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

TL;DR

This work extends the classical class-number relations to the higher-rank setting of ${\mathcal D}$-elliptic sheaves over global function fields. By deploying Briney’s intersection framework on level covers to obtain smooth moduli, a projection formula for Hecke cycles, and a rigid-analytic uniformization, the authors reduce intersection counts to combinatorics of optimal embeddings of imaginary orders into a division algebra. They prove that $i(\mathcal Z\cdot \mathcal Z_{\mathfrak a})$ equals a sum of modified Hurwitz class numbers $H^D$, including a volume term $H^D(0)$, thereby giving a higher-rank Kronecker–Hurwitz-type class-number relation. The results bridge geometric intersections on D-elliptic-sheaf moduli with arithmetic invariants tied to imaginary orders, and suggest connections to automorphic forms in the $r=2$ case via theta-series constructions. This advances the understanding of Siegel–Weil-type phenomena in the function-field, higher-rank landscape.

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.

Theorems & Definitions (71)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Definition 3.1
• Remark 3.2
• Definition 3.3
• Definition 3.4
• Remark 3.5
• Theorem 3.6
• Remark 3.7