Elliptic Curves over Local Fields

Abstract

In this chapter we study the group of rational points on an elliptic curve defined over a field which is complete with respect to a discrete valuation. We start with some basic facts concerning Weierstrass equations and “reduction modulo π”.This enables us to break our problem up into several pieces; and then by examining each piece individually, we will be able to deduce a great deal about the group of rational points as a whole. Unless explicitly stated otherwise, we will use the following notation.