An overview

Abstract

Elliptic curves are the first examples of complete group varieties. These are the central objects in the proof of the famous Fermat’s Last Theorem. In this part of the book various aspects of the theory of Elliptic curves are treated. Here we give a brief description of the contents of the articles in the order in which they appear. Firstly, there is a quick introductory article by D.S. Nagaraj and B. Sury, in which some basic notations of algebraic geometry is recalled and several elementary results about elliptic curves are treated. We hope that this chapter will serve as a reference to several basic results used in the other articles which appear in this part of the book. Next is an article by B. Sury on Elliptic curves over finite fields. This article contains among other things Riemann hypothesis and Weil conjectures for elliptic curves over finite fields. The article of R. Tandon treats the Nagell-Lutz theorem, which gives a necessary condition for a point of an elliptic curve defined over a number field to be a torsion point. The article of C.S. Rajan treats the weak Modrell-Weil theorem. This theorem asserts that for an elliptic curve E defined over a number field K, the abelian group E(K)/mE(K) is finite for all integers m ≥ 1. The article by D.S. Nagaraj and B. Sury contains a proof of Modrell-Weil theorem, which states that for an elliptic curve E defined over a number field K, the abelian group E(K) is finitely generated.