Density of modular forms with transcendental zeros

Dohoon Choi, Youngmin Lee, Subong Lim

Research output: Contribution to journalArticlepeer-review

Abstract

For an even positive integer k, let Mk,Z(SL2(Z)) be the set of modular forms of weight k on SL2(Z) with integral Fourier coefficients. Let Mk,Ztran(SL2(Z)) be the subset of Mk,Z(SL2(Z)) consisting of modular forms with only transcendental zeros on the upper half plane H except all elliptic points of SL2(Z). For a modular form f(z)=∑n=0af(n)e2πinz of weight k(f), let ϖ(f):=∑n=0rk(f)|af(n)|, where rk(f)=dimC⁡Mk(f),Z(SL2(Z))⊗C−1. In this paper, we prove that if k=12 or k≥16, then [Formula presented] as X→∞, where αk denotes the sum of the volumes of certain polytopes. Moreover, if we let MZ=∪k=0Mk,Z(SL2(Z)) (resp. MZtran=∪k=0Mk,Ztran(SL2(Z))) and φ is a monotone increasing function on R+ such that φ(x+1)−φ(x)≥Cx2 for some positive number C, then we prove [Formula presented]

Original languageEnglish
Article number125141
JournalJournal of Mathematical Analysis and Applications
Volume500
Issue number2
DOIs
Publication statusPublished - 2021 Aug 15

Keywords

  • Density
  • Modular forms
  • Transcendental zeros

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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