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Correlation of zeros of automorphic L-functions

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Abstract

We compute the n-level correlation of normalized nontrivial zeros of a product of Lfunctions: L(s, π1) … L(s, π k ), where πj, j = 1, …, k, are automorphic cuspidal representations of GL mj (ℚA). Here the sizes of the groups GL mj (ℚA) are not necessarily the same. When these L(s, π j ) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s, π j ) are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when m 1, …, m k ⩽ 4, but are under Hypothesis H in other cases.

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Authors and Affiliations

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

    Liu JianYa & YangBo Ye

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

    YangBo Ye

Authors
  1. Liu JianYa
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  2. YangBo Ye
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Correspondence to Liu JianYa.

Additional information

The first author was supported by the 973 Program, the National Natural Science Foundation of China (Grant No. 10531060), and Ministry of Education of China (Grant No. 305009). The second author was supported by the National Security Agency of USA (Grant No. H98230-06-1-0075). The United States government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.

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JianYa, L., Ye, Y. Correlation of zeros of automorphic L-functions. Sci. China Ser. A-Math. 51, 1147–1166 (2008). https://doi.org/10.1007/s11425-008-0085-0

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  • DOI: https://doi.org/10.1007/s11425-008-0085-0

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