Communications in Number Theory and Physics

Volume 16 (2022)

Number 3

On the mixed-twist construction and monodromy of associated Picard–Fuchs systems

Pages: 459 – 513



Andreas Malmendier (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.; and Department of Mathematics & Statistics, Utah State University, Logan, Ut., U.S.A.)

Michael T. Schultz (Department of Mathematics, Virginia Tech, Blacksburg, Va., U.S.A.)


We use the mixed-twist construction of Doran and Malmendier to obtain a multi-parameter family of K3 surfaces of Picard rank $\rho \geq 16$. Upon identifying a particular Jacobian elliptic fibration on its general member, we determine the lattice polarization and the Picard–Fuchs system for the family. We construct a sequence of restrictions that lead to extensions of the polarization by twoelementary lattices. We show that the Picard–Fuchs operators for the restricted families coincide with known resonant hypergeometric systems. Second, for the one-parameter mirror families of deformed Fermat hypersurfaces we show that the mixed-twist construction produces a non-resonant GKZ system for which a basis of solutions in the form of absolutely convergent Mellin–Barnes integrals exists whose monodromy we compute explicitly.


K3 surfaces, Picard–Fuchs equations, Euler integral transform

2010 Mathematics Subject Classification

14D05, 14J27, 14J28, 14J32, 32Q25, 33C60

A.M. acknowledges support from the Simons Foundation through grant no. 202367.

Received 25 August 2021

Accepted 2 May 2022

Published 4 October 2022