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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.)

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What about something like using the Weil pairing to cup two things in the $H^1$ and get an element of the Brauer group, then play with the local invariants? –  David Hansen Feb 3 '11 at 0:06
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I don't know if it's useful, but I've tended to believe that you can define a Faltings' height. Given a torsor $T$, we would need some regular model over the ring of integers and consider its global differentials as a metrized line bundle over the ring of integers. There's no problem with the metric, but a suitable model may be a bit problematic. –  Minhyong Kim Feb 3 '11 at 9:11
    
@Minhyong Kim ... and imagine if that had something to do with the answer of Marty below ! –  Chris Wuthrich Feb 3 '11 at 12:41

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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!

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This is a very interesting answer. Whatever you have defined is a more plausible candidate for a "height function" on an infinite torsion group than anything I would have thought of. As to what the set of guys of bounded depth looks like: the construction reminds me a little bit of what I called (by analogy) "Kolyvagin classes" in my paper "There are genus one curves of every index over every number field". The interesting question is how this depth behaves functorially, i.e., upon restriction to the WC group over a number field $K$. It's well worth thinking about. –  Pete L. Clark Feb 3 '11 at 12:53
    
Let me say though that poor Andre Neron is spinning in his grave from the phrase "choosing a Neron model". (Maybe you were thinking a little bit about Neron differentials when you wrote this?) –  Pete L. Clark Feb 3 '11 at 12:55
    
Why's that? By choosing a Neron model, I meant that we want a "good" model of $E$ over the DVR $Z_p$, so that we can talk about the nonsingular points over the special fibre, and define $E_p^0$. Now I haven't read any of Neron's work in its original -- am I saying something stupid? Should I use the work "minimal" in this case? "Weierstrass" for historical reasons? –  Marty Feb 3 '11 at 15:48
    
Oh - and I'll check out your "Kolyvagin classes" and think about the functoriality with respect to field extensions too. Thanks Pete! –  Marty Feb 3 '11 at 15:54
    
@Marty: all I mean is that the definition of a Neron model of a variety over $\mathbb{Q}_p$ involves a universal mapping property, so there is no choice involved: if it exists, it is unique up to canonical isomorphism. By a huge theorem of Neron, the Neron model of an abelian variety exists. –  Pete L. Clark Feb 3 '11 at 18:20

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