Taniyama-Shimura theorem

For any natural numberMathworldPlanetmath N1, define the modular group Γ0(N) to be the following subgroup of the group SL(2,) of integer coefficient matrices of determinantMathworldPlanetmath 1:


Let * be the subset of the Riemann sphere consisting of all points in the upper half plane (i.e., complex numbersMathworldPlanetmathPlanetmath with strictly positive imaginary partMathworldPlanetmath), together with the rational numbers and the point at infinity. Then Γ0(N) acts on *, with group action given by the operationMathworldPlanetmath


Define X0(N) to be the quotient of * by the action of Γ0(N). The quotient space X0(N) inherits a quotient topology and holomorphic structure from making it into a compact Riemann surface. (Note: * itself is not a Riemann surface; only the quotient X0(N) is.) By a general theorem in complex algebraic geometryMathworldPlanetmathPlanetmath, every compact Riemann surface admits a unique realization as a complex nonsingularPlanetmathPlanetmath projective curve; in particular, X0(N) has such a realization, which by abuse of notation we will also denote X0(N). This curve is defined over , although the proof of this fact is beyond the scope of this entry11Explicitly, the curve X0(N) is the unique nonsingular projective curve which has function fieldMathworldPlanetmath equal to (j(z),j(Nz)), where j denotes the elliptic modular jfunction. The curve X0(N) is essentially the algebraic curve defined by the polynomial equation ΦN(X,Y)=0 where ΦN is the modular polynomialPlanetmathPlanetmath, with the caveat that this procedure yields singularities which must be resolved manually. The fact that ΦN has integer coefficients provides one proof that X0(N) is defined over ..

Taniyama-Shimura Theorem (weak form): For any elliptic curveMathworldPlanetmath E defined over , there exists a positive integer N and a surjectivePlanetmathPlanetmath algebraic morphism ϕ:X0(N)E defined over .

This theorem was first conjectured (in a much more precise, but equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath formulation) by Taniyama, Shimura, and Weil in the 1970’s. It attracted considerable interest in the 1980’s when Frey [2] proposed that the Taniyama-Shimura conjecture implies Fermat’s Last Theorem. In 1995, Andrew Wiles [3] proved a special case of the Taniyama-Shimura theorem which was strong enough to yield a proof of Fermat’s Last Theorem. The full Taniyama-Shimura theorem was finally proved in 1997 by a team of a half-dozen mathematicians who, building on Wiles’s work, incrementally chipped away at the remaining cases until the full result was proved. As of this writing, the proof of the full theorem can still be found on http://abel.math.harvard.edu/ rtaylor/Richard Taylor’s preprints page.


  • 1 Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard; On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843–939
  • 2 Frey, G. Links between stable elliptic curves and certain Diophantine equationsMathworldPlanetmath. Ann. Univ. Sarav. 1 (1986), 1–40.
  • 3 Wiles, A. Modular elliptic curves and Fermat’s Last Theorem. Annals of Math. 141 (1995), 443–551.
Title Taniyama-Shimura theorem
Canonical name TaniyamaShimuraTheorem
Date of creation 2013-03-22 12:16:27
Last modified on 2013-03-22 12:16:27
Owner djao (24)
Last modified by djao (24)
Numerical id 14
Author djao (24)
Entry type Theorem
Classification msc 11F06
Classification msc 14H52
Synonym Taniyama-Shimura-Weil conjecture
Synonym Taniyama-Weil conjecture
Synonym Taniyama-Shimura conjecture
Synonym Taniyama-Shimura-Weil theorem
Related topic FermatsLastTheorem
Related topic ModularForms