One-level density for holomorphic cusp forms of arbitrary level

Abstract

In 2000 Iwaniec, Luo, and Sarnak proved for certain families of L-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Milićević, which is of use for other problems as well.

Analytic isolation of newforms of given level

“…9,Rmk. 4] in his work on newforms of level p ν , ν → ∞ after some involved calculations along the lines indicated following (1) (see also [4, §5], [3]). The proof given here by way of Theorem 1 is simpler in that it avoids explicit orthogonalization of the decomposition (1).…”