Modularity theorem

Modularity theorem Modularity theorem Field Number theory Conjectured by Yutaka Taniyama Goro Shimura Conjectured in 1957 First proof by Christophe Breuil Brian Conrad Fred Diamond Richard Taylor First proof in 2001 Consequences Fermat's Last Theorem The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) 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. Dopo, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.

Contenuti 1 Dichiarazione 1.1 Dichiarazioni correlate 2 Storia 3 generalizzazioni 4 Esempio 5 Appunti 6 Riferimenti 7 External links Statement This section needs additional citations for verification. Aiutaci a migliorare questo articolo aggiungendo citazioni a fonti affidabili. Il materiale non fornito può essere contestato e rimosso. Trova fonti: "Modularity theorem" – news · newspapers · books · scholar · JSTOR (Marzo 2021) (Scopri come e quando rimuovere questo messaggio modello) The theorem states that any elliptic curve over {displaystyle mathbf {Q} } can be obtained via a rational map with integer coefficients from the classical modular curve {stile di visualizzazione X_{0}(N)} per qualche numero intero {stile di visualizzazione N} ; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level {stile di visualizzazione N} . Se {stile di visualizzazione 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 {stile di visualizzazione N} , a normalized newform with integer {stile di visualizzazione q} -expansion, followed if need be by an isogeny.

Related statements The modularity theorem implies a closely related analytic statement: To each elliptic curve E over {displaystyle mathbf {Q} } we may attach a corresponding L-series. Il {stile di visualizzazione L} -series is a Dirichlet series, commonly written {stile di visualizzazione L(e,S)=somma _{n=1}^{infty }{frac {un_{n}}{n^{S}}}.} The generating function of the coefficients {stile di visualizzazione a_{n}} is then {stile di visualizzazione f(e,q)=somma _{n=1}^{infty }un_{n}q^{n}.} If we make the substitution {displaystyle q=e^{2pi itau }} we see that we have written the Fourier expansion of a function {stile di visualizzazione f(e,sì )} of the complex variable {displaystyle tau } , so the coefficients of the {stile di visualizzazione q} -series are also thought of as the Fourier coefficients of {stile di visualizzazione f} . The function obtained in this way is, remarkably, a cusp form of weight two and level {stile di visualizzazione 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, a sua volta, 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 1-dimensional 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 generale, isomorphic to it).

History Yutaka Taniyama[1] stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikkō. Goro Shimura and Taniyama worked on improving its rigor until 1957. André Weil[2] rediscovered the conjecture, and showed in 1967 that it would follow from the (conjectured) functional equations for some twisted {stile di visualizzazione L} -series 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 Taniyama–Shimura–Weil conjecture became a part of the Langlands program.

The conjecture attracted considerable interest when Gerhard Frey[3] suggested in 1986 that it implies Fermat's Last Theorem. He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply the existence of at least one non-modular elliptic curve. This argument was completed in 1987 when Jean-Pierre Serre[4] identified a missing link (now known as the epsilon conjecture or Ribet's theorem) in Frey's original work, followed two years later by Ken Ribet[5]'s completion of a proof of the epsilon conjecture.

Even after gaining serious attention, the Taniyama–Shimura–Weil conjecture was seen by contemporary mathematicians as extraordinarily difficult to prove or perhaps even inaccessible to proof.[6] Per esempio, Wiles's Ph.D. supervisor John Coates states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".

In 1995 Andrew Wiles, with some help from Richard Taylor, proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves, which he used to prove Fermat's Last Theorem,[7] and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond,[8] Conrad, Diamond & Taylor; and Breuil, Conrad, Diamond & Taylor; building on Wiles's work, they incrementally chipped away at the remaining cases until the full result was proved in 1999.[9][10] Ulteriori informazioni: Fermat's Last Theorem and Wiles's proof of Fermat's Last Theorem Once fully proven, the conjecture became known as the modularity theorem.

Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem. Per esempio: no cube can be written as a sum of two coprime {stile di visualizzazione n} -th powers, {stile di visualizzazione ngeq 3} . (Il caso {stile di visualizzazione n=3} was already known by Euler.) Generalizations 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. Tuttavia, Freitas, Le Hung & Siksek[11] proved that elliptic curves defined over real quadratic fields are modular.

Example For example,[12][13][14] the elliptic curve {displaystyle y^{2}-y=x^{3}-X} , with discriminant (and conductor) 37, is associated to the form {stile di visualizzazione f(z)=q-2q^{2}-3q^{3}+2q^{4}-2q^{5}+6q^{6}+cdot ,qquad q=e^{2pi iz}} For prime numbers ℓ not equal to 37, one can verify the property about the coefficients. così, for ℓ = 3, ci sono 6 solutions of the equation modulo 3: (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1); thus a(3) = 3 − 6 = −3.

The conjecture, going back to the 1950s, was completely proven by 1999 using ideas of Andrew Wiles, chi l'ha dimostrato 1994 for a large family of elliptic curves.[15] There are several formulations of the conjecture. Showing that they are equivalent was a main challenge of number theory in the second half of the 20th century. The modularity of an elliptic curve E of conductor N can be expressed also by saying that there is a non-constant rational map defined over Q, from the modular curve X0(N) to E. In particolare, the points of E can be parametrized by modular functions.

Per esempio, a modular parametrization of the curve {displaystyle y^{2}-y=x^{3}-X} è dato da[16] {stile di visualizzazione {inizio{allineato}X(z)&=q^{-2}+2q^{-1}+5+9q+18q^{2}+29q^{3}+51q^{4}+cdots \y(z)&=q^{-3}+3q^{-2}+9q^{-1}+21+46q+92q^{2}+180q^{3}+cdots end{allineato}}} dove, come sopra, q = exp(2πiz). The functions x(z) and y(z) are modular of weight 0 and level 37; in other words they are meromorphic, defined on the upper half-plane Im(z) > 0 and satisfy {stile di visualizzazione x!sinistra({frac {az+b}{cz+d}}Giusto)=x(z)} and likewise for y(z), for all integers a, b, c, d with ad − bc = 1 e 37|c.

Another formulation depends on the comparison of Galois representations attached on the one hand to elliptic curves, and on the other hand to modular forms. The latter formulation has been used in the proof of the conjecture. Dealing with the level of the forms (and the connection to the conductor of the curve) is particularly delicate.

The most spectacular application of the conjecture is the proof of Fermat's Last Theorem (FLT). Suppose that for a prime p ≥ 5, the Fermat equation {stile di visualizzazione a^{p}+b^{p}=c^{p}} has a solution with non-zero integers, hence a counter-example to FLT. Then as Yves Hellegouarch was the first to notice,[17] the elliptic curve {displaystyle y^{2}=x(x-a^{p})(x+b^{p})} of discriminant {displaystyle Delta ={frac {1}{256}}(abc)^{2p}} cannot be modular.[5] così, the proof of the Taniyama–Shimura–Weil conjecture for this family of elliptic curves (called Hellegouarch–Frey curves) implies FLT. The proof of the link between these two statements, based on an idea of Gerhard Frey (1985), is difficult and technical. It was established by Kenneth Ribet in 1987.[18] Notes ^ Taniyama 1956. ^ Weil 1967. ^ Frey 1986. ^ Serra 1987. ^ Salta su: a b Ribet 1990. ^ Singh 1997, pp. 203–205, 223, 226. ^ Wiles 1995a; Wiles 1995b. ^ Diamond 1996. ^ Conrad, Diamond & Taylor 1999. ^ Breuil et al. 2001. ^ Freitas, Le Hung & Siksek 2015. ^ For the calculations, see for example Zagier 1985, pp. 225–248 ^ LMFDB: ^ OEIS: ^ A synthetic presentation (in francese) of the main ideas can be found in this Bourbaki article of Jean-Pierre Serre. For more details see Hellegouarch 2001 ^ Zagier, D. (1985). "Modular points, modular curves, modular surfaces and modular forms". Arbeitstagung Bonn 1984. Appunti delle lezioni in matematica. vol. 1111. Springer. pp. 225–248. doi:10.1007/BFb0084592. ISBN 978-3-540-39298-9. ^ Hellegouarch, Yves (1974). "Points d'ordre 2ph sur les courbes elliptiques" (PDF). Diario di aritmetica. 26 (3): 253–263. doi:10.4064/aa-26-3-253-263. ISSN 0065-1036. SIG 0379507. ^ See the survey of Ribet, K. (1990b). "From the Taniyama–Shimura conjecture to Fermat's Last Theorem". Annales de la Faculté des Sciences de Toulouse. 11: 116–139. doi:10.5802/afst.698. References Breuil, Cristofe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001), "On the modularity of elliptic curves over Q: wild 3-adic exercises", Giornale della Società matematica americana, 14 (4): 843–939, doi:10.1090/S0894-0347-01-00370-8, ISSN 0894-0347, SIG 1839918 Conrad, Brian; Diamond, Fred; Taylor, Richard (1999), "Modularity of certain potentially Barsotti–Tate Galois representations", Giornale della Società matematica americana, 12 (2): 521–567, doi:10.1090/S0894-0347-99-00287-8, ISSN 0894-0347, SIG 1639612 Cornell, Gary; Argentiere, Joseph H.; Stevens, Glenn, eds. (1997), Modular forms and Fermat's last theorem, Berlino, New York: Springer-Verlag, ISBN 978-0-387-94609-2, SIG 1638473 Darmon, Henri (1999), "A proof of the full Shimura–Taniyama–Weil conjecture is announced" (PDF), Avvisi dell'American Mathematical Society, 46 (11): 1397–1401, ISSN 0002-9920, MR 1723249Contains a gentle introduction to the theorem and an outline of the proof. Diamond, Fred (1996), "On deformation rings and Hecke rings", Annali di matematica, Seconda serie, 144 (1): 137–166, doi:10.2307/2118586, ISSN 0003-486X, JSTOR 2118586, SIG 1405946 Freitas, Nuno; Le Hung, Bao V.; Siksek, Samir (2015), "Elliptic curves over real quadratic fields are modular", Scoperte matematiche, 201 (1): 159–206, arXiv:1310.7088, Bibcode:2015InMat.201..159F, doi:10.1007/s00222-014-0550-z, ISSN 0020-9910, SIG 3359051, S2CID 119132800 Frey, Gerhard (1986), "Links between stable elliptic curves and certain Diophantine equations", Annales Universitatis Saraviensis. Series Mathematicae, 1 (1): iv+40, ISSN 0933-8268, SIG 0853387 Mazur, Barry (1991), "Number theory as gadfly", Il mensile matematico americano, 98 (7): 593–610, doi:10.2307/2324924, ISSN 0002-9890, JSTOR 2324924, SIG 1121312 Discusses the Taniyama–Shimura–Weil conjecture 3 years before it was proven for infinitely many cases. Complicato, Kenneth A. (1990), "On modular representations of Gal(Q/Q) arising from modular forms", Scoperte matematiche, 100 (2): 431–476, Bibcode:1990InMat.100..431R, doi:10.1007/BF01231195, hdl:10338.dmlcz/147454, ISSN 0020-9910, SIG 1047143, S2CID 120614740 Stretto, JeanPierre (1987), "Sur les représentations modulaires de degré 2 de Gal(Q/Q)", Diario di matematica del duca, 54 (1): 179–230, doi:10.1215/S0012-7094-87-05413-5, ISSN 0012-7094, SIG 0885783 Shimura, Goro (1989), "Yutaka Taniyama and his time. Very personal recollections", Il Bollettino della London Mathematical Society, 21 (2): 186–196, doi:10.1112/blms/21.2.186, ISSN 0024-6093, SIG 0976064 Singh, Simone (1997), L'ultimo teorema di Fermat, ISBN 978-1-85702-521-7 Taniyama, Yutaka (1956), "Problema 12", Sugaku (in Japanese), 7: 269 English translation in (Shimura 1989, p. 194) Taylor, Richard; Wiles, Andrea (1995), "Ring-theoretic properties of certain Hecke algebras", Annali di matematica, Seconda serie, 141 (3): 553–572, CiteSeerX, doi:10.2307/2118560, ISSN 0003-486X, JSTOR 2118560, SIG 1333036 Weil, André (1967), "Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen", Annali matematici, 168: 149–156, doi:10.1007/BF01361551, ISSN 0025-5831, SIG 0207658, S2CID 120553723 Wiles, Andrea (1995un), "Modular elliptic curves and Fermat's last theorem", Annali di matematica, Seconda serie, 141 (3): 443–551, CiteSeerX, doi:10.2307/2118559, ISSN 0003-486X, JSTOR 2118559, SIG 1333035 Wiles, Andrea (1995b), "Modular forms, elliptic curves, and Fermat's last theorem", Atti del Congresso Internazionale dei Matematici, vol. 1, 2 (Zurigo, 1994), Basilea, Boston, Berlino: Birkhauser, pp. 243–245, SIG 1403925 External links Darmon, H. (2001) [1994], "Shimura–Taniyama conjecture", Enciclopedia della matematica, EMS Press Weisstein, Eric W. "Taniyama–Shimura Conjecture". 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