Correlation of zeros of automorphic L-functions

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 GLm j (Q A ). Here the sizes of the groups GLm j (Q A ) 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, . . . , m k 4, but are under Hypothesis H in other cases.