Constructing a CM Mumford fourfold from Shioda's fourfold

TL;DR

This paper investigates the relationship between Shioda's fourfold and Mumford's construction, demonstrating that Shioda's fourfold cannot be a special case but can be adapted to construct a CM Mumford fourfold with specific properties.

Contribution

The authors show that Shioda's fourfold is not a special case of Mumford's construction and provide a method to modify its Hodge structure to build a CM Mumford fourfold with 03 multiplication.

Findings

Shioda's fourfold cannot be realized as a Mumford fourfold.

A modified Hodge structure enables construction of a CM Mumford fourfold.

Explicit basis for the period matrix of the new fourfold is provided.

Abstract

Shioda proved that the Jacobian $A_S$ of the curve $y^2 = x^9 -1$ is a 4-dimensional CM abelian variety with codimension 2 Hodge cycles not generated by divisors. It was noted by Shioda that this behavior resembles the abelian varieties constructed by Mumford. We prove that Shioda's fourfold $A_S$ cannot be realized as a special case of Mumford's construction. However, by modifying its Hodge structure, we construct a basis for computing the period matrix of a CM Mumford fourfold with multiplication by $\sqrt{-3}$.