Managing Editor: Ranicki, Andrew
null Brüdern, Jörg / Bruinier, Jan Hendrik / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Gallavotti, Giovanni / Garnier, Josselin / Neeb, Karl-Hermann / Noguchi, Junjiro / Segal, Dan / Freydoon, Shahidi / Shigekawa, Ichiro / Sogge, Christopher D. / Strambach, Karl
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1Institute for Mathematics and Computer Sciences, Tsuda University, 2-1-1 Tsuda-machi, Kodaira-shi, Tokyo 187-8577, Japan; and Department of Mathematics, Stanford University, Building 380, Stanford, California 94305, USA
Citation Information: Forum Mathematicum. Volume 24, Issue 1, Pages 85–98, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: 10.1515/form.2011.050, January 2012
Publication History:
The new bound toward the generalized Ramanujan conjecture for
GL n (𝐀 𝐐 ) is given by Kim–Sarnak [4].
We generalize it to that for
GL n (𝐀 F ) with 
F being an imaginary quadratic field.
Applying the obtained bound to the automorphic representation of 
GL 5
gives the following estimate for the first positive eigenvalue of the Laplacian on 
L 2 ( 3 ), where 
is any congruence subgroups of 
SL 2 ( F )
with 
F the ring of integers of 
F:
1 ()0.952.
Keywords.: Generalized Ramanujan conjecture; Ramanujan–Selberg conjecture; imaginary quadratic fields; symmetric square L$L$-function
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