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Hecke operators acting on optimal embeddings in indefinite quaternion algebras

Volume 204 / 2022

James Rickards Acta Arithmetica 204 (2022), 347-367 MSC: Primary 11R52; Secondary 11F11, 11Y40, 16H05. DOI: 10.4064/aa210723-11-7 Published online: 18 August 2022


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.


  • James RickardsDepartment of Mathematics
    University of Colorado Boulder
    2300 Colorado Ave
    Boulder, CO 80309, USA

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