Nonvanishing of self-dual $L$-values via spectral decomposition of shifted convolution sums

Abstract

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.