On the monodromy of elliptic surfaces

There have been several constructions of family of varieties with exceptional monodromy group [ 2 ], [ 8 ]. In most cases, these constructions give Hodge structures with high weight (Hodge numbers spread out). N. Katz was the first to obtain Hodge structures with low weight (Hodge numbers equal to (2, 3, 2)) and geometric monodromy group G 2 . In this article I will give an explicit proof of Katz’s result by finding all the monodromies in a given basis.