Proportion of modular forms with transcendental zeros for general levels

Dohoon Choi; Youngmin Lee; Subong Lim; Jaegwang Ryu

doi:10.3792/pjaa.99.004

Proportion of modular forms with transcendental zeros for general levels

Dohoon Choi, Youngmin Lee, Subong Lim, Jaegwang Ryu

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let Γ be a congruence subgroup such that (Formula Presented) for some positive integer N. For a positive integer k, let (Formula Presented) be the set of modular forms of weight k on Γ with integral Fourier coefficients. Let (Formula Presented) be the set of common zeros in the upper half plane H of all the modular forms of weight k on Γ. In this note, we prove that the density of modular forms in (Formula Presented) with an algebraic zero (Formula Presented) is zero.

Original language	English
Pages (from-to)	19-22
Number of pages	4
Journal	Proceedings of the Japan Academy Series A: Mathematical Sciences
Volume	99
Issue number	2
DOIs	https://doi.org/10.3792/pjaa.99.004
Publication status	Published - 2023

Bibliographical note

Publisher Copyright:
© 2023 The Japan Academy

Keywords

Modular form
density
transcendental zero

ASJC Scopus subject areas

General Mathematics

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10.3792/pjaa.99.004

Cite this

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author = "Dohoon Choi and Youngmin Lee and Subong Lim and Jaegwang Ryu",

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AU - Lim, Subong

AU - Ryu, Jaegwang

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N2 - Let Γ be a congruence subgroup such that (Formula Presented) for some positive integer N. For a positive integer k, let (Formula Presented) be the set of modular forms of weight k on Γ with integral Fourier coefficients. Let (Formula Presented) be the set of common zeros in the upper half plane H of all the modular forms of weight k on Γ. In this note, we prove that the density of modular forms in (Formula Presented) with an algebraic zero (Formula Presented) is zero.

AB - Let Γ be a congruence subgroup such that (Formula Presented) for some positive integer N. For a positive integer k, let (Formula Presented) be the set of modular forms of weight k on Γ with integral Fourier coefficients. Let (Formula Presented) be the set of common zeros in the upper half plane H of all the modular forms of weight k on Γ. In this note, we prove that the density of modular forms in (Formula Presented) with an algebraic zero (Formula Presented) is zero.

KW - Modular form

KW - density

KW - transcendental zero

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JF - Proceedings of the Japan Academy Series A: Mathematical Sciences

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Proportion of modular forms with transcendental zeros for general levels

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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