Iwasawa theory

The Iwasawa theory is within mathematics in the field of number theory a theory for the determination of the ideal class group of infinite body towers, whose Galois group is isomorphic to the -adic numbers . The theory was established in the 1950s by Kenkichi Iwasawa for the investigation of circular dividing bodies . In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to Abelian varieties . In addition, Ralph Greenberg proposed an Iwasawa theory for motives . ${\displaystyle p}$

situation

Iwasawa's initial observation was that there are body towers in algebraic number theory whose Galois group is isomorphic to the additive group of -adic numbers. This group is often written multiplicatively and denoted by; it is the inverse limit of the (additive) groups ${\displaystyle p}$ ${\displaystyle \Gamma }$

${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$,

where is a fixed prime number and iterates through the natural numbers . ${\displaystyle p}$ ${\displaystyle n}$

example

Be a primitive -th root of unity and consider the body tower ${\displaystyle \zeta =\zeta _{p}}$ ${\displaystyle p}$

${\displaystyle K=\mathbf {Q} (\zeta )\subset K_{1}\subset K_{2}\subset \cdots \subset \mathbf {C} ,}$

where denotes the body generated by a primitive -th root of unity (note the indexing). Be the union of all these bodies. Then the Galois group is isomorphic to , since the Galois groups of over are equal . An interesting Galois module (ie an Abelian group on which the Galois surgery ) results when looking at the - twist the ideal class groups of the participating number field . Be the continuous torsion of the ideal class groups of having designated. These are connected to one another by standard images for and form a directed system. The group then operates on the Inverse Limes . In addition, there is a module on the pro-finite group ring (this observation goes back to Jean-Pierre Serre ). This ring, also called Iwasawa algebra , is regular and two-dimensional, and it is possible to broadly classify its modules. ${\displaystyle K_{n}}$ ${\displaystyle p^{n+1}}$ ${\displaystyle K_{\infty }}$ ${\displaystyle Gal(K_{\infty }/K)}$ ${\displaystyle \Gamma }$ ${\displaystyle K_{n}}$ ${\displaystyle K}$ ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle K_{n}}$ ${\displaystyle I_{n}}$ ${\displaystyle I_{m}\rightarrow I_{n}}$ ${\displaystyle m>n}$ ${\displaystyle \Gamma }$ ${\displaystyle I}$ ${\displaystyle I}$ ${\displaystyle \mathbb {Z} _{p}[[\Gamma ]]}$

The motivation here was that the torsion of the ideal class group of , as Kummer recognized, was a major obstacle to a proof of Fermat's Great Theorem . In this context, Kummer called a prime number regular if it did not divide the class number of . Iwasawa's idea was to study this torsion systematically with infinite Galois theory. With these methods Iwasawa was able to describe the torsions numerically. This is the content of Iwasawa's theorem. ${\displaystyle p}$ ${\displaystyle K_{\infty }}$ ${\displaystyle \mathbf {Q} (\zeta _{p})}$ ${\displaystyle p}$

Iwasawa's theorem

As above, let a body tower be given whose Galois group are the -adic numbers, and be the order of the -torsion of . Then there are integers , and so that for the relationship sufficiently large holds. ${\displaystyle K_{n}}$ ${\displaystyle p}$ ${\displaystyle p^{e_{n}}\,}$ ${\displaystyle p}$ ${\displaystyle I_{n}}$ ${\displaystyle \mu }$ ${\displaystyle \lambda }$ ${\displaystyle \nu }$ ${\displaystyle n}$ ${\displaystyle e_{n}=\mu p^{n}+\lambda n+\nu }$

Proof idea

Due to class field theory there is an extension of such that , namely, is the maximum unbranched -abelian extension of . The union of the then forms a body which is the maximum unbranched Abelian pro- -extension of . One then considers the Galois group , which is the inverse limit of the groups which appear as the quotient of . As an Abelian pro group, the group has the structure of a module. In addition, the Galois operates on which characterized a is modulus (that is, a Iwasawa module ). Structural investigations and the classification down to pseudo-isomorphisms of all Iwasawa modules lead to asymptotic estimates for the orders of and thus of . ${\displaystyle L_{n}}$ ${\displaystyle K_{n}}$ ${\displaystyle I_{n}\simeq Gal(L_{n}/K_{n})}$ ${\displaystyle L_{n}}$ ${\displaystyle p}$ ${\displaystyle K_{n}}$ ${\displaystyle L_{n}}$ ${\displaystyle L_{\infty }}$ ${\displaystyle p}$ ${\displaystyle K_{\infty }}$ ${\displaystyle X=Gal(L_{\infty }/K_{\infty })}$ ${\displaystyle Gal(L_{n}/K_{n})}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle p}$ ${\displaystyle \mathbb {Z} _{p}}$ ${\displaystyle Gal(K_{\infty }/K)}$ ${\displaystyle X}$ ${\displaystyle \mathbb {Z} _{p}[[T]]}$ ${\displaystyle Gal(L_{n}/K_{n})}$ ${\displaystyle I_{n}}$

Further developments and main assumption

In the 1960s, a fundamental connection was discovered between the module theory developed by Iwasawa on the one hand and p-adic L-functions on the other hand, which were defined by Tomio Kubota and Heinrich-Wolfgang Leopoldt . These functions are defined on the basis of Bernoulli numbers by means of interpolation and represent p-adic analogies to the Dirichlet L-functions. The so-called main assumption of the Iwasawa theory says that these two approaches (module theory and interpolation) add -adic L-functions define, agree with each other. This conjecture was proved in 1984 by Barry Mazur and Andrew Wiles for the rational numbers and later for all totally real number fields by Andrew Wiles. This evidence was based on Ken Ribet's proof of the reverse of Herbrand's theorem . In 2014 Chris Skinner and Eric Urban succeeded in proving the main conjecture for certain families of lace shapes. While the work of Mazur and Wiles can be seen as dealing with the case of GL (1) over or over a general totally real number field, Skinner-Urban solved the case of GL (2) . ${\displaystyle p}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$

literature

• John Coates , Ramdorai Sujatha : Cyclotomic Fields and Zeta Values. Springer, Berlin et al. 2006, ISBN 3-540-33068-2 .
• Ralph Greenberg : Iwasawa Theory - Past & Present. In: Advanced Studies in Pure Mathematics. Vol. 30, 2001, ZDB -ID 47532-4 , pp. 335-385.
• Serge Lang : Cyclotomic Fields (= Graduate Texts in Mathematics. Vol. 59). Springer, New York NY et al. 1978, ISBN 0-387-90307-0 .
• Barry Mazur , Andrew Wiles : Class Fields of Abelian Extensions of Q. In: Inventiones Mathematicae . Vol. 76, No. 2, 1984, pp. 179-330, doi : 10.1007 / BF01388599 .
• Chris Skinner , Eric Urban Sur les deformations p-adiques des formes de Saito-Kurokawa. In: Académie des Sciences Paris. Comptes Rendus Mathematique. Vol. 335, No. 1, 2002, ISSN  1631-073X , pp. 581-586, doi : 10.1016 / S1631-073X (02) 02540-2 .
• Lawrence C. Washington : Introduction to Cyclotomic Fields (= Graduate Texts in Mathematics. Vol. 83). 2nd Edition. Springer, New York NY et al. 1997, ISBN 0-387-94762-0 .
• Andrew Wiles : The Iwasawa Conjecture for Totally Real Fields. In: Annals of Mathematics . Vol. 131, No. 3, 1990, pp. 493-540.

Individual evidence

1. ↑ Christopher Skinner, Eric Urban: The Iwasawa Main Conjectures for GL ₂ . (PDF; 1.5 MB) Preprint. Retrieved July 30, 2013 .  Published in Inv. Math., Volume 194, 2014, pp. 1-277