Hecke operators acting on optimal embeddings in indefinite quaternion algebras

TL;DR

This paper investigates how Hecke operators act on optimal embeddings in indefinite quaternion algebras, linking algebraic and geometric structures to produce modular forms of weight two.

Contribution

It introduces a new framework connecting Hecke actions on optimal embeddings with intersection theory and modular form generation.

Findings

Hecke operators induce a natural action on optimal embeddings.

A generating series of intersection numbers forms a classical modular form.

The approach bridges algebraic embeddings with geometric and analytic structures.

Abstract

We explore a natural action of Hecke operators acting on formal sums of optimal embeddings of real quadratic orders into Eichler orders. By associating an optimal embedding to its root geodesic on the corresponding Shimura curve, we can consider the signed intersection number of pairs of embeddings. Using the Hecke operators and the intersection pairing, we construct a generating series that is demonstrated to be a classical modular form of weight two.