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Nonvanishing of L-functions, the Ramanujan Conjecture, and Families of Hecke Characters

Published online by Cambridge University Press:  20 November 2018

Valentin Blomer  and
Farrell Brumley
Valentin Blomer
Affiliation:
Universität Göttingen, Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen, e-mail: blomer@uni-math.gwdg.de
Farrell Brumley
Affiliation:
Institut Galilée, Université Paris 13, 93430 Villetaneuse, France, e-mail: brumley@math.univ-paris13.fr
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Abstract

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We prove a nonvanishing result for families of $\text{G}{{\text{L}}_{n}}\times \text{G}{{\text{L}}_{n}}$ Rankin–Selberg $L$-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on $\text{G}{{\text{L}}_{n}}$. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.

Keywords

11F7011M41nonvanishingautomorphic formsHecke characters, Ramanujan conjecture
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 1 , 01 February 2013 , pp. 22 - 51
Copyright
Copyright © Canadian Mathematical Society 2013

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