Hecke operators acting on optimal embeddings in indefinite quaternion algebras
Volume 204 / 2022
Acta Arithmetica 204 (2022), 347-367
MSC: Primary 11R52; Secondary 11F11, 11Y40, 16H05.
DOI: 10.4064/aa210723-11-7
Published online: 18 August 2022
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 2.
