The Arithmetic and Geometry of a Class of Algebraic Surfaces of General Type and Geometric Genus One

• Main Author: Lyons, Christopher Michael
• Format: Others
• Published: 2010
• Online Access: https://thesis.library.caltech.edu/5861/5/clyons_thesis.pdf; Lyons, Christopher Michael (2010) The Arithmetic and Geometry of a Class of Algebraic Surfaces of General Type and Geometric Genus One. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/67VN-D890. https://resolver.caltech.edu/CaltechTHESIS:05272010-144845323 <https://resolver.caltech.edu/CaltechTHESIS:05272010-144845323>

<p>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.</p> <p>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.</p>