Statement Aaron Silberstein 1

Anabelian geometry is a grand synthesis of number theory, algebraic geometry, algebraic topology, and group theory. Parallel to the situation in algebraic topology where to a space, one attaches a fundamental group (after fixing a basepoint), in algebraic geometry one attaches to every algebraic variety X (and a geometric point x) its étale fundamental group πét 1 (X,x). Similar to the topological fundamental group, πét 1 is functorial in the sense that every map of varieties endowed with geometric points, (X,x)→ (X ′, x′) gives rise to a map πét 1 (X,x)→ πét 1 (X ′, x′) which respects composition of functions. The étale fundamental group generalizes and synthesizes the Galois theory of fields and the fundamental group in topology. The aim of anabelian geometry is to offer a new, powerful tool to study problems in arithmetic and algebraic geometry by translating them into group-theoretical problems via the étale fundamental group. In his Esquisse d’un Programme [Gro97b] and letter to Faltings [Gro97a], Grothendieck put forward a program for translating questions in arithmetic and algebraic geometry into questions about profinite groups via πét 1 . This program was named Grothendieck’s Anabelian Program, and the “yoga” of Grothendieck’s anabelian geometry is that if the fundamental group πét 1 (X,x) is “rich enough”, then it should encode much of the information about X as a variety; such varieties X should be called anabelian in the sense of Grothendieck. Some of the questions central to anabelian geometry are: