Galoisove reprezentacije pridružene eliptičkim krivuljama

Gužvić, Tomislav (2016) Galoisove reprezentacije pridružene eliptičkim krivuljama. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

In the first chapter we defined endomorphisms of the elliptic curve $C$ and proved their basic properties. We introduced isogenies and showed how properties of endomorphisms hold for isogenies. We mentioned the Frobenius endomorphism and give a few examples of isogenies. In the second chapter we defined the height function and showed some important properties. We also introduced the Descent theorem and stated Mordel's theorem. In the third chapter we developed basic division polynomials theory and we showed that $C[n] \simeq \mathbb{Z}/n \mathbb{Z} \bigoplus \mathbb{Z}/n \mathbb{Z}$. The fourth chapter is the main part of this work. We defined the action of the Galois group $Gal(K/\mathbb{Q})$ on points on $C$ and showed basic properties. We showed that coordinates of points of finite order are algebraic over $\mathbb{Q}$ and that the field extension induced by the $x$ and $y$ coordinates of points that are of finite order $n$ is Galois. We defined the mod $n$ Galois representation attached to the curve $C$ and showed that it is a monomorphism. We defined the Borel and Cartan subgroup of $GL(\mathbb{F}_p)$ and showed some basic properties. We determined in which maximal subgroups of $GL(\mathbb{F}_p)$ can the image of non-surjective mod $p$ Galois representation be. We defined the Weil pairing and showed in theorem 4.15 some properties of mod $p$ Galois representation.

Item Type: Thesis (Diploma thesis)
Supervisor: Najman, Filip
Date: 2016
Number of Pages: 29
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 21 Oct 2016 11:33
Last Modified: 21 Oct 2016 11:33
URI: http://digre.pmf.unizg.hr/id/eprint/5211

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