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Arithmetic properties of the Shimura–Shintani–Waldspurger correspondence

With an appendix by Brian Conrad

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Abstract

We prove that the theta correspondence for the dual pair \(({\widetilde{\textup{SL}_2}}, \textit{PB}^\times)\), for B an indefinite quaternion algebra over ℚ, acting on modular forms of odd square-free level, preserves rationality and p-integrality in both directions. As a consequence, we deduce the rationality of certain period ratios of modular forms and even p-integrality of these ratios under the assumption that p does not divide a certain L-value. The rationality is applied to give a direct construction of isogenies between new quotients of Jacobians of Shimura curves, completely independent of Faltings’ isogeny theorem.

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  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Kartik Prasanna

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Prasanna, K. Arithmetic properties of the Shimura–Shintani–Waldspurger correspondence. Invent. math. 176, 521–600 (2009). https://doi.org/10.1007/s00222-008-0169-z

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