The Arithmetric of Elliptic Curves: Mordell's Theorem
Specialeforsvar ved Nicolas Bru Frantzen
Titel: The Arithmetric of Elliptic Curves: Mrodell's Theorem
Abstrakt: The main focus in this thesis will be on proving Mordell's theorem. We shall first study some basic properties of elliptic curves, that can be found in any introduction to the arithmetic on elliptic curves. Furthermore we will introduce the notion of the resultant of two polynomials that will serve as a tool when later dealing with heights on the projective plane. We will extend the definition of heights on the projective plane to heights on elliptic curves and introduce the Néron-Tate height function (or the canonical height function). We shall then introduce the $0$th and $1$st cohomology group (in the classical way) and carry this definition over to cohomology of the infinite Galois group $\Gal(\kalg/k)$ for some perfect field $k$. In order to define the Galois group of an infinite field extension, we will need to introduce the Krull topology so that we may extend the fundamental theorem of Galois theory. We shall introduce the Selmer- and Tate-Shafarevich groups and prove that the Selmer group considered on a finite Galois extension of $\Q$ is finite. From this we easily deduce that the Selmer group $\selmer$ is finite, and then prove the Finite Basis theorem for elliptic curves over $\Q$. Using only a special case of The Weak Finite Basis theorem, we will use the theory obtained earlier from the Néron-Tate height function to finally prove Mordell's Theorem. Furthermore we shall explain how to obtain the Weak Finite Basis theorem (for elliptic curves over a number field) and the Mordell-Weil Theorem. Finally we may apply Mordell's theorem on an elliptic curve $E$ over $\Q$, so that the fundamental theorem of finitely generated abelian groups implies that $E(\Q)$ has a rank. We shall then briefly study a relationship between the rank of particularly simple elliptic curves and the congruent number problem.
Vejleder: Ian Kiming
Censor: Tom Høholdt, DTU