We give a complete answer to the local-global divisibility problem for algebraic tori. In particular, we prove that given an odd prime p, if T is an algebraic torus of dimension r < p − 1 defined over a number field k, then the local-global divisibility by any power p n holds for T (k). We also show that this bound on the dimension is best possible, by providing a counterexample of every dimension r p − 1. Finally, we prove that under certain hypotheses on the number field generated by the coordinates of the p n -torsion point of T , the local-global divisibility still holds for tori of dimension less than 3(p − 1).

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.

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.