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

## TL;DR

This paper develops new integral and spectral methods to analyze the shifted convolution problem for $ ext{GL}_n$ automorphic forms over totally real fields, leading to novel subconvexity bounds for their $L$-functions.

## Contribution

It introduces integral presentations reducing the problem to $ ext{GL}_2$, constructs liftings of automorphic forms, and derives the first uniform subconvexity bounds for $ ext{GL}_n$ with $n \,\geq\, 3$.

## Key findings

- Derived new bounds for the shifted convolution problem in dimensions $n \,\geq\, 3$.
- Established a uniform subconvexity bound for $ ext{GL}_n$-automorphic $L$-functions.
- Connected the problem to $ ext{GL}_2$ techniques via integral and spectral methods.

## Abstract

Fix $n \geq 2$ an integer, and $F$ be a totally real number field. We reduce the shifted convolution problem for $L$-function coefficients of $\operatorname{GL}_n({\bf{A}}_F)$-automorphic forms to the better-understood setting of $\operatorname{GL}_2({\bf{A}}_F)$. The key idea behind this reduction is to use the classical projection operator $\mathbb P^n_1 \varphi$ 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 $P_2({\bf{A}}_F)$ of $\operatorname{GL}_2({\bf{A}}_F)$ or its two-fold metaplectic cover $\overline{P}_2({\bf{A}}_F)$. We then construct liftings of these mirabolic forms to $\operatorname{GL}_2({\bf{A}}_F)$ and its two-fold metaplectic cover $\overline{G}({\bf{A}}_F)$ to justify expanding the underlying forms into linear combinations of Poincar\'e series. Decomposing each of the Poincar\'e series spectrally then allows us to derive completely new bounds for the shifted convolution problem in dimensions $n \geq 3$. As an application, we derive a uniform subconvexity bound for $\operatorname{GL}_n({\bf{A}}_F)$-automorphic $L$-functions twisted by Hecke characters. This uniform level-aspect subconvexity estimate appears to the the first of its kind for dimensions $n \geq 3$.

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Source: https://tomesphere.com/paper/1903.07284