Intersections of Hecke correspondences on the modular varieties of $\mathcal{D}$-elliptic sheaves

TL;DR

This paper explores the intersection properties of Hecke correspondences on modular varieties associated with $ abla$-elliptic sheaves in a higher-rank context, linking intersection numbers to class numbers of imaginary orders.

Contribution

It provides a higher-rank analogue of the classical class number relation by expressing intersection numbers as combinations of modified Hurwitz class numbers.

Findings

Intersection numbers expressed via Hurwitz class numbers

Establishes a higher-rank class number relation

Links geometric intersections to algebraic class numbers

Abstract

This paper studies the intersections of Hecke correspondences on the modular varieties of $\mathcal{D}$ -elliptic sheaves in the higher-rank setting, where $\mathcal{D}$ is a "maximal order" in a central division algebra $D$ over a global function field $k$. Assuming that $\dim_k(D) = r^2$, where $r$ is a prime distinct from the characteristic of $k$, we express the intersection numbers of Hecke correspondences as suitable combinations of modified Hurwitz class numbers of "imaginary orders". This result establishes a higher-rank analogue of the classical class number relation.