Lambda-rings and Hilbert's 12th Problem
Hilbert's 12th Problem asks whether it's possible to generate in an explicit way all the extensions L of a number field K which are Galois with abelian Galois group. It had been known since Kronecker that one could do this using special values of the exponential function (roots of unity) when K is the field of rational numbers, or special values of elliptic and modular functions when K is an imaginary quadratic field. Since the mid-20th century, this has typically been expressed in the language of commutative algebraic groups with large endomorphism rings, and partial results were established in some new cases by Shimura and others. In this talk, I'll review some of this history and then explain a new framework for Hilbert's 12th Problem based on a generalization of the notion of lambda-ring in algebraic K-theory. This talk reports on joint work with Bart de Smit.
Dr James Borger, Australian National University