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Zeros of Automorphic L-Functions and Noncyclic Base Change

  • Conference paper
Number Theory

Part of the book series: Developments in Mathematics ((DEVM,volume 15))

Abstract

Let π be an automorphic irreducible cuspidal representation of GLm over a Galois (not necessarily cyclic) extension E of ℚ of degree ℓ. We compute the n-level correlation of normalized nontrivial zeros of L(s, π). Assuming that π is invariant under the action of the Galois group Gal(E/ℚ), we prove that it is equal to the n-level correlation of normalized nontrivial zeros of a product of ℓ distinct L-functions L(s, π1) ... L(s, πℓ) attached to cuspidal representations π1, ..., πℓ of GLm over ℚ. This is done unconditionally for m = 1,2 and for m = 3,4 with the degree ℓ having no prime factor ≤ (m2 + 1)/2. In other cases, the computation is made under a conjecture of bounds toward the Ramanujan conjecture over E, and a conjecture on convergence of certain series over prime powers (Hypothesis H over E and ℚ). The results provide an evidence that π should be (noncyclic) base change of ℓ distinct cuspidal representations π1,..., πℓ of GLm (ℚA), if it is invariant under the Galois action. A technique used in this article is a version of Selberg orthogonality for automorphic L-functions (Lemma 6.2 and Theorem 6.4), which is proved unconditionally, without assuming π and π1,..., πℓ being self-contragredient.

Supported by China NNSF Grant Number 10125101.

Project sponsored by the USA NSA under Grant Number MDA904-03-1-0066. The United States Government is authorized to reproduce and distributed reprints notwithstanding any copyright notation herein.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Shandong University, Jinan, Shandong, 250100, China

    Jianya Liu

  2. Department of Mathematics, The University of Iowa, Iowa City, Iowa, 52242-1419, USA

    Yangbo Ye

Authors
  1. Jianya Liu
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  2. Yangbo Ye
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Editor information

Editors and Affiliations

  1. Northwest University, Xi’an, P.R. China

    Wenpeng Zhang

  2. Nagoya University, Nagoya, Japan

    Yoshio Tanigawa

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© 2006 Springer Science + Bussiness Media, Inc.

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Liu, J., Ye, Y. (2006). Zeros of Automorphic L-Functions and Noncyclic Base Change. In: Zhang, W., Tanigawa, Y. (eds) Number Theory. Developments in Mathematics, vol 15. Springer, Boston, MA. https://doi.org/10.1007/0-387-30829-6_10

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