Rankin-Selberg L-functions in cyclotomic towers, III

Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a totally imaginary quadratic extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times \operatorname{GL}_2$ Rankin-Selberg $L$-functions associated to $\pi$ times representations induced from Hecke characters of $K$ which are ramified only at a given prime ideal $\mathfrak{p}$ of $F$. More specifically, we use spectral decompositions of shifted convolution sums and relations to Fourier-Whittaker coefficients of genuine and non-genuine metaplectic forms to obtain nonvanishing estimates, averaging over primitive ring class characters of a given exact order. When $\pi$ corresponds to a holomorphic Hilbert modular form of arithmetic weight $k \geq 2$, we then derive finer results from the rationality theorems of Shimura, together with the existence of suitable $\mathfrak{p}$-adic $L$-functions. This allows us to generalize the theorems of Rohrlich, Vatsal, and Cornut-Vatsal to this setting. Finally, in a self-contained appendix, we explain how to use these results to deduce bounds for Mordell-Weil ranks of the associated $\operatorname{GL}_2$-type abelian varieties via existing Iwasawa main conjecture divisibilities.