Fields generated by torsion points of elliptic curves

Let K be a field of characteristic char(K) not equal 2,3 and let epsilon be an elliptic curve defined over K. Let m be a positive integer, prime with char(K) if char(K) not equal 0; we denote by epsilon[m] the rn-torsion subgroup of epsilon and by K-m := K(epsilon[m]) the field obtained by adding to K the coordinates of the points of epsilon[m]. Let P-i := (x,, y(i)) (i = 1,2) be a Z-basis for epsilon[m]; then K-m = K (x(1), y(1), x(2), y(2)). We look for small sets of generators for K-m inside {x(1), y(1), x(2), P-2, zeta(m)} trying to emphasize the role of zeta(m) (a primitive m-th root of unity). In particular, we prove that K-m = K (x(1), zeta(m), y(2)), for any odd m >= 5. When m = p is prime and K is a number field we prove that the generating set {x(1), zeta(p), y(2)} is often minimal, while when the classical Galois representation Gal(K-p/K) -> GL(2)(Z/pZ) is not surjective we are sometimes able to further reduce the set of generators. We also describe explicit generators, degree and Galois groups of the extensions K-m/K for m = 3 and m = 4. (C) 2016 Elsevier Inc. All rights reserved.