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- Title
Determination of normalized extremal quasimodular forms of depth 1 with integral Fourier coefficients.
- Authors
Nakaya, Tomoaki
- Abstract
The main purpose of this paper is to determine all normalized extremal quasimodular forms of depth 1 whose Fourier coefficients are integers. By changing the local parameter at infinity from q = e 2 π i τ to the reciprocal of the elliptic modular j -function, we prove that all normalized extremal quasimodular forms of depth 1 have a hypergeometric series expression and that integrality is not affected by this change of parameters. Furthermore, by transforming these hypergeometric series expressions into a certain manageable form related to the Atkin(-like) polynomials and using the lemmas that appeared in the study of p -adic hypergeometric series by Dwork and Zudilin, the integrality problem can be reduced to the fact that a polynomial vanishes modulo a prime power, which we prove. We also prove that all extremal quasimodular forms of depth 1 with appropriate weight-dependent leading coefficients have integral Fourier coefficients by focusing on the hypergeometric expression of them.
- Subjects
FOURIER integrals; MODULAR forms; HYPERGEOMETRIC series; DIFFERENTIAL equations; INTEGERS
- Publication
International Journal of Number Theory, 2024, Vol 20, Issue 3, p641
- ISSN
1793-0421
- Publication type
Article
- DOI
10.1142/S1793042124500337