p-adic equidistribution of modular geodesics and of Heegner points on Shimura curves

TL;DR

This paper develops a p-adic analogue of Duke's Theorem, demonstrating the equidistribution of modular geodesics and Heegner points in p-adic spaces associated with Shimura curves, extending classical results into the p-adic setting.

Contribution

It introduces a p-adic version of Duke's Theorem and proves equidistribution results for Heegner points on Shimura curves, expanding the understanding of p-adic dynamics in arithmetic geometry.

Findings

Established a p-adic equidistribution theorem for modular geodesics.

Proved equidistribution of Heegner points in p-adic spaces.

Extended classical equidistribution results to the p-adic context.

Abstract

We propose a p-adic version of Duke's Theorem on the equidistribution of closed geodesics on modular curves. Our approach concerns quadratic fields split at p as well as a p-adic covering of the modular curve. We also prove an equidistribution result of Heegner points in the p-adic space attached to Shimura curves.