Kummer Theory

Abstract

In this chapter, we investigate the nature of the extension obtained by extracting m-th roots of rational points. More precisely, let A be an elliptic curve defined over the number field K. Let Q∈A _K . We look into the field K(P), where P is some point such that mP = Q,where misa positive integer. Let A _m as usual denote the group of points of period m on A. We shall first assume that A _m ⊂ A _K We give a computational proof thatA_K/2A _K is finite, making explicit use of the duplication formulas for points on A. This is the proof given by Weil [We 1] eight years after Mordell gave his first proof of the finite generation of AK, see also the account given in Mordell’s book [Mo]. Next we give a second proof depending on more algebraic number theory and reduction mod various primes, but involving no computations and also applicable to abelian varieties.