椭圆圈
Elliptic Loops
AI总结：
本文定义了局部环上的椭圆圈，证明其为幂结合交换代数圈，并分析了其在特定环和条件下的分层结构、结合性及群结构。
摘要：
给定局部环 $(R,\mathfrak{m})$ 和椭圆曲线 $E(R/\mathfrak{m})$，我们将椭圆圈定义为 $\mathbb{P}^2(R)$ 中在典范模 $\mathfrak{m}$ 约化下投射到 $E$ 的点，并赋予其一种延伸该曲线加法运算的操作。尽管满足该曲线 Weierstrass 方程的点子集构成一个群，但证明表明这些更大的对象是幂结合交换代数圈，且很少具有完全结合性。当椭圆圈没有阶为 $3$ 的点时，其仿射部分可由定义在 $R$ 上的单参数椭圆曲线族的分层获得，我们将其称为层。当 $\mathfrak{m}^e$ 对小的 $e \in \mathbb{Z}$ 值消失时，确立了更强的结合性性质。当底层环为 $R = \mathbb{Z}/p^e\mathbb{Z}$ 时，椭圆圈的无穷远部分由两个元素生成，层的群结构得以确立，且具有相同投射和相同阶的点具有几何描述。
英文摘要：
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.
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