# 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