Nguyen, Kim:

Explicit arithmetic of Brauer groups : ray class fields and index calculus

Essen (2001)
Dissertation / Fach: Mathematik
Fakultät für Mathematik
Zentrale wissenschaftliche Einrichtungen » Institut für Experimentelle Mathematik (IEM) Essen
Essen, Univ., Diss., 2001
Abstract:
In this thesis we examine the arithmetic of Brauer groups of local and global fields. Although Brauer groups are well studied from a theoretical point of view, no one has yet addressed the question of making this theory explicit. We propose to do exactly this in the case of relative Brauer groups. Let L=K be a local extension of degree l. Then the invariant map induces an isomorphism Br(L=K) ' Z=lZ. The first natural question is to compute this invariant map explicitly for a given element A 2 Br(K=L). In doing this we show that this problem is intimately related to the arithmetic of the underlying finite field. This motivates the following approach: calculate a local invariant map at a ramified place p via the Hasse{Brauer{Noether local{global principle by relating it to the invariant map at other (unramified) places q 6 = p. We show that { using the concept of smoothness { this leads to algorithms which are known as index calculus methods in order to compute the discrete logarithm in finite fields. Moreover we show how this approach links the question of solving the discrete logarithm in finite fields to the problem of solving discrete logarithms in the Galois group of certain global extensions. In order to apply the local global principle, we need to construct or at least prove the existence of global extensions with prescribed ramation and order. Except in the cases of K = Q and K an imaginary quadratic field we provide results about extensions of this kind in the case that K is a CM field. Using these results we are able to modify a well known algorithm in the case of discrete logarithms in certain subgroups of Fpn. We also give an interpretation of the function field sieve in the setting of Brauer groups. This interpretation explains a notable difference between number field sieve and function field sieve. Finally we link the discrete logarithm problem on abelian varieties to the arithmetic of Brauer groups using the Tate pairing.