Integral presentations of the shifted convolution problem and subconvexity estimates for ${\mathrm{GL}}_n$-automorphic $L$-functions

Abstract

Fix $n\ge 2$ an integer, and $F$ be a totally real number field. We reduce the shifted convolution problem for $L$-function coefficients of ${\mathrm{GL}}_n({\mathbf{A}}_F)$-automorphic forms to the better-understood setting of ${\mathrm{GL}}_2({\mathbf{A}}_F)$. The key idea behind this reduction is to use the classical projection operator ${\mathbb{P}}_1^n\phi$ together with properties of its Fourier-Whittaker expansion. This allows us to derive novel integral presentations for the shifted convolution problem as Fourier-Whittaker coefficients of certain $L^2$-automorphic forms on the mirabolic subgroup $P_2({\mathbf{A}}_F)$ of ${\mathrm{GL}}_2({\mathbf{A}}_F)$ or its two-fold metaplectic cover ${\overline{P}}_2({\mathbf{A}}_F)$. We then construct liftings of these mirabolic forms to ${\mathrm{GL}}_2({\mathbf{A}}_F)$ and its two-fold metaplectic cover $\overline{G}({\mathbf{A}}_F)$ to justify expanding the underlying forms into linear combinations of Poincaré series. Decomposing each of the Poincaré series spectrally then allows us to derive completely new bounds for the shifted convolution problem in dimensions $n\ge 3$. As an application, we derive a uniform subconvexity bound for ${\mathrm{GL}}_n({\mathbf{A}}_F)$-automorphic $L$-functions twisted by Hecke characters. This uniform level-aspect subconvexity estimate appears to the the first of its kind for dimensions $n\ge 3$.

This paper is withdrawn, at least temporarily, due to a gap in deriving bounds from the L^2-decomposition of the non-\Z-finite lifted mirabolic forms Φfor the shifted convolution problem in ranks n \geq 3. While the setup leading to integral presentations (+ applications) is correct, the derivation of bounds via decompositions starting in §4.3 is not. We intend to post a revised version later