Harmonic theta series and the Kodaira dimension of $\mathcal{A}_6$

TL;DR

This paper constructs a basis of harmonic theta series for Siegel cusp forms of degree 6 and weight 14, demonstrating that the moduli space al_6 has non-negative Kodaira dimension, advancing understanding of its geometric properties.

Contribution

It provides an explicit basis of harmonic theta series for degree 6 Siegel cusp forms and links boundary vanishing properties to the Kodaira dimension of al_6.

Findings

Constructed a basis of harmonic theta series for degree 6 cusp forms.

Identified a form with boundary vanishing order 2.

Proved al_6 has non-negative Kodaira dimension.

Abstract

We construct a basis of the space ${\text S}_{14}({\text{Sp}}_{12}({\mathbb Z}))$ of Siegel cusp forms of degree $6$ and weight $14$ consisting of harmonic theta series. One of these functions has vanishing order $2$ at the boundary which implies that the Kodaira dimension of $\mathcal{A}_6$ is non-negative.