Division fields of elliptic curves with minimal ramification
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Alvaro Lozano-Robledo [1]
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[1] University of Connecticut
University of Connecticut
Town of Mansfield, Estados Unidos
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- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 31, Nº 4, 2015, págs. 1311-1332
- Idioma: inglés
- DOI: 10.4171/RMI/870
- Texto completo no disponible (Saber más ...)
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Resumen
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).
