Progress In Galois Theory

The legacy of Galois was the beginning of Galois theory as well as group theory. From this common origin, the development of group theory took its own course, which led to great advances in the latter half of the 20th cen­ tury. It was John Thompson who shaped finite group theory like no-one else, leading the way towards a major milestone of 20th century mathematics, the classification of finite simple groups. After the classification was announced around 1980, it was again J. Thomp­ son who led the way in exploring its implications for Galois theory. The first question is whether all simple groups occur as Galois groups over the rationals (and related fields), and secondly, how can this be used to show that all finite groups occur (the 'Inverse Problem of Galois Theory'). What are the implica­ tions for the stmcture and representations of the absolute Galois group of the rationals (and other fields)? Various other applications to algebra and number theory have been found, most prominently, to the theory of algebraic curves (e.g., the Guralnick-Thompson Conjecture on the Galois theory of covers of the Riemann sphere).

A recent trend in the field of Galois theory is to tie the previous theory of curve coverings (mostly of the Riemann sphere) and Hurwitz spaces (moduli spaces for such covers) with the theory of algebraic curves and their moduli spaces. A general survey of this is given in the article by Voelklein. Further exemplifications come in the articles of Guralnick on automorphisms of modular curves in positive characteristic, of Zarhin on the Galois module structure of the 2-division points of hyperelliptic curves and of Krishnamoorthy, Shashka and Voelklein on invariants of genus 2 curves.

Abhyankar continues his work on explicit classes of polynomials in characteristic p>0 whose Glaois groups comprise entire families of Lie type groups in characteristic p. In his article, he proves a characterization of sympletic groups required for the identification of the Galois group of certain polynomials. 

The more abstract aspects come into play when considering the totality of Galois extensions of a given field.  This leads to the study of absolute Galois groups and (profinite) fundamental groups. Haran and Jarden present a result on the problem of finding a group-theoretic characterization of absolute Galois groups. In a similar spirit, Boston studies infinite p-extensions of number fields unramified at p and makes a conjecture about a group-theoretic characterization of their Galois groups. He notes connections with the Fontaine-Mazur conjecture, knot theory and quantum field theory.  Nakamura continues his work on relationships between the absolute Galois group of the rationals and the Grothendieck-Teichmüller group. Finally, Fried takes us on a tour of places where classical tropics like modular curves and j-line covers connect to the genus zero problems which was the starting point of the Guralnick-Thompson Conjecture.

Audience

This volume is suitable for graduate students and researchersin the field.