Determination of normalized extremal quasimodular forms of depth 1 with integral Fourier coefficients

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 $q=e^{2\pi i \tau}$ 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.