Torsion Points on Elliptic Curves: Nagell-Lutz, Galois Actions, and Finite Constraints — E8 Intelligence Research

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