Octavic theta series

2014/08/20 by Eberhard Freitag, Freitag, Eberhard, Riccardo Salvati Manni +1
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.1408.4537

arxiv created 2014/08/20 · arxiv updated 2017/10/11

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 H₁₀. 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 H₁₀ 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.

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