The arithmetic and geometry of a class of algebraic surfaces of general type and geometric genus one

Abstract

We study of a class of algebraic surfaces of general type and geometric genus one, with a view toward arithmetic results. These surfaces, called CC surfaces here, have been classified over the complex numbers by Catanese and Ciliberto. At the heart of our work is a large monodromy result for a family containing all members of a large subclass of CC surfaces, called the admissible CC surfaces. This result is obtained by an analysis of degenerations of admissible CC surfaces. We apply this monodromy theorem to prove the Tate and semisimplicity conjectures for all admissible CC surfaces over finitely-generated fields of characteristic zero, which are statements about the Galois representations on their cohomology. We also apply the theorem to produce an example of an algebraic cycle on a Shimura variety of orthogonal type that is not contained in any proper special subvariety; this we do by using the period map of the aforementioned family. Finally, we deduce the existence of complex CC surfaces with the minimum possible Picard number.