Canonical Heights on Elliptic Curves in Characteristic p
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Let k = double-struck F signq(t) be the rational function field with finite constant field and characteristic p ≥ 3, and let K/k be a finite separable extension. For a fixed place v of k and an elliptic curve E/K which has ordinary reduction at all places of K extending v, we consider a canonical height pairing 〈 , 〉v: E(Ksep) x E(Ksep) → ℂxv which is symmetric, bilinear and Galois equivariant. The pairing 〈 , 〉∞ for the "infinite" place of k is a natural extension of the classical Néron-Tate height. For v finite, the pairing 〈 , 〉.v plays the role of global analytic p-adic heights. We further determine some hypotheses for the nondegeneracy of these pairings.
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