Elliptic Curves and Modular Forms | The Proof of Fermat's Last Theorem

Number theorists study elliptic curves y² = x³ + ax + b over finite fields F_p (integers mod prime p) rather than rationals ℚ. This simplification makes counting rational points computationally feasible while preserving deep arithmetic structure.

The Hasse-Weil bound |ε_p| ≤ 2√p constrains error terms in elliptic curve point counts over finite fields. This fundamental inequality represents a special case of the Riemann hypothesis for elliptic curve zeta functions.

Modular forms are complex functions on the upper half-plane satisfying specific symmetries under the modular group SL(2,ℤ). These objects appear unrelated to elliptic curves yet encode identical arithmetic information through the modularity theorem.

The Taniyama-Shimura-Weil conjecture, now the modularity theorem, asserts every elliptic curve over ℚ is modular. Andrew Wiles proved it for semistable curves in 1995, completing the proof of Fermat’s Last Theorem after 358 years.