经典亏格一模型上的直接群律
Direct Group Laws on Classical Genus-One Models
- College of Science, North China University of Technology（华北理工大学科学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在多种经典亏格一曲线模型上通过剩余交点构造直接群律，并利用亏格一曲线无非平凡单极点有理函数证明结合性，且覆盖所有特征。
AI中文摘要：
本文通过剩余交点在短Weierstrass三次曲线、曲线$(u^2+u)(v^2+v)=d$、Edwards曲线、Hessian曲线、Jacobi四次曲线、Jacobi交曲线、Huff曲线以及对称QRT双二次曲线上构造群律。直线给出平面三次曲线上的运算，而平面给出$\PP^3$中两个二次曲面光滑交线上的运算。结合性通过比较三个直线或平面方程乘积来证明。在消去公共零点后，剩余零点的相等性由以下基本事实得出：亏格一的光滑射影曲线不允许具有单个简单极点的非常值有理函数。$(u^2+u)(v^2+v)=d$的公式在所有特征下均有效，并在特征2和3中显式写出；其他模型在不同于2和3的特征下处理。
英文摘要：
This exposition constructs group laws by residual intersection on short Weierstrass cubics, the curves $(u^2+u)(v^2+v)=d$, Edwards curves, Hessian curves, Jacobi quartics curves, Jacobi intersections curves, Huff curves, and a symmetric QRT biquadratic. Lines give the operation on plane cubics, while planes give the operation on smooth intersections of two quadrics in $\PP^3$. Associativity is proved by comparing two products of three line or plane equations. After the common zeros are cancelled, equality of the remaining zeros follows from the elementary fact that a smooth projective curve of genus one admits no nonconstant rational function with one simple pole. The formulas for $(u^2+u)(v^2+v)=d$ are valid in every characteristic and are written explicitly in characteristics $2$ and $3$; the other models are treated in characteristic different from $2$ and $3$.
