Let be the subset of the Riemann sphere consisting of all points in the upper half plane (i.e., complex numbers with strictly positive imaginary part), together with the rational numbers and the point at infinity. Then acts on , with group action given by the operation
Define to be the quotient of by the action of . The quotient space 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 is.) By a general theorem in complex algebraic geometry, every compact Riemann surface admits a unique realization as a complex nonsingular projective curve; in particular, has such a realization, which by abuse of notation we will also denote . This curve is defined over , although the proof of this fact is beyond the scope of this entry11Explicitly, the curve is the unique nonsingular projective curve which has function field equal to , where denotes the elliptic modular –function. The curve is essentially the algebraic curve defined by the polynomial equation where is the modular polynomial, with the caveat that this procedure yields singularities which must be resolved manually. The fact that has integer coefficients provides one proof that is defined over ..
This theorem was first conjectured (in a much more precise, but equivalent formulation) by Taniyama, Shimura, and Weil in the 1970’s. It attracted considerable interest in the 1980’s when Frey  proposed that the Taniyama-Shimura conjecture implies Fermat’s Last Theorem. In 1995, Andrew Wiles  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 : 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 equations. 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.
|Date of creation||2013-03-22 12:16:27|
|Last modified on||2013-03-22 12:16:27|
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