Graded rings of modular forms of rational weights

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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.

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Author's contributions


The author learned the possibility to use the Castelnuovo–Mumford criterion to the problem here from Professor G. Cornelissen. The author would like to thank him for his kind suggestion.

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Correspondence to Tomoyoshi Ibukiyama.

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Partially supported by Japan Society for the Promotion of Science KAKENHI Grant Numbers JP27247001 and JP19K03424

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Ibukiyama, T. Graded rings of modular forms of rational weights. Res. number theory 6, 8 (2020).

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  • Modular form
  • Rational weight
  • Graded ring

Mathematics Subject Classification

  • 11F03
  • 11F11
  • 11F27