2006/12/04 by Chris A. Kurth, Линг Лонг, Kurth, Chris +1
Mathematics · #11F11 #11F30 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.math/0612100
openalex publication_date 2006/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider modular forms for finite index subgroups of the modular group whose Fourier coefficients are algebraic. It is well-known that the Fourier coefficients of any holomorphic modular form for a congruence subgroup (with algebraic coefficients) have bounded denominators. It was observed by Atkin and Swinnerton-Dyer that this is no longer true for modular forms for noncongruence subgroups and they pointed out that unbounded denominator property is a clear distinction between modular forms for noncongruence and congruence modular forms. It is an open question whether genuine noncongruence modular forms (with algebraic coefficients) always satisfy the unbounded denominator property. Here, we give a partial positive answer to the above open question by constructing special finite index subgroups of SL2(Z) called character groups and discuss the properties of modular forms for some groups of this kind.