Title: 
Explicit p-adic Heights for Hyperelliptic Curves

Abstract: 
In the literature, there are several definitions of p-adic
height pairings on abelian varieties defined over number fields.
Algorithms for computing p-adic heights allow one to compute p-adic
regulators, some of which fit into p-adic versions of Birch and
Swinnerton-Dyer conjecture, and play a crucial role in carrying out the
quadratic Chabauty method. 

Some of the definitions were given by Schneider, Mazur-Tate and
Nekovář. For Jacobians of curves, there is a fourth definition due to
Coleman-Gross, which is particularly convenient to work with for two
reasons:  It can be described solely in terms of the curve, and is, by
definition, a sum of local height pairings at each finite place and the
local components at the places above p are given in terms of Vologodsky
integrals of non-holomorphic differential forms.

In this talk, we present an algorithm to compute the Coleman--Gross
p-adic height pairing on Jacobians of hyperelliptic curves of arbitrary
reduction type. We illustrate this algorithm with a numerical example
computed in Magma. We also discuss the relations among the p-adic
height pairings we touched upon.

This talk constitutes some parts of joint work with Francesca Bianchi
and J. Steffen Müller.