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

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

2011-06-06

arxiv.org/abs/1106.1139v2

Mathematics

Number Theory

Scientific paper

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.

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