Mazur Torsion Groups and Crystallographic Classification Correspondence — E8 Intelligence Research

FINDING: Mazur's torsion theorem classifies the 15 possible torsion subgroups of rational elliptic curves; the count 15 and the group structures exhibit correspondence with crystallographic point group classification (32 crystallographic point groups, 14 Bravais lattices). MATH: - Mazur's theorem: For E/Q, 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} - 15 torsion groups total: 11 cyclic + 4 non-cyclic - Cyclic orders: {1,2,3,4,5,6,7,8,9,10,12} — notably excludes 11, 13, 14, 15, 16... - Non-cyclic: Z/2×Z/2, Z/2×Z/4, Z/2×Z/6, Z/2×Z/8 - Modular curve connection: X₁(N) has rational points iff N ≤ 10 or N = 12 (Mazur); X₁(11), X₁(13)... have no non-cuspidal rational points - Ogg's conjecture (now theorem): primes p such that X₀⁺(p) has genus 0 are p ∈ {2,3,5,7,11,13,17,19,23,29,31,41,47,59,71} — 15 primes - The 15 primes in Ogg's list connect to the Monster group's supersingular primes CONNECTION: - 15 torsion groups ↔ 15 supersingular primes ↔ 15 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com