Integrality of a ratio of Petersson norms and level-lowering congruences. (English) Zbl 1158.11026

Summary: We prove integrality of the ratio \(\langle f, f\rangle/\langle g, g\rangle\) (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 \(\text{GL}(2)\) with the same Hecke eigenvalues as \(g\) and \(\langle\cdot, \cdot\rangle\) 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.


11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F85 \(p\)-adic theory, local fields
11G18 Arithmetic aspects of modular and Shimura varieties
11G15 Complex multiplication and moduli of abelian varieties
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