岩澤理論

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數論中，岩澤理論是理想類群中的Galois module理論，由岩澤健吉於1950年代提出，是單位根理論的一部分。1970年代初，Barry Mazur considered generalizations of 岩澤理論 to 阿貝爾簇. 到1990年代初，拉爾夫·格林伯格將岩澤理論應用到motives.

[编辑] 方程式

岩澤理論Iwasawa's starting observation was that there are towers of fields in 代數數論, having 伽羅瓦群 isomorphic with the additive group of p進數. That group, usually written Γ in the theory and with multiplicative notation, can be found as a subgroup of Galois groups of infinite field extensions (which are by their nature pro-finite groups). The group $Γ$ is the inverse limit of the additive groups $\mathbf Z/p^n \mathbf Z$ , where p is the fixed 質數 and $n = 1,2, \dots$ . We can express this by Pontryagin duality in another way: Γ is dual to the 離散群 of all $p$ -power 單位根 in the 複數.

[编辑] 例子

設 $ζ$ be a primitive $p$ -th root of unity and consider the following tower of number fields:

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

where $K n$ is the field generated by a primitive $p n + 1$ -th root of unity. This tower of fields has a union $L$ . Then the Galois group of $L$ over $K$ is isomorphic with $Γ$ ; because the Galois group of $K n$ over $K$ is $\mathbf Z/p^n \mathbf Z$ . In order to get an interesting Galois module here, Iwasawa took the ideal class group of $K n$ , and let $I n$ be its $p$ -torsion part. There are norm mappings $I_m \rightarrow I_n$ when $m > n$ , and so an inverse system. Letting $I$ be the inverse limit, we can say that $Γ$ acts on $I$ , and it is desirable to have a description of this action.

The motivation here was undoubtedly that the $p$ -torsion in the ideal class group of $K$ had already been identified by 恩斯特‧庫默爾 as the main obstruction to the direct proof of Fermat's last theorem. Iwasawa's originality was to go 'off to infinity' in a novel direction. In fact $I$ is a module over the group ring $\mathbf Z_p [[\Gamma]]$ . This is a well-behaved ring (regular and two-dimensional), meaning that it is quite possible to classify modules over it, in a way that is not too coarse.

[编辑] 歷史

From this beginning, in the 1950s, a substantial theory has been built up. A fundamental connection was noticed between the module theory, and the p-adic L-functions that were defined in the 1960s by 久保田 and Leopoldt. The latter begin from the 伯努利數, and use 插值 to define p-adic analogues of the Dirichlet L-functions. It became clear that the theory had prospects of moving ahead finally from 庫默爾's century-old results on regular primes.

The main conjecture of Iwasawa theory was formulated as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was eventually proved by Barry Mazur and 安德魯·懷爾斯 for Q, and for all totally real number fields by 安德魯·懷爾斯. These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (so-called Herbrand-Ribet theorem).

More recently, also modeled upon Ribet's method, Chris Skinner and Eric Urban have announced a proof of a main conjecture for GL(2). A more elementary proof of the Mazur-Wiles theorem can be obtained by using 歐拉系統s as developed by Kolyvagin (see Washington's book). Other generalizations of the main conjecture proved using the Euler system method have been obtained by Karl Rubin, amongst others.

[编辑] 參考文獻

Greenberg, Ralph, Iwasawa Theory - Past & Present, Advanced Studies in Pure Math. 30 (2001), 335-385. Available at [1].
Coates, J. and Sujatha, R., Cyclotomic Fields and Zeta Values, Springer-Verlag, 2006
Lang, S., Cyclotomic Fields, Springer-Verlag, 1978
Washington, L., Introduction to Cyclotomic Fields, 2nd edition, Springer-Verlag, 1997
Barry Mazur and 安德魯·懷爾斯 (1984). "Class Fields of Abelian Extensions of Q". Inventiones Mathematicae 76 (2): 179-330.

Andrew Wiles (1990). "The Iwasawa Conjecture for Totally Real Fields". Annals of Mathematics 131 (3): 493-540.

Chris Skinner and Eric Urban (2002). "Sur les deformations p-adiques des formes de Saito-Kurokawa". C. R. Math. Acad. Sci. Paris 335 (7): 581-586.

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