Title: Elliptic curves with complex multiplication and applications to class field theory
Abstract: One of the aims of algebraic number theory is to describe the field of
algebraic numbers and the extensions of number fields. This problem appears
as the 12° of the 23 Hilbert's problems, and is essentially an extension of
the Kronecker-Weber theorem, from the field of rational numbers to a
generic number field. Although the problem is still open, the particular
case of quadratic imaginary fields is completely understood, thanks to the
theory of elliptic curves with complex multiplication.
The purpose of this dissertation is to introduce some definitions and
properties of elliptic curves (in Chapter 1), of the complex multiplication
on them (in Chapter 2), of the class field theory (in Chapter 3) and then
to give a characterization of the maximal abelian extension and then of any
abelian extension of quadratic imaginary fields, with some other
interesting properties about elliptic curves with complex multiplication
(in Chapter 4).
Publication Year: 2020
Publication Date: 2020-02-21
Language: en
Type: article
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