The Homomorphism Form of Birational Anabelian Geometry

Abstract

Let K be a number field, and let K' be a separable closure of K, which is unique up to isomorphism. One may define the absolute Galois group of K as G_K = G(K'/K). The cohomology of the absolute Galois group can be studied using class field theory, which Neukirch used to show that some information about the primes of K is encoded in G_K, and is preserved by topological isomorphism of absolute Galois groups. Neukirch’s construction allowed Uchida to show that a topological isomorphism between absolute Galois groups determines a unique isomorphism of separable closures, a result now known as the birational anabelian Isom-Form. Uchida also obtained some partial results on a variation of the Isom-Form where isomorphisms are replaced with homomorphisms, known as the birational anabelian Hom-Form. More recently, Saïdi and Tamagawa obtained results on the encoding of primes in the maximal m-step solvable quotient G^m_K of G_K, and they used this result on the encoding of primes to obtain an “m-step” version of the Isom-Form. In this thesis, we build on some ideas used by Uchida to prove his partial results for the birational anabelian Hom-Form, combining them with the work of Saïdi and Tamagawa to determine a condition for which a continuous homomorphism σ_m between m-step solvably closed Galois groups determines some correspondence between primes. We then prove that under some conditions it is possible to recover an injection of fields from σ_m. We also prove that we are able to find conditions for which the injection we recover is uniquely determined, and use this result and the previous one to construct an m-step birational anabelian Hom-Form. Finally, we show that when one of the number fields in our homomorphism is Q, we can define the Hom-Form using our previous result by requiring weaker conditions.