A Quick Introduction to Algebraic Geometry and Elliptic Curves

Abstract

In this volume, there are articles on the following topics in elliptic curves: Mordell-Weil theorem, Nagell-Lutz theorem, Thue’s theorem, Siegel’s theorem, l-adic representation attached to an elliptic curve over a number field, Weil conjectures for elliptic curves over finite fields, p-adic theta functions and Tate curves and Complex Multiplication. In these articles, the basic theory of elliptic curves is assumed. As an introduction to the basics, there are now many good texts available. The standard texts are Silverman’s book [S] and Cassels’s book [C]. However, for the sake of self-containment and easy reference, we present here a very brief review of the basic background and theory by assuming some basic knowledge of field theory. We shall start with the basic definitions in algebraic geometry for which one could consult any standard text (for instance, [M]). Some proofs of results on elliptic curves have been sketched here. We have benefitted from a set of unpublished lecture notes of an Instructional conference on elliptic curves held at the Tata Institute of Fundamental Research, Mumbai in 1991. For more details one may consult [S].