I hope that the moderators will miss this post. I will not swear that I am an ace in mathematics, of course I understand this issue, or rather it will be possible to say that I understand to a certain extent, that is not bad, but only.

Ну в чём-то разбираться и быть асом-это не синоним. Здесь будет что-то вроде общих сведений, …Therefore, I would like to warn, even if I would know much more, that is, if I would have been an ace in mathematics, anyway, it would seem impossible answer all the questions, at least here.Please don't ask questions.;-)))). Короче,эта тема знакома мне пусть даже и немного,поэтому себе позволю сказать о ней хотя бы «1,5 слова» и хотя бы на уровня простых формулировок. Не думаю , что получится очень уж хорошо, но что нибудь попытаюсь изобразить. Well, in general, I will do a sketch ;-)))), ну «Теория полей класса»,or « the class field theory » ,hm,…. this term is generally used synonymously with the study of all the abelian extensions of algebraic number fields, or more generally global fields ; an abelian extension being Galois extension with Galois group that is an abelian group ( по-рууски это связано с изучение всех абелевых расширений полей алгебраических чисел, или в более общем плане глобальных полей) . Roughly speaking, the class field theory is a major branch of algebraic number theory.. Tate ‘s cohomology groups are used in the class field theory, and they are a slightly modified form of the usual cohomology groups of a finite group, they combine the homology and the cohomology into one sequence .. Tate ‘s cohomology groups were invented by John Tate. J. T. (with E. Artin) gave a new treatment of global class field theory, using techniques of group cohomology applied to the idele class group. And he also (together with J. Lubin) extended the reach of Galois cohomology: Poitou-Tate duality, the Tate-Shafarevich group, and relations with algebraic K-theory. And he has modified (or rather he presented a somewhat different) the local class field theory by the use of formal groups, creating the Lubin-Tate local theory of complex multiplication… Tate's invention of rigid analytic spaces spawned the whole field of rigid analytic geometry. And a p-adic analogue of Hodge theory was found by J.T... Now the p-adic analogue of Hodge theory is called the Hodge-Tate theory. The Hondge-Tate theory provides a complete description of abelian varieties over finite fields, and the isogenies between them. . . Well here it is appropriate to make such a clarification that actually the Tate module of an abelian group, is a module constructed from an abelian group. Quite often, this construction is made in the following situation: «G» is a commutative group scheme over a field «K» ... When the abelian group is the group of roots of unity in a separable closure Ks of K, the p-adic Tate module is sometimes referred to as the Tate module. Still here should be aware that an abelian variety G over a field K, the Ks-valued points of G are an abelian group. The p-adic Tate module G is a Galois representation (of the absolute Galois group, GK, of K). The classical results on abelian varieties show that if K has characteristic 0, then Tp (G) is a free module over Z of rank 2d, where d is the dimension of G. It is still free, but the rank may take any value from 0 to d.In the case where p is not equal to the characteristic of K. The p-adic Tate module of G is the dual of the etale cohomology. Well, the usual long exact sequence of Tate cohomology groups can get from the short exact sequence of G-modules ... Well, in special cases of the Tate conjecture can be phrased in terms of Tate’s modules. And once again sorry I have to lubricate even a primitive scheme, everything is much more complicated than I can portray here ….Tate's theorem gives conditions for multiplication by a cohomology class to be an isomorphism between cohomology groups. This theorem will be presented in a simpler version here, because it will hardly be able to present it in another way(Obviously ,I apologize for the inevitable inaccuracies, and warn that this theorem is transmitted in a very «schematic form», and all ,that passed in a chart form ,can not claim to be correct, under any circumstances, it is better to climb on the links. This scheme is only a primitive demonstration of the theorem ... I will neither prove nor substantiate this theory because it will be very problematic …):-Assume that «K» is a module over a finite group G and «k» is an element of H (G, K), such that for every subgroup E of G :H1 (E, K) ….H2 (E, K),and that is generated by Res (k) which has order E.. The finite group G defines Tate's cohomology with «k» in module («Z») is indicated here, as H ( in n) (G, Z) . The «cup product» with «k» is an isomorphism :- H (in n) (G, Z ).--.> H ( in (n +2)) (G, K).. For all «n»; thus the graded Tate cohomology of «K»is isomorphic to the Tate cohomology with integral coefficients, with the degree shifted by 2. And even when we apply the case n =- 2 of Tate's theorem to a class formation, we find that there is an isomorphism .I.e., аs u probably understand, this is the most primitive image , I dont know how it is extended ;-( ,and whether it can explain at least something to someone, because the primitive images are different, but I know this scheme, it is the most simple and primitive, according to my opinion its most readily depicted here, without resorting to the findings. ( By that « how is this scheme extended(?), I dont know », I mean that, I dont know how many people ,from those who are able to understand the topic being discussed in the article, know exactly the scheme . Это конечно должно было быть куда более сложней)…Please dont strongly increases for this explanation. Still, it does not seem possible to describe everything in detail, but I dont even try to do it here, but the details are most important in real life ... . Well, okay, I offer this link cdsweb.cern.ch/record/1222949 А,что про вливание Тейта на великую теорему Ферма??? On the other hand, Tate’s results in the theory of elliptic curves useful in proving the famous theorem of Fermat(it ‘s called Fermat's Last Theorem,and Fermat left no proof of the conjecture for all «n», but he did prove the special case n = 4.). Fermat's theorem was proved by Andrew Wiles(A.W.is a рrofessor of the Princeton University). Well, this theorem goes something like this: - for «n» greater than two does not exist such integers a, b and c, which holds a(in n )+ b(in n) = c(in n).А можно я не буду про теорему Ферма это так рассужения в сторону и тоже «для общего развития» ,её все знают(т.е. о ней слышали.Only the general principle can be shown here and no more, and maybe even less ;-))) Unfortunately, there may be possible to give only superficial definitions, and an everyone is not only an expert, but also a layman can say about these definitions, that they are naive and primitive .. ;-))))….Для большего нужны нам ссылки. Ладно можно было не умничать;-)))),а дать много ссылок, больше чем одну/две,ну да ладно «частым цитированием» это не назовёшт никто …Ну в этих ссылке можно немножко порыться…. Ладно , не буду больше умничать

And I must warn the readers that links w

Далее я предложил порыться в ссылках . Я надеюсь, что эти ссыли всё таки продут сюда.http://cdsweb.cern.ch/record/122294.And I must warn readers that the contents of the links will not coincide with the information in my previous post Perhaps , it would be not difficult to tell at least part of the contents from those links, but I would not want to do it for some reason .( т.е. ссылки не совпадают во всём с тем, что я рассказывал, но лучше смотрите их )
I hope ,that the hurts and
the misunderstandings will not be… http://www.greenlees.staff.shef.ac.uk/preprints.html. .I also thought that many of the visitors of this site would be able to sort things out themselves and draw their own conclusions about the contents of these links .Я думаю, что сами разберуться.