Communications in Number Theory and Physics

Volume 12 (2018)

Number 4

Picard–Fuchs equations of families of QM abelian surfaces

Pages: 829 – 856



Amnon Besser (Department of Mathematics, Ben-Gurion University of the Negev, Be’er-Sheva, Israel)

Ron Livné (Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel)


We describe an algorithm for computing the Picard–Fuchs equation for a family of twists of a fixed elliptic surface. We then apply this algorithm to obtain the equations for several examples, which come from families of Kummer surfaces over Shimura curves, as studied in our previous work. We use this to find correspondences between the parameter spaces of our families and Shimura curves.

Received 20 December 2015

Accepted 2 August 2018

Published 14 January 2019