Modularity theorem

The theorem that every elliptic curve over the rational numbers is associated with a weight-2 modular form of level equal to its conductor, linking arithmetic geometry with complex analysis.

Elliptic curve over the rational numbers: A nonsingular cubic curve defined over the rational numbers, equipped with a specified point and an algebraic group law. These are the curves whose arithmetic the theorem matches to modular forms.
Taniyama–Shimura conjecture: The conjecture that elliptic curves over the rational numbers are parametrized by modular forms. This conjecture was the historical form of the result now called the modularity theorem.
Elliptic curve: A smooth projective curve of genus one together with a chosen point, carrying a natural algebraic group structure. Rational elliptic curves are a special case of this broader geometric object.
Fermat's Last Theorem: The theorem that no positive integers satisfy xⁿ + yⁿ = zⁿ for any integer exponent n greater than 2. Semistable modularity, combined with Ribet's theorem, yields its proof.
Modular form: A complex analytic function on the upper half-plane that transforms predictably under a subgroup of integer matrices and satisfies additional regularity conditions. The theorem assigns a weight-2 form to each rational elliptic curve.
Gerhard Frey: A German mathematician known for proposing the elliptic curve now called the Frey curve in connection with Fermat's Last Theorem. His curve suggested how a Fermat counterexample would conflict with modularity.
Modular curve: An algebraic curve whose points classify elliptic curves equipped with specified level structure. Its geometry provides a setting for relating elliptic curves to modular forms.
Modularity lifting theorem: A theorem that promotes modularity of a residual Galois representation to modularity of a related characteristic-zero representation under hypotheses. Lifting methods formed the technical core of Wiles's semistable modularity proof.
Conductor (elliptic curve): A positive integer measuring the primes at which an elliptic curve has bad reduction and the severity of that reduction. It determines the level of the modular form associated with the curve.
Jean-Pierre Serre: A French mathematician whose work includes conjectures linking Galois representations, modular forms, and elliptic curves. His epsilon conjecture sharpened the proposed route from Fermat's equation to nonmodularity.

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