Small scale distribution of zeros and mass of modular forms

TL;DR

This paper investigates the small-scale distribution of zeros and mass of holomorphic Hecke cusp forms on the modular surface, proving equidistribution at certain scales and analyzing zeros near the cusp under hypotheses.

Contribution

It provides a new effective proof of Rudnick's theorem, establishes an effective Quantum Unique Ergodicity result, and confirms conjectures about zeros on geodesics near the cusp.

Findings

Zeros equidistribute within shrinking hyperbolic balls at specified rates.

A positive proportion of zeros near the cusp lie on two vertical geodesics.

Conditional lower bounds on zeros on geodesics assuming the Generalized Lindelöf Hypothesis.

Abstract

We study the behavior of zeros and mass of holomorphic Hecke cusp forms on $SL_2(\mathbb Z) \backslash \mathbb H$ at small scales. In particular, we examine the distribution of the zeros within hyperbolic balls whose radii shrink sufficiently slowly as $k \rightarrow \infty$. We show that the zeros equidistribute within such balls as $k \rightarrow \infty$ as long as the radii shrink at a rate at most a small power of $1/\log k$. This relies on a new, effective, proof of Rudnick's theorem on equidistribution of the zeros and on an effective version of Quantum Unique Ergodicity for holomorphic forms, which we obtain in this paper. We also examine the distribution of the zeros near the cusp of $SL_2(\mathbb Z) \backslash \mathbb H$. Ghosh and Sarnak conjectured that almost all the zeros here lie on two vertical geodesics. We show that for almost all forms a positive proportion of zeros high in the cusp do lie on these geodesics. For all forms, we assume the Generalized Lindel\"of Hypothesis and establish a lower bound on the number of zeros that lie on these geodesics, which is significantly stronger than the previous unconditional results.