Non-split Sums of Coefficients of GL(2)-Automorphic Forms

TL;DR

This paper investigates smoothed sums of coefficients of GL(2) automorphic forms, providing sharp error bounds and identifying main terms related to the symmetric square L-function, especially in the case of dihedral forms.

Contribution

It introduces a precise analysis of non-split sums of automorphic coefficients, establishing uniform error bounds and explicit main term identification for square-free levels.

Findings

Error term is sharp and uniform in parameters.

Main term linked to the residue of the symmetric square L-function.

No main term unless specific conditions on $d$ and $\

Abstract

Given a cuspidal automorphic form $\pi$ on $\GL_2$, we study smoothed sums of the form $\sum_{n\in\mathbb{N}} a_{\pi}(n^2+d)W(\frac{n}{Y})$. The error term we get is sharp in that it is uniform in both $d$ and $Y$ and depends directly on bounds towards Ramanujan for forms of half-integral weight and Selberg eigenvalue conjecture. Moreover, we identify (at least in the case where the level is square-free) the main term as a simple factor times the residue as $s=1$ of the symmetric square L-function $L(s,\Msym^2\pi)$. In particular there is no main term unless $d>0$ and $\pi$ is a dihedral form.