Is there a canonical height on the Weil-Chatelet group?

Question

Jon Hanke and I were just chatting and realized we didn't know the answer to the following question. If E is an elliptic curve over a number field, is there in any sense a "canonical height" on the Weil-Chatelet group H^1(G_Q,E)? (Note that we have no clearly articulated criteria for what "canoncial" means or what kind of compatibility with the group law one should demand -- since the group is torsion, any exact imitation of the usual definition will give everything height 0.)

Answer 1

In my opinion, instead of a "height" on the Weil-Chatelet group, one should consider a "depth", using the local duality between the points on an elliptic curve and the elements of the Weil-Chatelet group. Working over $Q$ for simplicity, there is a Pontrjagin duality between locally compact abelian groups: $$WC_p \times E_p \rightarrow U(1),$$ where $WC_p = H^1(G_p, E)$ and $G_p$ is the absolute Galois group of $Q_p$ ($WC_p$ is the local Weil-Chatelet group at $p$) and $E_p = E(Q_p)$ is the group of points on $E$ over $Q_p$.

At the real place, the duality is between $WC_R$ and $\pi_0(E_R)$. But one might as well define an "extended Weil-Chatelet group" (whose interpretation is not clear to me) by letting $WC_R'$ be the Pontrjagin dual of $E_R$. There is an exact sequence of locally compact abelian groups: $$1 \rightarrow WC_R \rightarrow WC_R' \rightarrow Z \rightarrow 1,$$ to explain why I'd call $WC_R'$ an extended Weil-Chatelet group.

Now, there is a filtration on $E_p$, which can be used to define the local height. Namely, choosing a Neron model, and letting $E_p^0$ denote the preimage of the nonsingular points in the special fibre, and $E_p^1$ the preimage of the identity element in the special fibre, there is a further filtration: $$E_p^0 \supset E_p^1 \supset E_p^2 \supset \cdots.$$ When $E$ has nonsingular reduction at $p$, the local height function can be defined by $\lambda_p(e) = n \log(p)$, if $e \in E_p^n$ and $e \not \in E_p^{n+1}$.

So, the natural thing on Weil-Chatelet groups is the dual increasing filtration. Define $WC_p^n$ to be the annihilator of $E_p^n$ with respect to local duality. This increasing filtration on $WC_p^n$ has been studied by McCallum ("Tate duality and wild ramification", Math. Ann. 288 (1990) 553-558) and Yamazaki ("On Tate duality for Jacobian varieties", J. of Number Theory 99 (2003) 298-306); it has interpretations in terms of splitting fields and Brauer groups, I recall.

Now, if $w \in WC_p$, define the depth of $w$, written $d_p(w)$ to be the smallest $n$ for which $w \in WC_p^n$. If $w \in WC_Q$ (the global Weil-Chatelet group), then $d_p(w) = 0$ for almost all $p$, and one could define a global depth of $w$ by $d(w) = \prod_p p^{d_p(w)}$ (mapping $w$ to appropriate local Weil-Chatelet groups). For example, elements of Sha will have global depth zero (not accounting for archimedean places).

Now I haven't worked out the archimedean details, or primes of bad reduction, but using Pontrjagin duality and the Fourier transform it shouldn't be so terrible to write down the "local depths" in these cases. The resulting global depth function will have the nice properties that elements of Sha will have depth zero, and for any $N$, there will only be finitely many (assuming finiteness of Sha, and using Poitou-Tate duality) elements of $WC_Q$ of depth bounded by $N$.

Now, if any period-index specialists (you know who you are) out there would like to describe in more detail the elements of $WC_Q$ of depth bounded by $N$, I'd love to know more!