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Journal of the European Mathematical Society

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Volume 20, Issue 7, 2018, pp. 1595–1627
DOI: 10.4171/JEMS/794

Published online: 2018-05-15

Small scale distribution of zeros and mass of modular forms

Stephen Lester[1], Kaisa Matomäki[2] and Maksym Radziwiłł[3]

(1) Queen Mary University of London, UK, and Tel Aviv University, Israel
(2) University of Turku, Finland
(3) McGill University, Montreal

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 $1/\mathrm {log} \: k$. 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 Lindelöf 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.

Keywords: Zeros of modular forms, mass equidistribution, automorphic forms

Lester Stephen, Matomäki Kaisa, Radziwiłł Maksym: Small scale distribution of zeros and mass of modular forms. J. Eur. Math. Soc. 20 (2018), 1595-1627. doi: 10.4171/JEMS/794