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Siegel modular forms of degree three and invariants of ternary quartics

Reynald Lercier, Christophe Ritzenthaler

Abstract

We determine the structure of the graded ring of Siegel modular forms of degree 3. It is generated by 19 modular forms, among which we identify a homogeneous system of parameters with 7 forms of weights 4, 12, 12, 14, 18, 20 and 30. We also give a complete dictionary between the Dixmier-Ohno invariants of ternary quartics and the above generators.

Siegel modular forms of degree three and invariants of ternary quartics

Abstract

We determine the structure of the graded ring of Siegel modular forms of degree 3. It is generated by 19 modular forms, among which we identify a homogeneous system of parameters with 7 forms of weights 4, 12, 12, 14, 18, 20 and 30. We also give a complete dictionary between the Dixmier-Ohno invariants of ternary quartics and the above generators.

Paper Structure

This paper contains 11 sections, 7 theorems, 42 equations, 6 tables.

Table of Contents

  1. Introduction and main results
  2. Review of Tsuyumine's construction of Siegel modular forms
  3. Theta functions and theta constants
  4. Siegel modular forms
  5. A minimal set of generators for modular forms of degree $3$
  6. Fundamental set of modular forms
  7. Module of relations between the generators
  8. A homogeneous system of parameters
  9. A dictionary between modular forms and invariants of quartics
  10. Modular forms in terms of invariants
  11. Invariants in terms of modular functions

Key Result

Proposition \oldthetheorem

Let $z \in \mathbb{C}^g$, $\tau \in \mathbb{H}_{g}$, $\left[{\substack{\varepsilon_{1}\\\varepsilon_{2}}}\right] \in \mathbf{M}_{2,g}(\mathbb{Z})$, then and

Theorems & Definitions (15)

  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem: Transformation formula Igusa72manni1989 Cosset11
  • Theorem \oldthetheorem: Tsuyumine Tsuyumine86
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • ...and 5 more