Jared Asuncion (IMB) : « Explicit Construction of Abelian Extensions of Number Fields »
Given a number field K, the twelfth problem of Hilbert asks to
construct all abelian extensions of K by adjoining special values of
particular analytic functions. In this talk, we will discuss the
only two cases in which this problem is completely solved, namely when K is the field of rational numbers and when K is an imaginary
quadratic number field.
The talk will begin with recalling the necessary definitions from
algebraic number theory, including the definition of an abelian
extension. We will also define elliptic curves, as an algebraic
structure and analytically as a complex torus. After these
preliminaries, we will state the main theorems of complex
multiplication, which allow us to explicitly solve Hilbert’s twelfth problem for
an imaginary quadratic number field K.
Towards the end, we will briefly mention the case of CM fields, and how it
relates to the case of the imaginary quadratic number field.