naa a22 4500 510782744 CHVBK 20180411083252.0 cr unu---uuuuu 180411e20130601xx s 000 0 eng 10.1007/s11856-012-0112-2 doi (NATIONALLICENCE)springer-10.1007/s11856-012-0112-2 Non-split sums of coefficients of GL (2)-automorphic forms [Elektronische Daten] [Nicolas Templier, Jacob Tsimerman] 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. Hebrew University Magnes Press, 2013 Templier Nicolas Department of Mathematics, Fine Hall, Washington Road, 08544-1000, Princeton, NJ, USA aut Tsimerman Jacob Department of Mathematics, Fine Hall, Washington Road, 08544-1000, Princeton, NJ, USA aut Israel Journal of Mathematics Springer US; http://www.springer-ny.com 195/2(2013-06-01), 677-723 0021-2172 195:2<677 2013 195 11856 https://doi.org/10.1007/s11856-012-0112-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11856-012-0112-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Templier Nicolas Department of Mathematics, Fine Hall, Washington Road, 08544-1000, Princeton, NJ, USA aut NATIONALLICENCE 700 1- Tsimerman Jacob Department of Mathematics, Fine Hall, Washington Road, 08544-1000, Princeton, NJ, USA aut NATIONALLICENCE 773 0- Israel Journal of Mathematics Springer US; http://www.springer-ny.com 195/2(2013-06-01), 677-723 0021-2172 195:2<677 2013 195 11856 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer