Modularity set

The modularity theorem (formerly the Taniyama-Shimura conjecture ) is a mathematical theorem about elliptic curves and modular shapes . It was suspected by Yutaka Taniyama and Gorō Shimura in 1958 and proven by Christophe Breuil , Brian Conrad , Fred Diamond and Richard Taylor in 2001 , after Andrew Wiles had already shown the most important (and most difficult) case of semi-stable curves in 1995. The theorem and its proof are considered to be one of the great mathematical advances of the 20th century. One consequence of the modularity theorem is Fermat's great theorem . Today, the modularity theorem is seen as a special case of the much more general and important Serre conjecture about Galois representations . Building on the work of Andrew Wiles, this was proven in 2006 by Chandrashekhar Khare , Jean-Pierre Wintenberger and Mark Kisin .

The statement of the modularity theorem

The complex-analytical version

The group

${\displaystyle \Gamma _{0}(N)=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{Sl}}_{2}(\mathbb {Z} )\ |\ c\equiv 0\mod N\right\}}$

(a congruence subgroup of the module group , N is also called the level of the associated module form) operates on the upper half-level through Möbius transformation . The quotient space is a non- compact Riemann surface . By adding certain points from (the so-called peaks), one can compactize and thus obtain a compact Riemann surface ( module curve ). The complex-analytic version of the conjecture states that for every elliptic curve over ( a grid), with the associated value of the j-function , one and a non-constant holomorphic mapping of Riemann surfaces ${\displaystyle {\mathfrak {H}}}$ ${\displaystyle Y_{0}(N)=\Gamma _{0}(N)\backslash {\mathfrak {H}}}$ ${\displaystyle \mathbb {Q} \cup \{i\infty \}}$ ${\displaystyle Y_{0}(N)}$ ${\displaystyle X_{0}(N)}$ ${\displaystyle E=\mathbb {C} /\Lambda }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \Lambda }$ ${\displaystyle j(E)\in \mathbb {Q} }$ ${\displaystyle N\in \mathbb {N} }$

${\displaystyle X_{0}(N)\rightarrow E}$

exists. The number N is called the (modular) guide of E. The module curve parameterizes the elliptic curve.

An elliptic curve for which the statement given here is true is called modular.

The complex-analytic version of the theorem is very weak and a priori no number-theoretic statement. The actual modularity theorem makes statements for elliptic curves defined over the rational numbers and states that all elliptic curves are over modular. ${\displaystyle \mathbb {Q} }$

L-series version

The following version of the conjecture makes a statement about elliptic curves above . ${\displaystyle \mathbb {Q} }$

Let be an elliptic curve over with L-series (for its definition see conjecture by Birch and Swinnerton-Dyer ). Then there is a (the guide) and a modular form with . Here, the Hecke-L-series is from (for the definition see the connection between module forms and Dirichlet series ). ${\displaystyle E}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle L(E,s)}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle f\in S_{2}(N)}$ ${\displaystyle L_{f}(s)=L(E,s)}$ ${\displaystyle L_{f}(s)}$ ${\displaystyle f}$

From the theory of modular forms one can easily deduce from this that it has an analytical continuation and a functional equation. This plays an important role in the well-defined nature of the Birch and Swinnerton-Dyer conjecture . ${\displaystyle L(E,s)}$

Algebraic-geometric version

From the theory of the Riemann surfaces (or a version of the GAGA theorem) it follows that the module curve can be defined as a scheme via . One can show that even a scheme is over . The modularity theorem now postulates a surjective morphism for every elliptic curve ${\displaystyle X_{0}(N)}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle X_{0}(N)}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle E}$

${\displaystyle X_{0}(N)\rightarrow E}$

of algebraic curves over a N . ${\displaystyle \mathbb {Q} }$

Representation theory version

Be a modular form. After deep sentences by Pierre Deligne , Jean-Pierre Serre and Robert Langlands , one can f a two-dimensional Galois representation ${\displaystyle f\in S_{k}(N)}$

${\displaystyle \rho _{f,p}:{\text{Gal}}({\overline {\mathbb {Q} }}/\mathbb {Q} )\rightarrow {\text{GL}}_{2}(\mathbb {Q} _{p})}$

assign ( is the algebraic conclusion of in ). Here is the absolute Galois group on the left and the general linear group of the two-dimensional vector space over the field of the p-adic numbers on the right . Likewise, one can assign each elliptical curve E via such a Galois representation . ${\displaystyle {\overline {\mathbb {Q} }}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \rho _{E,p}}$

In this case, the modularity theorem says that for every elliptic curve there is E over a prime p and a modular form for an N such that and are equivalent. ${\displaystyle \mathbb {Q} }$ ${\displaystyle f\in S_{2}(N)}$ ${\displaystyle \rho _{E,p}}$ ${\displaystyle \rho _{f,p}}$

This is the version proven by Wiles.

Outline of the connection between Taniyama-Shimura and Fermat

Fermat's last theorem states that there are no positive integer solutions to the equation for n greater than 2. Ever since the French mathematician Pierre de Fermat claimed in 1637 that they had found evidence of this statement - but without giving it or leaving it in his written records - mathematicians have sought evidence of this statement. The search for a proof of Fermat's last theorem has the number theory dominated for more than two centuries and important elements, such as the theory of ideals of Ernst Kummer , were designed to prove the proposition. ${\displaystyle a^{n}+b^{n}=c^{n}}$

The Saarbrücken mathematician Gerhard Frey made a conjecture in 1986 about a connection between Fermat's last theorem and the Taniyama-Shimura conjecture: If one assumes that Fermat's last theorem is wrong and that there are actually solutions to the equation , then the elliptic curve is probably not modular. Jean-Pierre Serre proved this except for one remainder, the epsilon conjecture, which Ken Ribet proved in 1990 and thus showed that this so-called Frey curve (which Yves Hellegouarch had already considered before) is actually not modular (he used the so-called " Level-lowering ”method, where“ Level ”denotes the level of the module forms under consideration). ${\displaystyle a^{p}+b^{p}=c^{p}}$ ${\displaystyle y^{2}=x(x-a^{p})(x+b^{p})}$

In other words, if Fermat's last theorem is wrong, so is the Taniyama-Shimura conjecture; if the Taniyama-Shimura conjecture is correct, then Fermat's last sentence must also be correct. It suffices to show that the Taniyama-Shimura conjecture holds for semi-stable elliptic curves over the rational numbers. With a semi-stable elliptic curve over the rational numbers, there are only poor semi-stable type reductions. Bad reduction modulo p means that the curve defined over the finite field of integers mod p (the reduction of mod p) becomes singular. If the singularity is a colon and not a tip , it is called a semi-stable type. In this case, in the equation for the elliptic curve with a cubic polynomial with three different roots, at most two zeros coincide with reduction mod p. Good reduction means that all three zeros are different on reduction mod p. The elliptic curve is semi-stable if it only has good reductions or if the bad reductions are semi-stable. ${\displaystyle E}$ ${\displaystyle E_{p}}$ ${\displaystyle E}$ ${\displaystyle y^{2}=f(x)}$ ${\displaystyle f(x)}$ ${\displaystyle \mathbb {Q} }$

Since the Frey curve is semi-stable, the proof of Fermat's last theorem follows from the version of the modularity theorem proved by Wiles.

Meaning for math

The Taniyama-Shimura theorem is an example of the unification of mathematics; This refers to the establishment of connections between areas of mathematics that were previously regarded as completely different, which enables mathematicians to translate problems that cannot be solved in one area into an equivalent problem in another area and, if necessary, to solve them there. In this case, the standardization takes place through the theory of modular forms , which also made clear their outstanding importance for number theory in the Langlands program .

Original work

The following three publications contain the proof of the modularity theorem:

• Andrew Wiles : Modular Elliptic Curves and Fermat's last theorem (PDF; 10.7 MB). Annals of Mathematics 141: 443-551 (1995)
• Richard Taylor , Andrew Wiles : Ring-theoretic properties of certain Hecke algebras . Annals of Mathematics 141: 553-572 (1995).
• 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 the modularity theorem.)

In the following publication, Fermat's last theorem is reduced to the modularity theorem:

• Ribet, KA From the Taniyama-Shimura Conjecture to Fermat's Last Theorem . Annales de la faculté des sciences de Toulouse, 5e série, 11.1 (1990), p. 116-139

literature

• Gary Cornell, Joseph H. Silverman , Glenn Stevens (Eds.): Modular forms and Fermat's last theorem , Springer, 1997
• Fred Diamond , Jerry Shurman: A first course in modular forms (= Graduate Texts in Mathematics 228). Corrected 3rd printing. Springer, New York NY 2007, ISBN 978-0-387-23229-4 . (Chapter 9: Galois representations)
• Gerd Faltings : The proof of Fermat's last theorem by R. Taylor and A. Wiles, Notices American Mathematical Society, 1995, No. 7, PDF

Popular science:

• Simon Singh : Fermat's last sentence. The adventurous story of a mathematical puzzle (= dtv 33052). 14th edition. Deutscher Taschenbuch-Verlag, Munich 2010, ISBN 978-3-423-33052-7 .
• Simon Singh, Kenneth Ribet : The solution to Fermat's riddle. In spectrum of science. 1, 98, ISSN  0170-2971 , p. 96 ff.