Division fields of elliptic curves with minimal ramification
Division fields of elliptic curves with minimal ramification
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具有最小分枝的椭圆曲线的除域
DOI：
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发表时间：
2015
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通讯作者：
Álvaro Lozano
中科院分区：
文献类型：
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作者：
Álvaro Lozano
设E是定义在Q上的椭圆曲线，p是素数，n≥1.众所周知，椭圆曲线E的pnpn次除法域Q（E[pn]）包含所有pn次单位根。由此得出伽罗瓦扩张Q（E[pn]）/Q在p上分歧，并且Q（E[pn]）的任何素数P位于pp之上的分歧指数e（p，Q（E [pn]）/Q）可被φ（pn）整除。本文的目标是构造椭圆曲线E/Q，使得e（p，Q（E[pn]）/Q）恰好是φ（pn），并且使得Q（E[pn]）/Q的Galois群尽可能大，即，同构于GL（2，Z/pnZ）。
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).