  We obtain nonvanishing estimates for central values of certain self-dual
Rankin-Selberg $L$-functions on $\operatorname{GL}_2({\bf{A}}_F) \times
\operatorname{GL}_2({\bf{A}}_F)$, and more generally
$\operatorname{GL}_r({\bf{A}}_F) \times \operatorname{GL}_2({\bf{A}}_F)$ for $r
\geq 2$ an integer over $F$ a totally real number field, contingent on the best
known approximations towards the generalized Lindel\"of hypothesis for
$\operatorname{GL}_2({\bf{A}}_F)$-automorphic forms in the level aspect, as
well as the best known approximations to the generalized Ramanujan conjecture
hypothesis for $\operatorname{GL}_2({\bf{A}}_F)$-automorphic forms. We proceed
by developing a spectral approach to the shifted convolution problem for
coefficients of $\operatorname{GL}_2({\bf{A}}_F)$-automorphic forms, accessing
he higher-rank case through the classical projection operator $\mathbb P^r_1$
and the way it respects Fourier-Whittaker expansions. In the course of deriving
our results, we supply the required nonvanishing hypothesis for recent work of
Darmon-Rotger to bound Mordell-Weil ranks of elliptic curves in number fields
cut out by tensor products of two odd, two-dimensional Artin representations
whose product of determinants is trivial. This in particular allows us to
deduce bounds (on average) for Mordell-Weil ranks of elliptic curves in ring
class extensions of real quadratic fields which had not been accessible
previously.
