The Picard numbers of elliptic surfaces with many symmetries (Q1074678)

In this paper the Picard numbers of several families of elliptic surfaces are computed. This is equivalent to the difficult problem of determining the rank of the Mordell-Weil group of certain elliptic curves over function fields. The method is to study the action induced by automorphisms of these surfaces on a relevant part of the cohomology. The cohomology classes are represented by certain inhomogeneous differential equations - on which the effect of the action is easily understood.