The purpose of this article is to generalize some results of Vatsal on studying the special values of Rankin-Selberg L-functions in an anticyclotomic $\mathbb{ Z}_{ p}$-extension. Let $g$ be a cuspidal Hilbert modular form of parallel weight (2,...,2) and level $\mathcal{ N}$ over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant $\mathcal{ D}$. We study the $l$-adic valuation of the special values $L(g,\chi,\frac{ 1}{ 2})$ as \chi varies over the ring class characters of K of $\mathcal{ P}$-power conductor, for some fixed prime ideal $\mathcal{ P}$. We prove our results under the only assumption that the prime to $\mathcal{ P}$ part of $\mathcal{ N}$ is relatively prime to $\mathcal{ D}$.
Categories: math.NT