The arithmetic of elliptic curves
The arithmetic of elliptic curves
复制标题
DOI：
10.1007/bf01389745
复制
发表时间：
1974-09
影响因子：
3.1
通讯作者：
J. Tate
中科院分区：
文献类型：
--
作者：
J. Tate
亏格 0 曲线（例如，平面中的直线和二次曲线）之后是亏格 1 曲线，或“椭圆”曲线（例如，平面三次曲线或三空间中二次曲面的交点）。椭圆曲线是阿贝尔簇的第一个例子。它们的有限阶点给出了 6tale 上同调群的第一个重要例子。伽罗瓦群对这些的作用既导致了复乘法的经典理论，也导致了非阿贝尔扩张系统，其中可能包含非阿贝尔类域论的线索。椭圆曲线与模形式理论以多种方式密切相关。在前面的部分中，我尝试简要介绍该主题的基础知识，并尽可能使用明确的公式来绕过一般理论的部分。后面的部分是对最近工作的概述，重点是三个主要主题：（1）有理点问题，Shafarevitch群以及Birch和Swinnerton-Dyer的猜想。（2）模曲线和Weil的令人震惊的想法，即有理域上的每条椭圆曲线都是“模”。（3）塞尔定理，即从椭圆曲线上的有限阶点获得的伽罗瓦群“与”一样大尽可能”。我希望能够在这里传达这些进步的一些想法，并通过上一节中讨论的数字示例来说明它们。 w 2. Weierstrass 模型 在这些讲座中，我们将使用术语“椭圆曲线”来表示 1 维的阿贝尔簇，或者相同的是，具有 0 点的属 1 的不可约非奇异投影代数曲线，它是群律的原点。任何这样的曲线 E，在域 K 上定义，都具有以下形式的平面三次模型
After curves of genus 0 (eg lines and conics in the plane) come curves of genus 1, or" elliptic" curves (eg plane cubics or intersections of quadric surfaces in three-space). Elliptic curves are the first examples of abelian varieties. Their points of finite order give the first non-trivial examples of 6tale cohomology groups. The action of Galois groups on these leads both to the classical theory of complex multiplication as well as to systems of non-abelian extensions which may contain clues to non-abelian class field theory. Elliptic curves are intimately connected with the theory of modular forms, in more ways than one. In the early sections I have tried to give a brief introduction to the fundamentals of the subject, using explicit formulas to by-pass chunks of general theory when possible. The later sections are a survey of recent work with emphasis on three main topics:(1) The problem of rational points, the Shafarevitch group, and the conjecture of Birch and Swinnerton-Dyer.(2) Modular curves and Weil's astounding idea that every elliptic curve over the rational field is"'modular".(3) Serre's theorem that the Galois groups obtained from points of finite order on elliptic curves are" as big as possible". I hope to be able to convey some idea of these advances here, illustrating them by numerical examples discussed in the last section. w 2. Weierstrass ModelsIn these lectures we will use the term elliptic curee to mean an abelian variety of dimension 1, or, what is the same, an irreducible non-singular projective algebraic curve of genus 1 furnished with a point 0, the origin for the group law. Any such curve E, defined over a field K, has a plane cubic model of the form