Modularity Theorem: CM Elliptic Curves and Weight-2 Modular Forms — E8 Intelligence Research

FINDING: The Modularity Theorem (Taniyama-Shimura-Weil) establishes a bijection between elliptic curves over ℚ and weight-2 modular forms, with complex multiplication (CM) curves forming a distinguished subset linked to roots of unity and class field theory. | MATH: For an elliptic curve E/ℚ with conductor N, there exists a modular form f of weight 2 and level N such that L(E,s) = L(f,s). The CM case: End(E) ⊗ ℚ is an imaginary quadratic field K = ℚ(√-d); the associated modular form has complex multiplication by the ring of integers of K, and its Fourier coefficients a_p satisfy a_p = 0 for inert primes p, a_p = 2·Re(χ(p))·√p for split primes (χ is the Hecke character). The j-invariant of a CM curve is an algebraic integer; the singular moduli j(τ) for τ in an imaginary quadratic field generate the Hilbert class field. Roots of unity appear via the Weber function and the torsion structure: E[n] ≅ (ℤ/nℤ)², and the Galois action on torsion points factors through cyclotomic characters. | Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com