Jacobi Forms of Lattice Index I. Basic Theory

TL;DR

This paper develops the foundational theory of Jacobi forms of lattice index, exploring their connections with lattice arithmetic and Weil representations, providing structure theorems, dimension formulas, and explicit examples.

Contribution

It introduces the basic theory and structure theorems for Jacobi forms of lattice index, linking them to lattice arithmetic and Weil representations.

Findings

Derived explicit dimension formulas for Jacobi forms of lattice index

Established initial structure theorems for the theory

Provided concrete explicit examples of the forms

Abstract

This is the first one of a series of articles in which we develop the theory of Jacobi forms of lattice index, their close interplay with the arithmetic theory of lattices and the theory of Weil representations. We hope to publish this series eventually in an extended and combined way as a monograph. In this part we present the basic theory and first structure theorems. We deduce explicit dimension formulas and give non-trivial explicit examples