Division fields of elliptic curves with minimal ramification

  • Alvaro Lozano-Robledo

    University of Connecticut, Storrs, USA


Let EE be an elliptic curve defined over Q\mathbb Q, let pp be a prime number, and let n1n\geq 1. It is well-known that the pnp^n-th division field Q(E[pn])\mathbb Q(E[p^n]) of the elliptic curve EE contains all the pnp^n-th roots of unity. It follows that the Galois extension Q(E[pn])/Q\mathbb Q(E[p^n])/\mathbb Q is ramified above pp, and the ramification index e(p,Q(E[pn])/Q)e(p,\mathbb Q(E[p^n])/\mathbb Q) of any prime P\mathfrak P of Q(E[pn])\mathbb Q(E[p^n]) lying above pp is divisible by φ(pn)\varphi(p^n). The goal of this article is to construct elliptic curves E/QE/\mathbb Q such that e(p,Q(E[pn])/Q)e(p,\mathbb Q(E[p^n])/\mathbb Q) is precisely φ(pn)\varphi(p^n), and such that the Galois group of Q(E[pn])/Q\mathbb Q(E[p^n])/\mathbb Q is as large as possible, i.e., isomorphic to GL(2,Z/pnZ)\mathrm {GL}(2,\mathbb Z/p^n\mathbb Z).

Cite this article

Alvaro Lozano-Robledo, Division fields of elliptic curves with minimal ramification. Rev. Mat. Iberoam. 31 (2015), no. 4, pp. 1311–1332

DOI 10.4171/RMI/870