Octavic theta series

TL;DR

This paper establishes a connection between orthogonal modular forms related to a specific lattice and theta series, using a modular embedding into Siegel space to generate these forms as restrictions of theta series.

Contribution

It demonstrates that a large space of orthogonal modular forms can be obtained as restrictions of theta series via a modular embedding into Siegel space.

Findings

The 715-dimensional space of modular forms is related to theta series.

A modular embedding into Siegel space of degree 16 is constructed.

Orthogonal modular forms can be expressed as restrictions of theta series.

Abstract

Let L be the even unimodular lattice of signature (2,10), In the paper [FS] we considered the subgroup O(L)^+ of index two in the orthogonal group. It acts biholomorphically on a ten dimensional tube domain H_{10}. We found a 715 dimensional space of modular forms with respect to the principal congruence subgroup of level two O^+(L)[2]. It defines an everywhere regular birational embedding of the related modular variety into the 714 dimensional projective space. In this paper, we prove that this space of orthogonal modular forms is related to a space of theta series. The main tool is a modular embedding of H_{10} into the Siegel half space of degree 16. As a consequence the modular forms in the 715 dimensional space can be obtained as restrictions of the simplest among all theta series.