Correlation of zeros of automorphic L-functions  

作　　者：LIU JianYa1 & YE YangBo1,2 1 School of Mathematics,Shandong University,Jinan 250100, China 2Department of Mathematics,The University of Iowa,Iowa City,IA 52242-1419,USA 

出　　处：《Science China Mathematics》2008年第7期1147-1166,共20页中国科学：数学（英文版）

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

摘　　要：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 (?A). Here the sizes of the groups GL mj (?A) 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.We compute the n-level correlation of normalized nontrivial zeros of a product of L-functions:L(s,π1)···L(s,πk), where πj, j=1,...,k, are automorphic cuspidal representations of GLmj(QA). Here the sizes of the groups GLmj(QA) are not necessarily the same. When these L(s,πj) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s,πj) 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 m1,...,mk 4,but are under Hypothesis H in other cases.

关 键 词：automorphic L-function zero correlation Selberg’s orthogonality conjecture 11F70 11M26 11M41 

分 类 号：O152[理学—数学]