Let π be a cuspidal automorphic representation of GL2 over a totally real number field F. Let K be a totally imaginary quadratic extension of F
Nearby in the stack
. We estimate central values of the
GL2×GL2
Rankin-Selberg
L
-functions associated to
π
times representations induced from Hecke characters of
K
which are ramified only at a given prime ideal
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
π
corresponds to a holomorphic Hilbert modular form of arithmetic weight
k≥2
, we then derive finer results from the rationality theorems of Shimura, together with the existence of suitable
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
GL2
-type abelian varieties via existing Iwasawa main conjecture divisibilities.