Resumen de Division fields of elliptic curves with minimal ramification

Álvaro Lozano Robledo

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