Special values of anticyclotomic L-functions modulo λ

The purpose of this article is to generalize some results of Vatsal on the special values of Rankin-Selberg L-functions in an anticyclotomic -extension. Let g be a cuspidal Hilbert modular newform of parallel weight and level over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant . We study the l-adic valuation of the special values as varies over the ring class characters of K of -power conductor, for some fixed prime ideal . We prove our results under the only assumption that the prime to part of is relatively prime to D.