Integrality of a ratio of Petersson norms and level-lowering congruences

Abstract

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 ${\rm {\rm GL}}(2)$ with the same Hecke eigenvalues as $g$ and $\langle,\rangle$ 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.

Authors

Kartik Prasanna

Department of Mathematics
University of California
Los Angeles, CA 90095
United States