Octavic theta series

Let $L = \Pi_{2,10}$ be the even unimodular lattice of signature $(2,10)$. In the paper [FS] we considered a subgroup $\mathbb{O}^{+} (L)$ of index two in the orthogonal group $\mathbb{O} (L)$. It acts biholomorphically on a ten dimensional tube domain $\mathcal{H}_{10}$. We found a $715$ dimensional space of modular forms with respect to the principal congruence subgroup of level two $\mathbb{O}^{+} (L)[2]$. 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 $\mathcal{H}_{10}$ into the Siegel half space $\mathbb{H}_{16}$. As a consequence, the modular forms in the $715$ dimensional space can be obtained as restrictions of the theta constants, i.e the simplest among all theta series.

Keywords

theta series, octonions

2010 Mathematics Subject Classification

11F46

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Received 1 August 2015

Published 5 July 2017