Fields generated by torsion points of elliptic curves.

Let K be a field of characteristic char ( K ) ≠ 2 , 3 and let E be an elliptic curve defined over K . Let m be a positive integer, prime with char ( K ) if char ( K ) ≠ 0 ; we denote by E [ m ] the m -torsion subgroup of E and by K m : = K ( E [ m ] ) the field obtained by adding to K the coordinates of the points of E [ m ] . Let P i : = ( x i , y i ) ( i = 1 , 2 ) be a Z -basis for E [ 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 , y 2 , ζ m } trying to emphasize the role of ζ m (a primitive m -th root of unity). In particular, we prove that K m = K ( x 1 , ζ 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 , ζ p , y 2 } is often minimal, while when the classical Galois representation Gal ( K p / K ) → GL 2 ( Z / p Z ) 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 . [ABSTRACT FROM AUTHOR]