On 7-division fields of CM elliptic curves

Abstract

Let E be a CM elliptic curve defined over a number field K, with Weiestrass form y 3 = x 3 + bx or y 2 = x 3 + c. For every positive integer m, we denote by E [m] the m-torsion subgroup of E and by Km := K(E [m]) the m-th division field, i.e. the extension of K obtained by adding to it the coordinates of the points in E [m]. We classify all the fields K7 in terms of generators, degrees and Galois groups. We also show some applications to the Local-Global Divisibility Problem and to modular curves and Shimura curves.