Direct Group Laws on Classical Genus-One Models
Abstract
This exposition constructs group laws by residual intersection on short Weierstrass cubics, the curves (u2+u)(v2+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 PP3. 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 (u2+u)(v2+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 $3$.
Related
- Galois actions on surfaces and a higher genus Grothendieck-Teichmüller group
- The first chiral homology group in higher genus
- Failure of direct finiteness for group algebras in every odd characteristic
- Unique direct-sum decompositions of infinite-dimensional unitary representations of the infinite dihedral group via real CMV matrices
- Algebra objects in direct limit completions of compact Lie group duals and the classification of c=1 vertex operator algebras
BibTeX
@misc{wu2026direct,
title = {{Direct Group Laws on Classical Genus-One Models}},
author = {Hongfeng Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00029},
eprint = {2610.00029},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00029},
}
RIS
TY - UNPB
TI - Direct Group Laws on Classical Genus-One Models
AU - Hongfeng Wu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00029
UR - https://arxiv.org/abs/2610.00029
AB - 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 $3$.
ER -