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Published July 2018 | Submitted
Journal Article Open

Small scale distribution of zeros and mass of modular forms


We study the behavior of zeros and mass of holomorphic Hecke cusp forms on SL_2(ℤ)∖ℍ at small scales. In particular, we examine the distribution of the zeros within hyperbolic balls whose radii shrink sufficiently slowly as k→∞. We show that the zeros equidistribute within such balls as k→∞ 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(ℤ)∖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.

Additional Information

© 2018 EMS Publishing House. The second author was supported by the Academy of Finland grants no. 137883 and 138522. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 320755. The third author is grateful to Zeev Rudnick for inviting him to Tel-Aviv University where this work was started. We would also like to thank Misha Sodin and Zeev Rudnick for many stimulating and helpful discussions. Additionally, we are also grateful to Roman Holowinsky for sending us lecture notes from Henryk Iwaniec's course on QUE. We would also like to thank Matt Young for comments on an earlier draft of this paper and in particular for pointing out a better proof of Lemma 4.2.

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