In this paper we compute the
Picard numbers of several families of elliptic surfaces (see Example 1, §5 for a
typical result.) This is equivalent to the difficult problem of determining
the rank of the Mordell-Weil group of certain elliptic curves over function
fields. Our 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—our so-called
inhomogeneous de Rham cohomology—where the effect of the action is easily
understood.