-Weil40rank of the twist of $E$41by the quadratic character determining $K$ has Mordell-Weil rank one over42${\bf Q}$.43\bigskip4445Now what I would want to know is whether or not the $p$-adic46anti-cyclotomic height pairing47$$E(K) \times E(K) \to {\bf Q}_p$$48is identically trivial, or not. If $P,Q$ are points in $E(K)$, let this49pairing be denoted $<P,Q>$. I50should also say that this pairing is equivariant for complex conjugation,51where the action of52complex conjugation on the ``53${\bf Q}_p$" is multiplication by $-1$; so $$<{\bar P},{\bar Q}> = -54<P,Q>.$$ In particular, the55only possibly nontrivial piece of this pairing comes from taking the height56of some point that is57not an eigen-point for complex conjugation, and, in particular, not defined58over ${\bf Q}$.59\bigskip60Let me give you the recipe for computing $<P,Q>$. Put61$S =\{\pi, {\bar62\pi}63\}$ and for some of the notation I will use, let us just consult page 1464(and surround) of65Perrin-Riou's notes. In particular, she defines a subgroup of finite index66$E_S^{(1)}(K)\subset E(K)$ (this is defined by imposing conditions on the67reduction of points to68$S$) and gives a somewhat naive definition for the computation of the69pairing $<P,P>$ restricted70to points $P\in E_S^{(1)}(K)$. First consider the anti-cylotomic mapping71$${\tilde \rho}: {\rm ideals \ rel \ prime\ to\ } S \longrightarrow72{\bf Z}_p$$73defined as follows. Note that if we are given a principal ideal74relatively prime to $S$, $(\alpha)75\subset {\cal O}_K$, then $\alpha$ is well-defined modulo an $e$-th root of76$1$ where $e$ is77usually $2$, and in exceptional situations is $4$ or $6$. Therefore,78taking an appropriate power79of any ideal $I$ relatively prime to $S$ to make it principal, and then to80obliterate this finite81amibiguity, we render that power $I^{\nu}$ generated by a canonical82element, call it83$\alpha_\nu$, which is a unit mod $\pi$ and mod ${\bar \pi}$. Fix $$\lambda:84{\cal O}_\pi^* \to {\bf Z_p}$$ a ``logarithm" (we can assume that it is an85isomorphism between86${\cal O}_\pi^*$ mod its torsion with $ {\bf Z_p}$) and $${\bar \lambda}:87{\cal O}_{\bar\pi}^* \to {\bf Z_p}$$ its complex conjugate. Now put88$${\tilde \rho}(I): = {1\over \nu}\cdot (\lambda(\alpha_\nu) - {\bar89\lambda}(\alpha_\nu))90\in {1\over \nu}\cdot {\bf Z_p} \subset {\bf Q_p}.$$9192\bigskip9394OK, now write $P$ in ``coordinates", as in Perrin-Riou's manuscript.95Specifically, we write96$$P = ({x(p)\over e(P)^2}, {y(p)\over e(P)^3})$$ where one has a bit of97head-ache if these98``numerators and denominators" are really ideals rather than numbers, but99in any event, we100then put (as P.-R. does on page 14):101102$$H_2(P) = {1\over 2}{\tilde \rho}(e(P)^2\cdot x(p)),$$ noting that our103definition of the104subgroup $E_S^{(1)}(K)\subset E(K)$ is fashioned to guarantee that the105ideal $e(P)^2\cdot x(p)$ is106prime to $S$.107108\bigskip109110Now comes the standard ``Tate trick"; that is, we average, to define:111$$<P,P>:= {\rm lim}_{n \to {\cal 1}}{ H_2(p^n\cdot P)\over p^{2n}}.$$112113\bigskip114115Is it feasible to make some computations along the above lines?116117\bigskip118119Best,120\bigskip121122Barry123124