Dear William,

Thanks loads for the Sha computations, and I think I have an idea about why the bizarre behavior, but let me ponder a bit more this weekend. I also promised some precision regarding the other project I was hoping to interest you in; that is, the search for evidence for, or against, what I call the ”sign conjecture”. Here are some words about that. This sign conjecture is, in a particular instance, equivalent to the assertion that the p-adic anticyclotomic height pairing is not identically zero which brings me to the situation I suggest working in.

Fix E an elliptic curve over Q of conductor N, and of Mordell-Weil rank 2 (or any even positive number). Fix p a prime number with respect to which E has good ordinary reduction. Let K range through quadratic imaginary fields with the following properties:

(a) The discriminant of K is prime to Np (this can be significantly weakened). (b)The ideal pO_{K} splits to the product of two ideals which I will call π and π̄.

(c) The field K satisfies the “Heegner condition” with respect to N, which means that there is an ideal N ⊂ O_{K} such that O_{K}/N is a cyclic group of order N.

(d) The Mordell-Weil group of E over K (which is necessarily of odd rank) is of rank one more than the rank of E(Q). Equivalently, the Mordell-Weil rank of the twist of E by the quadratic character determining K has Mordell-Weil rank one over Q.

Now what I would want to know is whether or not the p-adic anti-cyclotomic height pairing

E(K) × E(K) → Q_{p}

is identically trivial, or not. If P, Q are points in E(K), let this pairing be denoted < P, Q >. I should also say that this pairing is equivariant for complex conjugation, where the action of complex conjugation on the “ Q_{p}” is multiplication by −1; so

< P̄, Q̄ >= − < P, Q > .

In particular, the only possibly nontrivial piece of this pairing comes from taking the height of some point that is not an eigen-point for complex conjugation, and, in particular, not defined over Q.

Let me give you the recipe for computing < P, Q >. Put S = {π, π̄} and for some of the notation I will use, let us just consult page 14 (and surround) of Perrin-Riou’s notes. In particular, she defines a subgroup of finite index E_{S}^{(1)}(K) ⊂ E(K) (this is defined by imposing conditions on the reduction of points to S) and gives a somewhat naive definition for the computation of the pairing < P, P > restricted to points P ∈ E_{S}^{(1)}(K). First consider the anti-cylotomic mapping

ρ̃: ideals rel prime to S ⟶ Z_{p} defined as follows. Note that if we are given a principal ideal relatively prime to S, (α) ⊂ O_{K}, then α is well-defined modulo an e-th root of 1 where e is usually 2, and in exceptional situations is 4 or 6. Therefore, taking an appropriate power of any ideal I relatively prime to S to make it principal, and then to obliterate this finite amibiguity, we render that power I^{ν} generated by a canonical element, call it α_{ν}, which is a unit mod π and mod π̄. Fix

λ : O_{π}^{∗} → Z_{p}

a “logarithm” (we can assume that it is an isomorphism between O_{π}^{∗} mod its torsion with Z_{p}) and

λ̄ : O∗ ; π̄ → Z_{p}

its complex conjugate. Now put

ρ̃(I) := \frac{1}{ν} · (λ(α_{ν}) − λ̄(α_{ν})) ∈ \frac{1}{ν} · Z_{p} ⊂ Q_{p}.

OK, now write P in “coordinates”, as in Perrin-Riou’s manuscript. Specifically, we write

P = (\frac{x(p)}{e(P)^{2}} , \frac{y(p)}{e(P)^{3}} )

where one has a bit of head-ache if these “numerators and denominators” are really ideals rather than numbers, but in any event, we then put (as P.-R. does on page 14):

H_{2}(P) = \frac{1}{2}ρ̃(e(P)^{2} · x(p)),

noting that our definition of the subgroup E_{S}^{(1)}(K) ⊂ E(K) is fashioned to guarantee that the ideal e(P)^{2} · x(p) is prime to S.

Now comes the standard “Tate trick”; that is, we average, to define:

< P, P >:= lim_{n→∞} \frac{H_{2}(p^{n} · P)}{p^{2n}}.

Is it feasible to make some computations along the above lines?

Best,

Barry