$p$-adic Heights on Curves

Abstract

In this paper, we will present a new construction of the p-adic height pairings of Mazur-Tate [MT] and Schneider [S], when the Abelian variety in question is the Jacobian of a curve.Our aim is to describe the local height symbol solely in terms of the curve, using arithmetic intersection theory at the places not dividing p and integrals of normalized differentials of the third kind (Green's functions) at the places dividing p.