Murmurations and ratios conjectures

Abstract

We introduce a new method for studying murmurations, based on random matrix theory. With this method, we exhibit murmurations or similar phenomena: assuming ratios conjectures, for elliptic curves ordered by height, quadratic twists of a fixed elliptic curve, and the inverse Mellin transform of the shifted second moment of $\hat{I}{\P }^{\prime }/\hat{I}\P$ on vertical lines; assuming GRH, for primitive quadratic Dirichlet characters, and holomorphic modular forms of prime level tending to infinity with sign and weight fixed; and unconditionally the inverse Mellin transform of the shifted second moment of $\hat{I}\P$ on vertical lines. We also present a generalization of our approach which relies only on the approximate functional equation in place of ratios conjectures.