# L-functions of Elliptic Curves with Complex Multiplication

Specialeforsvar: Erik Lange

Titel:  L-functions of Elliptic Curves with Complex Multiplication

Abstract: It has been conjectured that the $L$-function attached to an elliptic curve defined over a number field has an analytic continuation to the whole complex plane and satisfies a certain functional equation. The conjecture has turned out to hold true in a number of cases.
The aim of this thesis is to show that this conjecture is true for a certain family of elliptic curves defined over the field $\Q(\sqrt{-3})$, namely the curves of the form $y^2 = x^3 + D$, $D$ a non-zero integer. Everyone of these curves has complex multiplication (CM) by the ring of integers of $\Q(\sqrt{-3})$ and as such this result is already known in larger generality. Indeed, Deuring has proved that the conjecture on $L$-functions hold for \emph{any} elliptic curves with CM.
Since the $L$-function of an elliptic curve with CM can be written in terms of Hecke $L$-functions, the work of Hecke and Tate is relevant in this context. We give a detailed exposition of Tate's work on Hecke $L$-functions and deduce the results concerning elliptic curves from this.

Vejleder: Jasmin Matz
Censor:   Simon Kristensen, Aarhus Universitet