Picard-Fuchs system for family of Kummer surfaces as subsystem of GKZ hypergeometric system

We determine a simple expression of the Picard-Fuchs system for a family of Kummer surfaces for all principally polarized Abelian surfaces. It is given by a system of linear partial differential equations in three variables of rank five. Our results are based on a Jacobian elliptic fibration on Kummer surfaces and a GKZ hypergeometric system suited to the elliptic fibration.