# Integrality of a ratio of Petersson norms and level-lowering congruences | Annals of Mathematics

Abstract We prove integrality of the ratio ⟨f,f⟩/⟨g,g⟩ (outside an explicit finite set of primes), where g is an arithmetically normalized holomorphic newform on a Shimura curve, f is a normalized Hecke eigenform on GL(2) with the same Hecke eigenvalues as g and ⟨,⟩ denotes the Petersson inner product. The primes dividing this ratio are shown to be closely related to certain level-lowering congruences satisfied by f and to the central values of a family of Rankin-Selberg L-functions. Finally we give two applications, the first to proving the integrality of a certain triple product L-value and the second to the computation of the Faltings height of Jacobians of Shimura curves.

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### 数学年刊（Annals of Mathematics）

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