On class number relations, intersections, and GL(2)-tale over the function field side

TL;DR

This paper explores class number relations and divisor intersections on Drinfeld-Stuhler modular surfaces using harmonic theta series, linking class numbers to intersection theory and automorphic forms.

Contribution

It introduces a novel approach connecting class numbers, intersection numbers, and automorphic forms via harmonic theta series on function field modular surfaces.

Findings

Fourier coefficients relate to modified Hurwitz class numbers and intersection numbers.

Class numbers can be interpreted as a mass sum over CM points.

The generating function is realized as a metaplectic automorphic form.

Abstract

The aim of this paper is to study class number relations over function fields and the intersections of Hirzebruch-Zagier type divisors on the Drinfeld-Stuhler modular surfaces. The main bridge is a particular "harmonic" theta series with nebentypus. Using the strong approximation theorem, the Fourier coefficients of this series are expressed in two ways; one comes from modified Hurwitz class numbers and another gives the intersection numbers in question. An elaboration of this approach enables us to interpret these class numbers as a "mass sum" over the CM points on the Drinfeld-Stuhler modular curves, and even realize the generating function as a metaplectic automorphic form.