Enumeration of modular forms for $\hat{I}{“}_1(N)$

Abstract

This paper considers holomorphic modular forms for $\hat{I}{“}_1(N)$ of integral weight of the form $$f_{\mathbf{a}}^{(N)}(\ddot{I}„)=q^s(q^N;q^N{)}_{\infty }^{a_0}\prod _{j=1}^{\lfloor N/2\rfloor }(q^j,q^{N-j};q^N{)}_{\infty }^{a_j},\mathbf{a}=(a_1,\dots ,a_{\lfloor N/2\rfloor }),$$ for fixed $a_0=2k\in 2{\mathbb{Z}}_{\ge 0}$. We show that the number of relevant exponent vectors $\mathbf{a}$ is finite and characterize them in terms of the $\mathbb{Q}$-rational cuspidal divisor class group of $X_1(N)$. Effective procedures are given for counting the admissible exponents by enumerating the corresponding polytopes. This leads to formulas for the number of exponent vectors in terms of quasipolynomials in $k$.