A Brief Theory of Elliptic Curves

Sinha, Saloni and Kumar, Narasimha (2018) A Brief Theory of Elliptic Curves. Masters thesis, Indian Institute of Technology, Hyderabad.

Text
Thesis_Msc_MA_3954.pdf - Accepted Version
Restricted to Repository staff only until July 2022.
Download (10MB) | Request a copy

Abstract

The aim of this thesis is to de�ne elliptic curves and look at some of the interesting properties a special class of points, namely, rational points on these curves possess. The properties that we are looking at are not analytical, instead, they are arithmetic, and that's what makes these curves noteworthy. Rational points on elliptic curve form a group structure. In fact, they form a �nitely generated abelian group. This is done with the help of chord-tangent composition law. After establishing the abelian group structure, we are able to look at various elementary properties such as torsion subgroups of some particular elliptic curves. Finally, classi�cation of elliptic curves upto isomorphism, over �elds of di�erent characteristics is done. In the second part of the thesis, we begin by looking at a few families of elliptic curves. These families are classi�ed by certain parameters. We obtain these parameters by making a speci�c point on the curve to be of a prescribed order. Further, we study the reduction modulo p of an elliptic curve, which requires many elements from ring theory. In conclusion, Mordell-Weil's �nite generation theorem is stated and proved using certain results from group theory and constructions. I read the book Elliptic Curves by Dale Husem�oller and it has been the source of inspiration for this thesis. The framework of this thesis is based o� this book, and I have not contributed anything new except adding a few observations of my own. My thesis adviser suggested me some changes and corrections in this thesis. I have incorporated these changes to the best of my e�orts.

[error in script]

IITH Creators: