The bridge between number theory and complex analysis

Integrating modular forms over arcs in the complex unit disk produces values forming lattices in the complex plane. The Weierstrass ℘-function provides the higher-dimensional analog of sine/cosine for lattices. Mathematicians construct elliptic curves from these lattices using doubly-periodic functions.

Srinivasa Ramanujan discovered in 1916 a strange function Δ(q) by expanding q∏(1-qⁿ)²⁴ as a power series. Mathematicians Martin Eichler and Goro Shimura in the 1950s studied modular forms’ coefficients to uncover their deep meaning. Ramanujan noticed the coefficients satisfied multiplicative properties without proving why.

The 1967 Taniyama-Shimura conjecture proposed every elliptic curve over ℚ is modular. Eichler and Shimura showed how to attach elliptic curves to modular forms. The reverse direction—every curve comes from some modular form—seemed impossible. John Coates called it “beautiful though impossible to prove.”

Andrew Wiles worked in secret for seven years to prove enough of Taniyama-Shimura to imply Fermat’s Last Theorem. In September 1994, Wiles announced success proving all semi-stable elliptic curves over ℚ are modular. Breuil, Conrad, Diamond, and Taylor completed the full conjecture in 2001.