Mazur's Torsion Theorem as Cusp-Counting on Modular Curves X₁(N) — E8 Intelligence Research

FINDING: Mazur's torsion theorem classifies rational torsion on elliptic curves over ℚ into exactly 15 groups, with the 12 cyclic types corresponding to cusps of modular curves X₁(N) — the classification is a cusp-counting problem on modular curves. | MATH: Torsion subgroups E(ℚ)_tors ∈ {ℤ/nℤ : n = 1–10, 12} ∪ {ℤ/2ℤ × ℤ/2nℤ : n = 1–4}. Modular curve X₁(N) has genus 0 for N ∈ {1,...,10, 12} and its cusps correspond bijectively to the torsion types. The 12 cyclic orders: n = 1,2,3,4,5,6,7,8,9,10,12. Note 11 is excluded — X₁(11) has genus 1, not 0. | CONNECTION: The excluded values n = 11 and n ≥ 13 correspond to X₁(N) having positive genus. The critical threshold: genus(X₁(N)) = 0 iff N ≤ 10 or N = 12. The gap at 11 and the resumption at 12 is structurally significant — 12 = 2²·3 is the last highly composite number where φ(12) = 4 and the unit group (ℤ/12ℤ)* ≅ ℤ/2 × ℤ/2 has exponent 2. The 12 cusps of X(12) relate to the 12 vertices of the icosahedron via the icosahedral group A₅ ≅ PSL(2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com