Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T06:56:51.408Z Has data issue: false hasContentIssue false

Canonical Heights on Elliptic Curves in Characteristic p

Published online by Cambridge University Press:  04 December 2007

Matthew A. Papanikolas
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802 U.S.A. E-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Copyright
© 2000 Kluwer Academic Publishers