Equidistribution of cusp forms in the level aspect

TL;DR

This paper proves that holomorphic newforms of fixed weight and increasing squarefree level become uniformly distributed on the modular curve of level 1, confirming a conjecture for the level aspect using adapted methods from previous large weight results.

Contribution

It extends equidistribution results to the level aspect for fixed weight newforms, adapting techniques to handle the complexities of large squarefree levels.

Findings

Mass of newforms equidistributes on the modular curve of level 1

Confirmed the squarefree level case of a conjecture by Kowalski, Michel, and Vanderkam

Extended Watson's formula to newforms with level dividing but not equal to the level of other forms

Abstract

Let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing squarefree level q tending to infinity. We prove that the pushforward of the mass of f to the modular curve of level 1 equidistributes with respect to the Poincar\'{e} measure. Our result answers affirmatively the squarefree level case of a conjecture spelled out by Kowalski, Michel and Vanderkam (2002) in the spirit of a conjecture of Rudnick and Sarnak (1994). Our proof follows the strategy of Holowinsky and Soundararajan (2008) who show 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 (which on the surface makes sense only in the large weight limit), an evaluation of the p-adic integral needed to extend Watson's formula to the case of three newforms where the level of one divides but need not equal the common squarefree level of the other two, and some additional technical work in the problematic case that the level has many small prime factors.