Title:

p-adic sigma functions and heights on Jacobians of genus 2 curves

Abstract:

Let C be a curve of positive genus over the rationals with Jacobian J
and let p be an odd prime. A p-adic height pairing for J is a Qp-valued
symmetric bilinear pairing on J(Q) times itself. These objects have
arithmetic applications, such as in the study of integral and rational
points on C.

While there are several theoretical constructions of p-adic heights,
very few are explicit enough that can be computed. Our first goal is to
use p-adic sigma functions on the formal group of J to define explicitly
computable heights when the genus of C is 2, thereby extending a known
construction for elliptic curves.

Secondly, when the prime is of semistable ordinary reduction, we will
consider the problem of computing the “canonical” p-adic height.

If time permits, we will compare this construction to others, such as
the Coleman–Gross one.
The talk is partly based on joint work with Enis Kaya and Steffen
Müller.