With a view toward eventual arithmetic applications, I'll describe a result in complex algebraic geometry about a family of surfaces of general type with invariants p_g=q=1, K^2=3.  Specifically, I'll describe a result on the monodromy representation underlying this family, which comes from a study of the singular fibers in a larger family.  Time permitting, I'll also indicate how the monodromy is related to the Picard number of a generic surface in this family.