Canonical Heights on Elliptic Curves in Characteristic p

Abstract

Let $k = {\Bbb F}_q(t)$ be the rational function field with finite constant field and characteristic $p \geq 3$, 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 $\langle \phantom {x},\phantom {x}\! \rangle _v \colon E(K^{\rm {sep}}) \times E(K^{\rm {sep}}) \to {\Bbb C}_{v}^{\times }$ which is symmetric, bilinear and Galois equivariant. The pairing $\langle \phantom {x},\phantom {x}\! \rangle _\infty$ for the “infinite” place of k is a natural extension of the classical Néron–Tate height. For v finite, the pairing $\langle \phantom {x},\phantom {x}\! \rangle _v$ plays the role of global analytic p-adic heights. We further determine some hypotheses for the nondegeneracy of these pairings.