Cyclic algebras over local fields arising from elliptic curves

Abstract

Richard Brauer introduced the concept of the Brauer group of a field K to classify central simple algebras over K. An important example of a central simple algebra is a cyclic algebra which we can associate to a cyclic Galois extension over K and an element of the multiplicative group of K.

In this thesis, we study local fields and determine which elements of the Brauer group can be reached if we restrict ourselves to the subgroup of K* of discriminants of elliptic curves over K. We show that essentially any expected element of the Brauer group can be reached. Using local class field theory we also show that the answer differs when we look at elliptic curves over the ring of integers of a local field over the 2- and 3-adic numbers.