Equidistribution of cusp forms in the level aspect

Abstract

Let $f$ traverse a sequence of classical holomorphic newforms of fixed weight and increasing square-free level $q\to \infty$. We prove that the pushforward of the mass of $f$ to the modular curve of level $1$ equidistributes with respect to the Poincaré measure.

Our result answers affirmatively the square-free level case of a conjecture spelled out in 2002 by Kowalski, Michel, and Vanderkam in the spirit of a conjecture that Rudnick and Sarnak made in 1994.

Our proof follows the strategy of Holowinsky and Soundararajan, who showed in 2008 that newforms of level $1$ and large weight have equidistributed mass. The new ingredients required to treat forms of fixed weight and large level are an adaptation of Holowinsky’s reduction of the problem to one of bounding shifted sums of Fourier coefficients, a refinement of his bounds for shifted sums, an evaluation of the $p$-adic integral needed to extend Watson’s formula to the case of three newforms of not necessarily equal square-free levels, and some additional technical work in the problematic case that the level has many small prime factors.