Non-split sums of coefficients of GL(2)-automorphic forms

@article{Templier2011NonsplitSO,
title={Non-split sums of coefficients of GL(2)-automorphic forms},
author={Nicolas Templier and Jacob Tsimerman},
journal={Israel Journal of Mathematics},
year={2011},
volume={195},
pages={677-723}
}
• Published 6 June 2011
• Mathematics
• Israel Journal of Mathematics
Given a cuspidal automorphic form π on GL2, we study smoothed sums of the form $$\sum\nolimits_n {{a_\pi }({n^2} + d)V({n \over x})}$$. 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…
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