Non-vanishing of L-functions, the Ramanujan conjecture, and families of Hecke characters

Details Non-vanishing of L-functions, the Ramanujan conjecture, and families of Hecke characters Non-vanishing of L-functions, the Ramanujan conjecture, and families of Hecke characters

2011-04-06

arxiv.org/abs/1104.1067v1

Mathematics

Number Theory

Scientific paper

We prove a non-vanishing result for families of $\GL_n\times\GL_n$ 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 $\GL_n$. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.

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