Generalized Ramanujan conjecture over general imaginary quadratic fields

Maki Nakasuji

doi:10.1515/form.2011.050

Article

Generalized Ramanujan conjecture over general imaginary quadratic fields

Maki Nakasuji

Published/Copyright: January 19, 2012

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From the journal Forum Mathematicum Volume 24 Issue 1

Abstract.

The new bound toward the generalized Ramanujan conjecture for GLn(𝐀𝐐) is given by Kim–Sarnak [4]. We generalize it to that for GLn(𝐀F) with F being an imaginary quadratic field. Applying the obtained bound to the automorphic representation of GL5 gives the following estimate for the first positive eigenvalue of the Laplacian on L2(3), where is any congruence subgroups of SL2(F) with F the ring of integers of F: 1()0.952.

Keywords.: Generalized Ramanujan conjecture; Ramanujan–Selberg conjecture; imaginary quadratic fields; symmetric square -function

Received: 2009-03-10

Revised: 2010-02-24

Published Online: 2012-01-19

Published in Print: 2012-January

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Articles in the same Issue

https://doi.org/10.1515/form.2011.050

Keywords for this article

Generalized Ramanujan conjecture; Ramanujan–Selberg conjecture; imaginary quadratic fields; symmetric square -function