%0 Journal Article
%T Rankin-Selberg L-functions in cyclotomic towers, III
%A Jeanine Van Order
%J Mathematics
%D 2014
%I arXiv
%X Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a quadratic extension of $F$. Fix a prime ideal of $F$, and consider the set $X$ of all finite-order Hecke characters of $K$ unramified outside of this prime. We consider averages of central values of the Rankin-Selberg $L$-functions $L(s, \pi \times \pi(\mathcal{W}))$, where $\mathcal{W}$ ranges over characters in $X$, and $\pi(\mathcal{W})$ denotes the representation of $\operatorname{GL}_2$ associated to a $\mathcal{W} \in X$. In particular, we estimate the averages over such characters $\mathcal{W} \in X$ which arise from Dirichlet characters after composition with the norm homomorphism from $K$ to the rational number field, and determine that these do not vanish when the conductor of $\mathcal{W}$ is sufficiently large. When the representation $\pi$ is assumed to be a holomorphic discrete series of weight greater than one, and also the quadratic extension $K/F$ is assumed to be totally imaginary, we can derive stronger results extending those of Rohrlich via either the algebraicity theorem of Shimura or else the existence of a related $p$-adic interpolation series. Thus our results also apply to show the nontriviality of various constructions of $p$-adic interpolation series that appear in the literature, which had previously only been conjectured.
%U http://arxiv.org/abs/1410.4915v1