Asian Journal of Mathematics

Volume 21 (2017)

Number 3

Octavic theta series

Pages: 483 – 498

DOI: http://dx.doi.org/10.4310/AJM.2017.v21.n3.a4

Authors

Eberhard Freitag (Mathematisches Institut, Ruprecht-Karls-Universität, Heidelberg, Germany)

Riccardo Salvati Manni (Dipartimento di Matematica, Università “La Sapienza”, Roma, Italy)

Abstract

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

Full Text (PDF format)

Received 1 August 2015

Published 5 July 2017