Surfaces Associated to Zeros of Automorphic $L$-functions

Source abstract

Assuming the Riemann Hypothesis, Montgomery established results concerning pair correlation of zeros of the Riemann zeta function. Rudnick and Sarnak extended these results to automorphic $L$-functions and all level correlations. We discover additional geometric structures associated to the zeros of automorphic $L$-functions. In the case of pair correlation, these structures form certain surfaces which display Gaussian behavior. For triple correlation, these structures exhibit Laplace and Chi-squared distributions, revealing an unexpected phase transition.