The Modularity theorem asserts a deep equivalence between two central objects of number theory: elliptic curves defined over the field of rational numbers and certain complex-analytic functions called modular forms. Informally, it says that every elliptic curve over the rationals can be obtained from, or is "parametrized" by, a modular form of weight 2. This connection links arithmetic geometry, complex analysis and the theory of L-functions.

Statement and key concepts

More precisely, the theorem relates an elliptic curve E over Q to a weight-two newform f for a congruence subgroup of SL(2,Z) whose level equals the conductor of E. The equality is usually expressed through their L-series: the Hasse–Weil L-function of E matches the L-function of f. Another geometric formulation says there exists a nonconstant algebraic map from the modular curve X_0(N) to E, called a modular parametrization.

Historical development

The idea was first proposed in the mid-20th century by mathematicians including Yutaka Taniyama and Goro Shimura and later publicized by André Weil; for many years it remained a conjecture. A breakthrough came in the 1990s when methods from Galois representations and deformation theory were used to prove modularity for large classes of elliptic curves. The work of Andrew Wiles, Richard Taylor and subsequent collaborators completed the proof for semistable curves and later efforts extended it to all elliptic curves over Q.

Importance and consequences

The theorem has several major consequences and applications:

  • It provided the crucial ingredient to deduce Fermat's Last Theorem from properties of elliptic curves.
  • It established a concrete case of the Langlands philosophy by matching automorphic forms and Galois representations.
  • It motivates modern modularity lifting theorems and advances in the study of L-functions and arithmetic of elliptic curves.

Important technical notions appearing around the theorem include Hecke eigenforms, conductors and Galois representations attached to torsion points. Generalizations ask for modularity over larger number fields or for higher-dimensional varieties, which are active research directions in the Langlands program. For background on the topics mentioned here see modularity theorem, introductions to elliptic curves and materials on the field of rational numbers Q.