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The ring of Weyl invariant $E_8$ Jacobi forms

Kazuhiro Sakai

Abstract

We prove that the ring of Weyl invariant $E_8$ weak Jacobi forms is isomorphic to that of joint covariants of a binary sextic and a binary quartic form. The ring is therefore finitely generated. A minimal basis of generators is obtained from that already known for the ring of covariants.

The ring of Weyl invariant $E_8$ Jacobi forms

Abstract

We prove that the ring of Weyl invariant weak Jacobi forms is isomorphic to that of joint covariants of a binary sextic and a binary quartic form. The ring is therefore finitely generated. A minimal basis of generators is obtained from that already known for the ring of covariants.

Paper Structure

This paper contains 15 sections, 25 theorems, 89 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Weyl invariant $E_8$ Jacobi forms
  3. Meromorphic Jacobi forms and intersection of polynomial rings
  4. Seiberg--Witten curve and its coefficients
  5. Intersection of two polynomial rings
  6. New algorithm
  7. Invariants of binary forms and isomorphism
  8. Invariants of binary forms
  9. Trigrading
  10. Isomorphism
  11. Generators
  12. Basic meromorphic Jacobi forms
  13. Basic holomorphic Jacobi forms as polynomials
  14. Full list of generators
  15. Special functions

Key Result

Proposition 2.2

The above nine $A_i,B_j$ are algebraically independent over the ring of modular forms $\mathbb{C}[E_4,E_6]$.

Theorems & Definitions (53)

  • Definition 2.1
  • Remark
  • Proposition 2.2: Wang:2018fil, Sun:2021ije
  • Theorem 2.3: Sun and Wang Sun:2021ije
  • Proposition 3.1: Properties of $a_i,b_j$
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Proposition 3.5
  • ...and 43 more