Quaternionic Darmon points on p-adic tori and abelian varieties

TL;DR

This paper derives explicit formulas for p-adic logarithms of quaternionic Darmon points on p-adic tori and modular abelian varieties, enabling computations of Stark-Heegner type points on higher-dimensional abelian varieties.

Contribution

It provides the first explicit formulas for Stark-Heegner points on higher-dimensional abelian varieties, expanding the computational toolkit in this area.

Findings

Formulas for p-adic logarithms of Darmon points on p-adic tori and abelian varieties.

First treatment of Stark-Heegner points on higher-dimensional abelian varieties.

Formulas are suitable for explicit computations.

Abstract

We prove formulas for the p-adic logarithm of quaternionic Darmon points on p-adic tori and modular abelian varieties over Q having purely multiplicative reduction at p. These formulas are amenable to explicit computations and are the first to treat Stark-Heegner type points on higher-dimensional abelian varieties.