Small scale distribution of zeros and mass of modular forms

Stephen Lester, Kaisa Matomäki, Maksym Radziwill

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


We study the behavior of zeros and mass of holomorphic Hecke cusp forms on SL2(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 equidistribution of mass for holomorphic forms, which we obtain in this paper. We also examine the distribution of the zeros near the cusp of SL2(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.

Original languageEnglish (US)
Pages (from-to)1595-1627
Number of pages33
JournalJournal of the European Mathematical Society
Issue number7
StatePublished - 2018


  • Automorphic forms
  • Mass equidistribution
  • Zeros of modular forms

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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