Abelian varieties associated to Gaussian lattices

TL;DR

This paper constructs abelian varieties from Gaussian lattices with automorphisms, revealing a pattern in theta divisors similar to classical theory, and identifies cases with many vanishing thetanulls, including a notable example related to E8.

Contribution

It introduces a new association between unimodular lattices with automorphisms and abelian varieties, extending classical theta characteristic theory to this setting.

Findings

The configuration of i-invariant theta divisors follows a classical pattern.

The associated abelian varieties have numerous vanishing thetanulls.

Special case of E8 yields known vanishing thetanulls of a fourfold.

Abstract

We associate to a unimodular lattice L, endowed with an automorphism of square -1, a principally polarized abelian variety A:= L_R/L. We show that the configuration of i-invariant theta divisors of A follows a pattern very similar to the classical theory of theta characteristics; as a consequence we find that A has a high number of vanishing thetanulls. When L = E_8 we recover the 10 vanishing thetanulls of the abelian fourfold discovered by R. Varley.