Twisted Hilbert modular surfaces, arithmetic intersections and the Jacquet-Langlands correspondence

TL;DR

This paper investigates arithmetic intersections on twisted Hilbert modular surfaces and Shimura curves over real quadratic fields, linking geometric invariants to automorphic forms via the Jacquet-Langlands correspondence.

Contribution

It determines the degree of the top arithmetic Todd class for twisted Hilbert modular surfaces and relates it to the arithmetic volume of Shimura curves using advanced arithmetic geometry tools.

Findings

Computed the degree of the top arithmetic Todd class.

Connected arithmetic invariants to automorphic forms through the Jacquet-Langlands correspondence.

Applied the arithmetic Grothendieck-Riemann-Roch theorem in this context.

Abstract

We study arithmetic intersections on twisted (quaternionic) Hilbert modular surfaces and Shimura curves over a real quadratic field. Our first main result is the determination of the degree of the top arithmetic Todd class of an arithmetic twisted Hilbert modular surface. This quantity is then related to the arithmetic volume of a Shimura curve, via the arithmetic Grothendieck-Riemann-Roch theorem and the Jacquet-Langlands correspondence.