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

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Abstract

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, sym2 π). In particular there is no main term unless d > 0 and π is a dihedral form.

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  1. Department of Mathematics, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    Nicolas Templier & Jacob Tsimerman

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  1. Nicolas Templier
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  2. Jacob Tsimerman
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Correspondence to Nicolas Templier.

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Templier, N., Tsimerman, J. Non-split sums of coefficients of GL(2)-automorphic forms. Isr. J. Math. 195, 677–723 (2013). https://doi.org/10.1007/s11856-012-0112-2

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  • DOI: https://doi.org/10.1007/s11856-012-0112-2

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