Automorphic products of singular weight for simple lattices

TL;DR

This paper classifies simple lattices of square-free level and constructs all holomorphic Borcherds products of singular weight for these lattices, using eta products and cusp expansions.

Contribution

It provides a complete classification of simple lattices of certain signatures and explicitly constructs all associated automorphic products of singular weight.

Findings

Classified all simple even lattices of square-free level and signature (2,n) for n > 3.

Constructed vector valued modular forms using eta products.

Computed expansions of automorphic products at various cusps.

Abstract

We classify the simple even lattices of square free level and signature (2,n) for n > 3. A lattice is called simple if the space of cusp forms of weight 1+n/2 for the dual Weil representation of the lattice is trivial. For a simple lattice every formal principal part obeying obvious conditions is the principal part of a vector valued modular form. Using this, we determine all holomorphic Borcherds products of singular weight (arising from vector valued modular forms with non-negative principal part) for the simple lattices. We construct the corresponding vector valued modular forms by eta products and compute expansions of the automorphic products at different cusps.