Hecke operators acting on optimal embeddings in indefinite quaternion algebras

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.

18 pages. Significant rewrite from v1 (29pgs -> 18pgs): removed unnecessary material, streamlined the exposition and various proofs, and updated some notation

References in corpus (2)

• Computing intersections of closed geodesics on the modular curve
• Counting intersection numbers of closed geodesics on Shimura curves

Cited by in corpus (1)

• Kudla-Millson lift of toric cycles and restriction of Hilbert modular forms