Anabelian Intersection Theory

Abstract

Let F be a field finitely generated and of transcendence degree 2 over (\bar{\mathbb{Q}}). We describe a correspondence between the smooth algebraic surfaces X defined over (\bar{\mathbb{Q}}) with field of rational functions F and Florian Pop’s geometric sets of prime divisors on (Gal(\bar{F}/F)), which are purely group-theoretical objects. This allows us to give a strong anabelian theorem for these surfaces. As a corollary, for each number field K, we give a method to construct infinitely many profinite groups (\Gamma) such that (Out_{cont} (\Gamma)) is isomorphic to (Gal(\bar{K}/K)), and we find a host of new categories which answer the Question of Ihara/Conjecture of Oda-Matsumura.