 Modularity theorem

In mathematics the modularity theorem (formerly called the Taniyama–Shimura–Weil conjecture and several related names) states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's last theorem, and Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor extended his techniques to prove the full modularity theorem in 2001. The modularity theorem is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field. Most cases of these extended conjectures have not yet been proved.
Contents
Statement
The theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve
for some integer N; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level N. If N is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the conductor), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level N, a normalized newform with integer qexpansion, followed if need be by an isogeny.
The modularity theorem implies a closely related analytic statement: to an elliptic curve E over Q we may attach a corresponding Lseries. The Lseries is a Dirichlet series, commonly written
The generating function of the coefficients a_{n} is then
If we make the substitution
we see that we have written the Fourier expansion of a function f(τ,E) of the complex variable τ, so the coefficients of the qseries are also thought of as the Fourier coefficients of f. The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform (an eigenvector of all Hecke operators); this is the Hasse–Weil conjecture, which follows from the modularity theorem.
Some modular forms of weight two, in turn, correspond to holomorphic differentials for an elliptic curve. The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible Abelian varieties, corresponding to Hecke eigenforms of weight 2. The 1dimensional factors are elliptic curves (there can also be higher dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve (but not, in general, isomorphic to it).
History
Taniyama (1956) stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikko. Goro Shimura and Taniyama worked on improving its rigor until 1957. Weil (1967) rediscovered the conjecture, and showed that it would follow from the (conjectured) functional equations for some twisted Lseries of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form.
The conjecture attracted considerable interest when Frey (1986) suggested that the Taniyama–Shimura–Weil conjecture implies Fermat's Last Theorem. He did this by attempting to show that any counterexample to Fermat's Last Theorem would give rise to a nonmodular elliptic curve. However, his argument was not complete. The extra condition which was needed to link TaniyamaShimuraWeil to Fermat's Last Theorem was identified by Serre (1987) and became known as the epsilon conjecture. In the summer of 1986, Ribet (1990) proved the epsilon conjecture, thereby proving that the Taniyama–Shimura–Weil conjecture implied Fermat's Last Theorem. Wiles (1995), with some help from Richard Taylor, proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves, which was strong enough to yield a proof of Fermat's Last Theorem.
The full Taniyama–Shimura–Weil conjecture was finally proved by Diamond (1996), Conrad, Diamond & Taylor (1999), and Breuil et al. (2001) who, building on Wiles' work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the modularity theorem.
Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem. For example: no cube can be written as a sum of two coprime nth powers, n ≥ 3. (The case n = 3 was already known by Euler.)
References
 Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001), "On the modularity of elliptic curves over Q: wild 3adic exercises", Journal of the American Mathematical Society 14 (4): 843–939, doi:10.1090/S0894034701003708, ISSN 08940347, MR1839918
 Conrad, Brian; Diamond, Fred; Taylor, Richard (1999), "Modularity of certain potentially BarsottiTate Galois representations", Journal of the American Mathematical Society 12 (2): 521–567, doi:10.1090/S0894034799002878, ISSN 08940347, MR1639612
 Cornell, Gary; Silverman, Joseph H.; Stevens, Glenn, eds. (1997), Modular forms and Fermat's last theorem, Berlin, New York: SpringerVerlag, ISBN 9780387946092; 9780387989983, MR1638473, http://books.google.com/books?id=VaquzVwtMsC
 Darmon, Henri (1999), "A proof of the full ShimuraTaniyamaWeil conjecture is announced", Notices of the American Mathematical Society 46 (11): 1397–1401, ISSN 00029920, MR1723249, http://www.ams.org/notices/199911/commdarmon.pdfContains a gentle introduction to the theorem and an outline of the proof.
 Diamond, Fred (1996), "On deformation rings and Hecke rings", Annals of Mathematics. Second Series 144 (1): 137–166, doi:10.2307/2118586, ISSN 0003486X, MR1405946
 Frey, Gerhard (1986), "Links between stable elliptic curves and certain Diophantine equations", Annales Universitatis Saraviensis. Series Mathematicae 1 (1): iv+40, ISSN 09338268, MR853387
 Mazur, Barry (1991), "Number theory as gadfly", The American Mathematical Monthly 98 (7): 593–610, doi:10.2307/2324924, ISSN 00029890, MR1121312 Discusses the TaniyamaShimuraWeil conjecture 3 years before it was proven for infinitely many cases.
 Ribet, Kenneth A. (1990), "On modular representations of Gal(Q/Q) arising from modular forms", Inventiones Mathematicae 100 (2): 431–476, doi:10.1007/BF01231195, ISSN 00209910, MR1047143
 Serre, JeanPierre (1987), "Sur les représentations modulaires de degré 2 de Gal(Q/Q)", Duke Mathematical Journal 54 (1): 179–230, doi:10.1215/S0012709487054135, ISSN 00127094, MR885783
 Shimura, Goro (1989), "Yutaka Taniyama and his time. Very personal recollections", The Bulletin of the London Mathematical Society 21 (2): 186–196, doi:10.1112/blms/21.2.186, ISSN 00246093, MR976064
 Taniyama, Yutaka (1956), "Problem 12" (in Japanese), Sugaku 7: 269 English translation in (Shimura 1989, p. 194)
 Taylor, Richard; Wiles, Andrew (1995), "Ringtheoretic properties of certain Hecke algebras", Annals of Mathematics. Second Series 141 (3): 553–572, doi:10.2307/2118560, ISSN 0003486X, MR1333036
 Weil, André (1967), "Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen", Mathematische Annalen 168: 149–156, doi:10.1007/BF01361551, ISSN 00255831, MR0207658
 Wiles, Andrew (1995), "Modular elliptic curves and Fermat's last theorem", Annals of Mathematics. Second Series 141 (3): 443–551, ISSN 0003486X, JSTOR 2118559, MR1333035
 Wiles, Andrew (1995), "Modular forms, elliptic curves, and Fermat's last theorem", Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Basel, Boston, Berlin: Birkhäuser, pp. 243–245, MR1403925
External links
 Darmon, H. (2001), "Shimura–Taniyama conjecture", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/S/s120140.htm
 Weisstein, Eric W., "TaniyamaShimura Conjecture" from MathWorld.
Categories: Algebraic curves
 Riemann surfaces
 Modular forms
 Theorems in number theory
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