# Théorème de modularité

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. Plus tard, 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.

Contenu 1 Déclaration 1.1 Déclarations connexes 2 Histoire 3 Généralisations 4 Exemple 5 Remarques 6 Références 7 External links Statement This section needs additional citations for verification. Aidez-nous à améliorer cet article en ajoutant des citations à des sources fiables. Le matériel non sourcé peut être contesté et supprimé. Trouver des sources: "Théorème de modularité" – actualités · journaux · livres · universitaires · JSTOR (Mars 2021) (Découvrez comment et quand supprimer ce modèle de message) The theorem states that any elliptic curve over {style d'affichage mathbf {Q} } can be obtained via a rational map with integer coefficients from the classical modular curve {style d'affichage X_{0}(N)} for some integer {displaystyle N} ; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level {displaystyle N} . Si {displaystyle 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 {displaystyle N} , a normalized newform with integer {style d'affichage 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 {style d'affichage mathbf {Q} } we may attach a corresponding L-series. La {displaystyle L} -series is a Dirichlet series, commonly written {displaystyle L(E,s)=somme _{n=1}^{infime }{frac {un_{n}}{n^{s}}}.} The generating function of the coefficients {style d'affichage a_{n}} is then {style d'affichage f(E,q)=somme _{n=1}^{infime }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 {style d'affichage f(E,oui )} of the complex variable {style d'affichage tau } , so the coefficients of the {style d'affichage q} -series are also thought of as the Fourier coefficients of {style d'affichage f} . The function obtained in this way is, remarkably, a cusp form of weight two and level {displaystyle 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, à son tour, 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, en général, 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 {displaystyle 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] Par exemple, 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".

Dans 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] Plus d'informations: 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. Par exemple: no cube can be written as a sum of two coprime {displaystyle n} -th powers, {style d'affichage ngeq 3} . (L'affaire {displaystyle 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. Cependant, 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 {style d'affichage f(z)=q-2q^{2}-3q ^{3}+2q ^{4}-2q ^{5}+6q ^{6}+cdots ,qquad q=e^{2pi de}} For prime numbers ℓ not equal to 37, one can verify the property about the coefficients. Ainsi, for ℓ = 3, il y a 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, qui l'a prouvé dans 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. En particulier, the points of E can be parametrized by modular functions.

Par exemple, a modular parametrization of the curve {displaystyle y^{2}-y=x^{3}-X} est donné par[16] {style d'affichage {commencer{aligné}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{aligné}}} où, comme ci-dessus, 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 {style d'affichage x!la gauche({frac {az+b}{cz+d}}droit)=x(z)} and likewise for y(z), for all integers a, b, c, d with ad − bc = 1 et 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.