Bounded negativity of self-intersection numbers of Shimura curves on Shimura surfaces

TL;DR

This paper proves that Shimura curves on Shimura surfaces cannot serve as counterexamples to the bounded negativity conjecture, showing only finitely many have self-intersection numbers below any given bound, using a uniform approach based on equidistribution.

Contribution

It establishes a uniform finiteness result for Shimura curves with bounded self-intersection on all Shimura surfaces, extending previous results beyond specific cases.

Findings

Finitely many Shimura curves have self-intersection below any bound

The proof uses equidistribution, not just inequalities

Results apply uniformly to all Shimura surfaces

Abstract

Shimura curves on Shimura surfaces have been a candidate for counterexamples to the bounded negativity conjecture. We prove that they do not serve this purpose: there are only finitely many whose self-intersection number lies below a given bound. Previously, this result has been shown in [BHK+13] for compact Hilbert modular surfaces using the Bogomolov-Miyaoka-Yau inequality. Our approach uses equidistribution and works uniformly for all Shimura surfaces.