On 7-division fields of CM elliptic curves

Abstract

Let "Equation missing" be a CM elliptic curve defined over a number field K, with Weiestrass form $$y^3=x^3+bx$$ y 3 = x 3 + b x or $$y^2=x^3+c$$ y 2 = x 3 + c . For every positive integer m, we denote by "Equation missing" the m-torsion subgroup of "Equation missing" and by "Equation missing" the m-th division field, i.e. the extension of K generated by the coordinates of the points in "Equation missing". We classify all the fields $$K_7$$ K 7 . In particular we give explicit generators for $$K_7/K$$ K 7 / K and produce all the Galois groups $$\textrm{Gal}(K_7/K)$$ Gal ( K 7 / K ) . We also show some applications to the Local–Global Divisibility Problem and to modular curves.