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Fix an elliptic curve $E$ over ${\Bbb Q}$ (or if you prefer, something more general over something more general). For each extension $F$ of ${\Bbb Q}$, the Néron-Tate height pairing gives an inner product on $E(F)\otimes {\Bbb R}$. If one normalizes carefully, all these inner products fit together and give an inner product on $E(\overline{\Bbb Q})\otimes {\Bbb R}$, where $\overline{\Bbb Q}$ means the algebraic closure. This real pre-Hilbert space admits a canonical Cauchy closure, so let's write the resulting Hilbert space as $H_E$. Observe that $H_E$ carries a natural action by isometries of Gal$(\overline{\Bbb Q}/{\Bbb Q})$, so still has intrinsic arithemtic interest.

I have many questions about all this (and will thus feel happy receive your wide-ranging thoughts), but to keep this focused for mathoverflow, I ask: has anyone thought before about these Hilbert spaces associated to elliptic curves, explicitly (probably not) or (perhaps more likely) implicitly?

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Do you mean the height pairing? – Felipe Voloch Dec 14 '10 at 2:27
See ch. 5 of P. Fili's PhD thesis: – Felipe Voloch Dec 14 '10 at 2:31
@Felipe ...Sorry, yes. – David Feldman Dec 14 '10 at 2:33
It might be worth your while to click the "edit" link and change the name of the pairing to something like "Néron-Tate height pairing". The name "Tate pairing" is now typically used by cryptographers to refer to a map $E[m](k) \times E/E[m](k) \to \mu_m(k)$ for $k$ finite. – S. Carnahan Dec 14 '10 at 10:45

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