Torsion Points on Elliptic Curves: Nagell-Lutz, Galois Actions, and Finite Constraints — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Torsion subgroups of elliptic curves over number fields are finite and highly constrained by the Nagell-Lutz theorem, with deep links to Galois representations and crystallographic-type group actions. | MATH: For an elliptic curve E: y2 = x3 + ax + b with a,b ∈ ℤ, the Nagell-Lutz theorem states: if P=(x,y) is a torsion point of order n>1, then x,y ∈ ℤ and either y=0 (order 2) or y2 | 4a3 + 27b2 (the discriminant Δ). The torsion subgroup E(ℚ)tors is finite (Mordell-Weil) and, by Mazur's theorem, is one of 15 possible groups: ℤ/mℤ for m=1,…,10,12, or ℤ/2ℤ × ℤ/2mℤ for m=1,…,4. The formal group E(pℤp) has torsion isomorphic to ℤ/pnℤ or trivial, depending on the reduction type. | CONNECTION: The finite list of torsion subgroups (orders 1–10, 12) includes the numbers 2, 3, 4,
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026torsion,
title = {{Torsion Points on Elliptic Curves: Nagell-Lutz, Galois Actions, and Finite Constraints — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635515},
url = {https://zenodo.org/records/23635515},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Torsion Points on Elliptic Curves: Nagell-Lutz, Galois Actions, and Finite Constraints — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635515
UR - https://zenodo.org/records/23635515
AB - FINDING: Torsion subgroups of elliptic curves over number fields are finite and highly constrained by the Nagell-Lutz theorem, with deep links to Galois representations and crystallographic-type group actions. | MATH: For an elliptic curve E: y^2 = x^3 + ax + b with a,b ∈ ℤ, the Nagell-Lutz theorem states: if P=(x,y) is a torsion point of order n>1, then x,y ∈ ℤ and either y=0 (order 2) or y^2 | 4a^3 + 27b^2 (the discriminant Δ). The torsion subgroup E(ℚ)_tors is finite (Mordell-Weil) and, by Mazur's theorem, is one of 15 possible groups: ℤ/mℤ for m=1,…,10,12, or ℤ/2ℤ × ℤ/2mℤ for m=1,…,4. The formal group E(pℤ_p) has torsion isomorphic to ℤ/p^nℤ or trivial, depending on the reduction type. | CONNECTION: The finite list of torsion subgroups (orders 1–10, 12) includes the numbers 2, 3, 4, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -