论文标题
椭圆环
Elliptic Loops
论文作者
论文摘要
给定一个本地戒指$（r，\ mathfrak {m}）$和一个椭圆曲线$ e（r/\ mathfrak {m}）$，我们将椭圆循环定义为$ \ mathbb {p}^2（r）$ $ $ e $在规范模量下的$ \ $ \ mathfrak curive and cully and cully and cully cully cully cur cur curifts cullive cucters cully cur， 添加。尽管它们满足曲线WeierStrass方程的点子集是一个组，但这些较大的对象被证明是幂关联的Abelian代数循环，这些循环很少完全关联。当椭圆循环没有订单点$ 3 $时，其仿射零件将作为一个分层的分层，该分层定义了$ r $的椭圆曲线，我们称之为层。当$ \ mathfrak {m}^e $消失的$ e \ in \ mathbb {z} $中的小值时，建立了更强的关联属性。当基础环为$ r = \ mathbb {z}/p^e \ mathbb {z} $时，椭圆环的无穷大部分由两个元素生成，可以建立层结构，并具有相同的投影和相同顺序的点具有几何描述。
Given a local ring $(R,\mathfrak{m})$ and an elliptic curve $E(R/\mathfrak{m})$, we define elliptic loops as the points of $\mathbb{P}^2(R)$ projecting to $E$ under the canonical modulo-$\mathfrak{m}$ reduction, endowed with an operation that extends the curve's addition. While their subset of points satisfying the curve's Weierstrass equation is a group, these larger objects are proved to be power associative abelian algebraic loops, which are seldom completely associative. When an elliptic loop has no points of order $3$, its affine part is obtained as a stratification of a one-parameter family of elliptic curves defined over $R$, which we call layers. Stronger associativity properties are established when $\mathfrak{m}^e$ vanishes for small values of $e \in \mathbb{Z}$. When the underlying ring is $R = \mathbb{Z}/p^e\mathbb{Z}$, the infinity part of an elliptic loop is generated by two elements, the group structure of layers may be established and the points with the same projection and same order possess a geometric description.
