The m-Step Solvable Hom-Form of Birational Anabelian Geometry for Number Fields

Let K and L be number fields. A result by Uchida shows that a homomorphism from the maximal pro-solvable Galois groups of K to the one of L satisfying some condition is obtained from an embedding of the solvable closure of L in the solvable closure of K, and that this condition is satisfied when K is the field of rational numbers.

In this talk I will discuss some work done jointly with Mohamed Saidi, where we prove a result analogous to Uchida's for the maximal m-step solvable quotients (i.e. maximal quotients of derived length m) of the absolute Galois groups of K and L using a weaker condition than the one in Uchida's work, and a related unconditional version for the rational numbers.

I will also briefly discuss a work in progress attempt at providing more unconditional results.