Modular forms of orthogonal type and Jacobi theta-series

TL;DR

This paper explores Jacobi forms of half-integral index associated with positive definite lattices, providing examples, describing lifting methods, and constructing new reflective modular forms.

Contribution

It introduces a comprehensive study of Jacobi forms of half-integral index, including examples, lifting techniques, and new reflective modular forms construction.

Findings

Examples of Jacobi forms for root systems are provided.

Jacobi lifting for half-integral indices is described.

Additive lifting can produce new reflective modular forms.

Abstract

In this paper we consider Jacobi forms of half-integral index for any positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A_1=<2>). We give a lot of examples of Jacobi forms of singular and critical weights for root systems using Jacobi theta-series. We describe the Jacobi lifting for Jacobi forms of half-integral indices. In some case it gives additive lifting construction of new reflective modular forms.