Arithmetic properties of the Shimura-Shintani-Waldspurger correspondence

• 成果类型: Article
• 署名作者: Prasanna, Kartik
• 署名单位: University System of Maryland; University of Maryland College Park
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-008-0169-z
• 发表日期: 2009
• 页码: 521-600
• 关键词: half-integral weight; imaginary quadratic fields; automorphic l-functions; modular-forms; zeta-functions; cusp forms; fourier coefficients; special values; periods; CURVES

摘要

We prove that the theta correspondence for the dual pair ((SL2) over tilde, PBx), 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.