Hopf-Galois Structures on Extensions Arising from Division Points on Elliptic Curves

We provide a general construction of infinite Hopf-Galois extensions from a direct system of finite Hopf-Galois extensions. We then apply this framework to extensions generated by division points on elliptic curves. Specifically, for a number field K and an elliptic curve $E/K$ of rank at least $1$ with P∈ E(K) of infinite order, we consider for each integer n≥ 2 the extension Lₙ=K(Qₙ), where Qₙ∈ E(K̄) is such that nQₙ=P. We show that the finite-level extensions Lₙ/K need not be Hopf-Galois, and we give sufficient conditions for them to be so. We then prove that the infinite extension L_∞=n≥ 2Lₙ of K is potentially Hopf-Galois, meaning it becomes Hopf-Galois after a finite extension of the base field.