Arithmetic of the moduli of fibered algebraic surfaces with heuristics for counting curves over global fields

Abstract

The classical theory of algebraic surfaces is essential in both geometry and number theory. The study of fibrations lies at the heart of the Enriques-Kodaira classification of compact complex surfaces as well as the Mumford-Bombieri classification of algebraic surfaces in positive characteristic. In my work, I consider the moduli of fibered algebraic surfaces through the moduli of fibrations and produce its arithmetic invariants of motivic nature with the aspiration of finding relevant applications to number theory under the global fields analogy.