Graded rings of modular forms of rational weights

Abstract

In a previous paper, for any odd integer $N\ge 5$ we constructed $(N-1)/2$ modular forms of $\varGamma (N)$ of rational weight $(N-3)/2N$ and proved that the graded rings of modular forms of weight $\ell (N-3)/2N$ ($\ell \in {\mathbb {Z}}_{\ge 0}$) are generated by our forms for $N=5$, 7, 9. The proof was given by a direct calculation of the structure of the ring. In this paper, we generalize the result to cases when $N=11$ and 13 by using Castelnuovo–Mumford criterion on normal generation and Fujita criterion on relations of sections of invertible sheaves. For this purpose, it is needed to handle modular forms of small weight where Riemann Roch theorem has cohomological obstruction. Both rings for $N=11$ and 13 are generated by 5 and 6 generators with 15 and 35 concrete fundamental relations, respectively. These relations also give equations of the corresponding modular varieties. We will show that the similar claim does not hold for $N=15$ and $N=23$. We also give remarks on relations of some theta constants and further problems.