[Paper Review] $W(E_8)$-invariant Jacobi forms

Haowu Wang|arXiv (Cornell University)|Jan 25, 2018

Advanced Algebra and Geometry20 references3 citations

TL;DR

This paper establishes that all $W(E_8)$-invariant Jacobi forms are uniquely expressible as polynomials in nine algebraically independent holomorphic Jacobi forms introduced by Sakai, with coefficients in meromorphic modular forms. It further proves that the space of $W(E_8)$-invariant weak Jacobi forms of fixed index forms a free module over the ring of modular forms, and fully determines the module structure for index less than 5 by constructing explicit generators.

ABSTRACT

We investigate Jacobi forms invariant under the action of the Weyl group of root lattice $E_8$. Such Jacobi forms are called $W(E_8)$-invariant Jacobi forms. We prove that every $W(E_8)$-invariant Jacobi form can be expressed uniquely as a polynomial in nine algebraically independent holomorphic Jacobi forms introduced by Sakai with coefficients which are meromorphic modular forms. The space of $W(E_8)$-invariant weak Jacobi forms of fixed index is a free module over the ring of modular forms. When index is less than $5$, we determine the structure of the corresponding module and construct all generators.

Motivation & Objective

• To understand the algebraic structure of Jacobi forms invariant under the Weyl group $W(E_8)$ of the $E_8$ root lattice.
• To determine whether such $W(E_8)$-invariant Jacobi forms can be expressed in terms of a finite set of basic forms.
• To establish that the space of $W(E_8)$-invariant weak Jacobi forms of fixed index is a free module over the ring of modular forms.
• To explicitly construct generators for the module of $W(E_8)$-invariant weak Jacobi forms when the index is less than 5.

Proposed method

• The authors use the action of the Weyl group $W(E_8)$ to analyze the symmetry properties of Jacobi forms on the $E_8$ root lattice.
• They employ Sakai's nine algebraically independent holomorphic Jacobi forms as a generating set for the ring of $W(E_8)$-invariant Jacobi forms.
• The proof relies on showing that any $W(E_8)$-invariant Jacobi form can be uniquely written as a polynomial in these nine forms with meromorphic modular form coefficients.
• The structure of the module of weak Jacobi forms is analyzed using the theory of modular forms and the graded ring structure of Jacobi forms.
• For index less than 5, the authors compute the generators explicitly by analyzing the dimension and symmetry constraints of the space.

Experimental results

Research questions

• RQ1 Can every $W(E_8)$-invariant Jacobi form be expressed as a polynomial in a finite set of basic Jacobi forms with meromorphic modular coefficients?
• RQ2 Is the space of $W(E_8)$-invariant weak Jacobi forms of fixed index a free module over the ring of modular forms?
• RQ3 What is the explicit structure of the module of $W(E_8)$-invariant weak Jacobi forms when the index is less than 5?
• RQ4 How many generators are needed to span the space of $W(E_8)$-invariant weak Jacobi forms of index less than 5, and what are their explicit forms?

Key findings

• Every $W(E_8)$-invariant Jacobi form is uniquely expressible as a polynomial in nine algebraically independent holomorphic Jacobi forms introduced by Sakai.
• The coefficients in this polynomial representation are meromorphic modular forms.
• The space of $W(E_8)$-invariant weak Jacobi forms of fixed index forms a free module over the ring of modular forms.
• For index less than 5, the module structure is completely determined, and all generators are explicitly constructed.
• The number of generators and their degrees are fully characterized in the low-index range, providing a complete basis for these spaces.