K3 surfaces associated to Abelian Fourfolds of Mumford's Type

TL;DR

This paper investigates K3 surfaces associated with Mumford's special abelian fourfolds, computing their Hodge structures, and applying these results to elliptic fibrations, Mordell-Weil groups, and CM abelian fourfolds.

Contribution

It provides a canonical integral Hodge structure for these K3 surfaces and explores their geometric and arithmetic properties, including elliptic fibrations and CM structures.

Findings

Every such K3 surface admits an elliptic fibration.

Mordell-Weil groups have 2-torsion or trivial torsion.

Identifies quaternionic structures related to CM abelian fourfolds.

Abstract

Mumford constructed a family of abelian fourfolds with special stucture not characterized by endomorphism ring. Galluzzi showed that the weight 2 Hodge structure of such a variety decomposes into Hodge substructures via the action of Mumford-Tate group, one of which is of K3 type with Hodge number (1,7,1). We will compute the intersection form of such a Hodge structure and provide a canonical integral Hodge structure on this subspace. Furthermore, we shall show two applications of this invariant. Firstly, using the above formula we shall show that every K3 surface obtained in this way will admit an elliptic fibration. Comparing them with the list by Shimada, we will show that the Mordell-Weil group of all elliptic fibrations have either 2-torsion or trivial torsion group, and the conditions of the occurance of 2-torsions can be specified. Secondly, we shall use it to determine the quaternion that gives a special CM abelian fourfold that is derived equivalent to the Shioda fourfold.