Special values of anticyclotomic Rankin–Selberg $L$-functions

Abstract

In this article, we construct a class of anticyclotomic $p$-adic Rankin–Selberg $L$-functions for Hilbert modular forms, generalizing the construction of Brakočevic, Bertolini, Darmon and Prasanna in the elliptic case. Moreover, building on works of Hida, we give a necessary and sufficient criterion for the vanishing of the Iwasawa $\mu$-invariant of this $p$-adic $L$-function vanishes and prove a result on the non-vanishing modulo $p$ of central Rankin–Selberg $L$-values with anticyclotomic twists. These results have future applications to Iwasawa main conjecture for Rankin–Selberg convolution and Iwasawa theory for Heegner cycles.