Mazur's Torsion Theorem: 15 Groups, Not 12 Genus-0 Levels — E8 Intelligence Research

FINDING: Mazur's Torsion Theorem classifies the 15 possible torsion subgroups of rational elliptic curves; the modular-curve cusps that must be excluded correspond to genus-0 X₀(N) for N ∈ {1,…,10,12} plus the four "sporadic" N ∈ {11,13,14,15,16,18} — the "12 types" framing conflates the 12 genus-0 levels with the full 15-group classification. MATH: - Classification: E(ℚ)_tors ≅ ℤ/nℤ for n ∈ {1,2,3,4,5,6,7,8,9,10,12}, or ℤ/2ℤ × ℤ/2nℤ for n ∈ {1,2,3,4}. Total = 12 cyclic + 3 non-cyclic = 15 groups. - Modular curve criterion: a torsion subgroup G arises iff the corresponding modular curve X_G(ℚ) has a non-cuspidal rational point. Mazur's method: X₀(N)(ℚ) consists only of cusps for N ∈ {11,13,14,15,16,18} (the six "exceptional" levels), forcing the genus-0 levels {1,…,10,12} to carry all torsion. - Genus formula: g(X₀(N)) = 1 + N/12 · ∏(1 + 1/p) − ν_∞/2 − ν_2/4 − ν_3/3, where ν_∞ = Σ_{d|N} φ(gcd(d,N/d)), ν_2, ν_3 count elliptic points. g = 0 exactly for N ∈ {1,…,10,12,13,16,18,25}. - Cus Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com