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 ${\mathrm{GL}}_2({\mathbf{A}}_F)\times {\mathrm{GL}}_2({\mathbf{A}}_F)$, and more generally ${\mathrm{GL}}_r({\mathbf{A}}_F)\times {\mathrm{GL}}_2({\mathbf{A}}_F)$ for $r\ge 2$ an integer over $F$ a totally real number field, contingent on the best known approximations towards the generalized Lindelöf hypothesis for ${\mathrm{GL}}_2({\mathbf{A}}_F)$-automorphic forms in the level aspect, as well as the best known approximations to the generalized Ramanujan conjecture hypothesis for ${\mathrm{GL}}_2({\mathbf{A}}_F)$-automorphic forms. We proceed by developing a spectral approach to the shifted convolution problem for coefficients of ${\mathrm{GL}}_2({\mathbf{A}}_F)$-automorphic forms, accessing he higher-rank case through the classical projection operator ${\mathbb{P}}_1^r$ 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.

This paper is withdrawn, at least temporarily, due to a gap in deriving bounds from the L^2-decomposition of the non-\Z-finite lifted parabolic forms for the shifted convolution problem in ranks n \geq 3. These are used to bound the off-diagonal contributions in the average derived via approximate functional equations, and estimates are otherwise correct. We intend to post a revised version later

References in corpus (2)

• Height pairings on orthogonal Shimura varieties
• Rankin-Selberg L-functions in cyclotomic towers, III