turkmath.org

There are several definitions of p-adic height pairings on curves in the literature, and algorithms for computing them play a crucial role in, for example, carrying out the quadratic Chabauty method, which is a $p$-adic method that attempts to determine rational points on curves of genus at least two.

The $p$-adic height pairing constructed by Peter Schneider in 1982 is particularly important because the corresponding $p$-adic regulator fits into $p$-adic versions of Birch and Swinnerton-Dyer conjecture. In this talk, we present an algorithm to compute the Schneider $p$-adic height pairing on hyperelliptic Mumford curves. We illustrate this algorithm with a numerical example computed in the computer algebra system SageMath.

This talk is based on a joint work in progress with Marc Masdeu, J. Steffen Müller and Marius van der Put.