Effective bounds for the negativity of Shimura curves on Hilbert Modular Surfaces

TL;DR

This paper investigates the bounded negativity conjecture for Hilbert modular surfaces, providing explicit bounds for the negativity of special curves called Hirzebruch-Zagier curves.

Contribution

It offers the first explicit bounds for the negativity of Hirzebruch-Zagier curves on Hilbert modular surfaces, advancing understanding of the bounded negativity conjecture.

Findings

Established explicit negativity bounds for Hirzebruch-Zagier curves

Confirmed the bounded negativity conjecture in specific non-quaternionic cases

Enhanced understanding of curve negativity on Hilbert modular surfaces

Abstract

We study the bounded negativity conjecture for non-quaternionic Hilbert modular surfaces and give an explicit bound for the special case of Hirzebruch-Zagier curves on Hilbert modular surfaces.