Quaternions and Kudla's matching princple

TL;DR

This paper establishes new identities linking average representation numbers of definite quaternion algebras with degrees of Hecke correspondences on Shimura curves related to indefinite quaternion algebras.

Contribution

It introduces novel identities connecting quaternion algebra representation numbers and geometric degrees of Shimura curves, expanding understanding of their arithmetic and geometric relationships.

Findings

Proved identities relating quaternion algebra representation numbers and Shimura curve degrees.

Established connections between algebraic and geometric properties of quaternion algebras.

Enhanced the theoretical framework for understanding quaternionic automorphic forms.

Abstract

In this paper, we prove some interesting identities, among average representation numbers (associated to definite quaternion algebras) and `degree' of Hecke correspondences on Shimura curves (associated to indefinite quaternion algebras).