Modularity theorem

The Modularitätssatz (formerly the Taniyama - Shimura conjecture ) is a mathematical theorem on elliptic curves and modular forms. He was suspected in 1958 by Yutaka Taniyama and Shimura Gorō and proved in 2001 by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, having already Andrew Wiles had shown the most important (and most difficult ) case of semi-stable curves in 1995. The theorem and its proof are considered one of the great mathematical advances of the 20th century. One consequence of Modularitätssatzes is the Fermat's Last Theorem. Nowadays the Modularitätssatz is seen as a special case of much more general and more important Serre 's conjecture on Galois representations. This was building on the work of Andrew Wiles, 2006 by Chandrashekhar Khare, Jean -Pierre Wintenberger and Mark Kisin proved.

The statement of the Modularitätssatzes

The complex - analytic version

The group

Operates on the upper half plane by Möbius transformation. The quotient space is a non- compact Riemann surface. By adding certain points of ( the so-called spikes), you can kompaktifizieren and thus obtains a compact Riemann surface. The complex - analytic version of the conjecture states that for any elliptic curve over ( a grid ), with one and a non-constant holomorphic map of Riemann surfaces

Exists. The number N is called the (modular ) drivers of E.

An elliptic curve for which the given statement is true here is, modular. The Modularitätssatz states that all elliptic curves over are modular.

The complex - analytic version of the sentence is very weak and a priori no number-theoretic statement.

L- series version

The following version of the conjecture makes a statement about elliptic curves.

Be an elliptic curve over L- range. Then there is with a ( the leader ) and a modular form. In this case the Hecke L- series is from.

From the theory of modular forms one deduces easily that an analytic continuation and a functional equation has. This plays a major role in the well- definedness of the conjecture of Birch and Swinnerton - Dyer.

Algebraic - geometric version

From the theory of the Riemann space (or a version of the GAGA theorem ), it follows that the modular cam may be defined as a schema. It can be shown that even a scheme is. The Modularitätssatz now postulated for each elliptic curve a surjective morphism

Of algebraic curves over a N.

Theoretical representation version

Be a modular form. After deep sets of Pierre Deligne, Jean-Pierre Serre and Robert Langlands, you can f is a two-dimensional Galoisdarstellung

Assign. Here on the left is the absolute Galois group of the general linear group and to the right of the two-dimensional vector space over the field of p- adic numbers. Similarly, one can assign any elliptic curve E over such Galoisdarstellung.

The Modularitätssatz means in this case that there is for each elliptic curve E over a prime number p and a modular form of an N, and that are equivalent.

This is the version that was proved by Wiles.

Outline of the relation between Taniyama - Shimura and Fermat

Fermat's last theorem states that there is no positive integer solutions of the equation for n greater than 2. Since the French mathematician Pierre de Fermat had in 1637 claimed to have found a proof of this statement - without this, however, specify or to leave in his written records - mathematicians have been looking for a proof of this theorem. The search for a proof of Fermat's last theorem has influenced the number theory for more than two centuries and important building blocks, such as the theory of ideals were invented to prove the theorem.

The Saarbrücken mathematician Gerhard Frey presented in 1986 a presumption of a connection between Fermat's Last Theorem and the Taniyama - Shimura conjecture on: Assuming that Fermat's Last Theorem is false and there are indeed solutions of the equation, the elliptic curve is probably not modular. Ken Ribet showed in 1990 that this so-called Frey curve is not actually modular then ( he used the so-called " level -lowering " method).

In other words, if Fermat's Last Theorem is false, as well as the Taniyama - Shimura conjecture; is the Taniyama - Shimura conjecture, however, correctly, as well as Fermat's Last Theorem must be true.

Since the Frey curve is semistable, but the proof of Fermat's Last Theorem by Wiles proved from the version of the Modularitätssatzes.

Importance for mathematics

The Taniyama - Shimura theorem is an example of the unification of mathematics; including the establishment of relationships between former is understood as a completely different considered areas of mathematics, mathematicians enabled, problems that are not solvable in an area that translate into an equivalent problem in another area, where necessary, to resolve.

Sources

The following three papers contain evidence of Modularitätsatzes:

  • Andrew Wiles: Modular Elliptic Curves and Fermat 's last theorem (PDF, 10.7 MB). Annals of Mathematics 141 (1995), 443-551
  • Richard Taylor, Andrew Wiles: Ring- theoretic properties of Certain Hecke algebras. Annals of Mathematics 141 (1995), 553-572.
  • Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor: On the modularity of elliptic curves over Q: wild 3- adic exercises ( PDF, 1.1 MB), Journal of the American Mathematical Society 14 (2001), pp. 843-939. ( Contains the proof of Modularitätssatzes. )

In the following paper is Fermat's Last Theorem attributed to the Modularitätssatz:

  • Ribet, KA From the Taniyama - Shimura Conjecture to Fermat 's Last Theorem. Ann. Fac. Sci. Toulouse Math 11, 116-139, 1990a.
Richard Taylor (mathematician) Serre's modularity conjecture Jean-Pierre Wintenberger Birch and Swinnerton-Dyer conjecture Simon Singh
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