A higher rank shifted convolution problem with applications to L-functions
Valentin Blomer, Junxian Li
TL;DR
This work resolves a challenging higher-rank shifted convolution problem for ${\rm GL}(3) \times {\rm GL}(2)$ where one factor is the divisor function by deploying two intertwined delta-symbol methods to create a bilinear structure and apply Voronoi-type summations. The authors establish a power-saving bound of $x^{41/42+\varepsilon}$ for sums of $A(n,1)\tau(m)$ under a linear shift, uniformly in the shift and parameters, and also prove a flexible divisor-switching variant. As an application, they derive an asymptotic formula for a twisted moment of GL$(3)$ $L$-functions, averaged over primitive even/odd Dirichlet characters with moduli up to $Q$, obtaining a main term proportional to $Q^2$ and a nontrivial power-saving error. The approach combines Jutila's circle method with a refined Kloosterman-circle method, leverages GL$(3)$ Voronoi summation, and utilizes sharp bounds for Hecke eigenvalues and hyper-Kloosterman sums, highlighting a novel interaction between two circle methods in the high-rank setting.
Abstract
While several instances of shifted convolution problems for GL(3) x GL(2) have been solved, the case where one factor is the classical divisor function and one factor is a GL(3) Fourier coefficient has remained open. We solve this case in the present paper. The proof involves two intertwined applications of different types of delta symbol methods. As an application we establish an asymptotic formula for central values of L-functions for a GL(3) automorphic form twisted by Dirichlet characters to moduli q < Q.