论文标题
椭圆形和戒指
Elliptic groups and rings
论文作者
论文摘要
众所周知，人们可以使用所谓的和弦tangent Law \ cite {dale}和一个选择的点来定义椭圆曲线点上的Abelian组。但是，这一非常有和弦的定律使我们能够在椭圆曲线的点上定义一个更加模糊的代数结构，我们称之为椭圆形。在我们的曲线具有所谓的flex点（与切线的交集号为$ 3 $）的情况下，经典的Abelian Group和Elliptic Group具有相同的信息。但是，如果我们的曲线没有这样的观点（通常以$ \ mathbb {q} $的形式发生这种点），则Abelian组不足以恢复椭圆形组。 本文的目的是更详细地研究这种代数结构，它与阿贝尔群体的联系，最后甚至介绍了椭圆环的概念（椭圆类类别中的单体对象）。
As it is well known, one can define an abelian group on the points of an elliptic curve, using the so called chord-tangent law \cite{dale}, and a chosen point. However, that very chord-tangent law allows us to define a rather more obscure algebraic structure, which we call an elliptic group, on the points of an elliptic curve. In the cases when our curve has a so called flex point (intersection number with the tangent is $3$), the classical abelian group and the elliptic group carry the same information. However, if our curve does not have such a point (which often happens over $\mathbb{Q}$), the abelian group is not enough to recover the elliptic group. The aim of this paper is to study this algebraic structure in more detail, its connections to abelian groups and at the very end even introduce the notion of an elliptic ring (a monoid object in the category of elliptic groups).
