The class group clf of a number field f is an object of central importance in number theory. Local class field theory pan yan summer 2015 these are notes for a reading course with d. Iwasawa theory of elliptic curves with complex multiplication anna seigal 2nd may 2014 contents. Coleman, ralph greenberg, gustave solomon, larry washington, and eugene m. Classically iwasawa theory was concerned with the study of sizes of class. The homomorphism is called the local artin homomorphism. The goal of local class field theory is to classify abelian galois. Observe that iis a complete local noetherian ring and is an integrally. Iwasawa iw 8, iw 10 developed a theory of local units analogous to the global theory, taking projective limits, especially in the cyclotomic tower, and getting the structure of this projective limit modulo the closure of the cyclotomic units. In number theory, iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. Jurgen neukirch class field theory, and kenkichi iwasawa, local class field theory find, read and cite all the research you. Local class eld theory says that there is a homomorphism. We also prove a result about k 1 of a certain canonical localisation of the iwasawa algebra of \\mathcalg\, which occurs in the formulation of the main conjectures of noncommutative. Sources for the history of algebraic number theory and class field theory.
The intention is to give an overview of some topics in. Introduction to the arithmetic theory of automorphic. The book is almost selfcontained and is accessible to any reader with a basic background in algebra and topological groups. The relation of yukawas nonlocal field theory 1 to the ordinary local field theory is investigated. Local class field theory is a theory of abelian extensions of socalled local fields, typical examples of which are the padic number fields. Class field theory describes the abelian extensions of a local or global. Nonabelian starktype conjectures and noncommutative iwasawa theory shinichi kobayashi. Class field theory is the description of abelian extensions of global fields and local.
Iwasawa theory of elliptic curves with complex multiplication. If ais a 2dimensional noetherian integrally closed local ring with. However, in favorable situations it is possible to visualize these di. In this work, we investigate other uniform propgroups which are realizable as galois groups of towers of number. Now, note that by combining the properties of the local artin. Zpextensions and stating the main conjecture of iwasawa theory for totally real fields. Thus, in a few wellchosen words, shimura stakes out the territory he proposes to survey in the book, including parts of classfield theory, the theory of elliptic curves, abelian varieties at least to some extent, and the theory of modular forms and automorphic functions, of course. In mathematics, class field theory is the branch of algebraic number theory concerned with the abelian extensions of number fields, global fields of positive characteristic, and local fields.
You should have a vague understanding of the use of complex multiplication to generate abelian extensions of imaginary quadratic fields first, in order. The main conjecture of iwasawa theory for totally real. Local class field theory oxford mathematical monographs by kenkichi iwasawa author visit amazons kenkichi iwasawa page. On the iwasawa decomposition of a symplectic matrix. These are notes for a course in local class field theory taught at caltech winter term of 2008. Then we consider the iwasawa theory of class groups of abelian extensions of f. Lubintate formal groups and local class field theory.
Iwasawa theory for elliptic curves with complex multiplication. We start by recalling some notation from iwasawa theory. Open questions and recent developments in iwasawa theory in honor of ralph greenbergs 60th birthday boston university june 17, 2005. Backgroundmaterial is presented, though in moreof a fact gatheringframework. In general, the explicit determination of h f, let alone the structure of cl f as a. Compatibility of local and global langlands correspondences. Local class field theory university of british columbia. Notes on noncommutative iwasawa theory hachimori, yoshitaka and ochiai, tadashi, asian journal of mathematics, 2010.
In the early 1970s, barry mazur considered generalizations of iwasawa theory to abelian varieties. Among other things they proved a remarkable euler characteristic formula for the selmer group, studied ranks respectively torsionproperties and projective dimensions of standard local and global iwasawa modules. Kazuya kato iwasawa theory and generalizations icm 2006. Hilbert modular forms and iwasawa theory haruzo hida. Iwasawatate on functions and lfunctions june 6, 2011 the product formula is the assertion that the idele norm is trivial on the diagonal copy of q inside j, that is, 1 jaj y v jaj v for a2q thus, the idele norm descends to a character jq. This book provides a readable introduction to local class field theory, a theory of algebraic extensions. This is done via the theory of residual selmer groups developed in section 2.
Lectures on padic lfunctions by kenkichi iwasawa 1972 local class field theory kenkichi iwasawa 1986 isbn 0195040309. Iwasawa 2017 will be held to commemorate the 100th anniversary of kenkichi iwasawas birth. It is a finite abelian group, and its order hf is known as the class. The theory had its origins in the proof of quadratic reciprocity by gauss at the end of the 18th century. It covers abelian extensions in particular of socalled local fields, typical examples of which are the padic number fields. When we pass to the language of ideles later, all equivalent ideal groups will merge into a. It began as a galois module theory of ideal class groups, initiated by kenkichi iwasawa, as part of the theory of cyclotomic fields. Kazuya kato function field analogues of compactifications of period domains duration. Historically, local class field theory branched off from global, or classical. Hence, by local class field theory, every galois 2cocycle over the field f splits9. A particularly important point is to determined the relation between the type the local conditions at primes dividing p of the selmer group considered in relation with the signature of the unitary group from which the eisentein ideal is defined.
There are undoubtably mistakes in these notes, and they are the authors alone. A field is called local if it is complete with respect to a discrete valuation and has a finite residue field. The lemma follows upon combining this with the equality. Representation theory of gln over nonarchimedean local. Introduction in this section we shall give the setting which leads to a formulation of iwasawas theorem 1. It has been an object of intense study since the nineteenth. In iwasawa theory, one of the central questions is the study of the iwasawa main conjecture, which relates the characteristic ideal of the selmer group of a motive to its padic lfunction when it exists. Journal of number theory 27, 238252 1987 on iwasawa ainvariants of imaginary abelian fields kuniaki horie department of mathematics, tokyo metropolitan university, fukazawa, setagayaku, tokyo 158, japan communicated by p. The student project will be on writting a detailed account on one of the steps of the arguments presented in the lectures. A nonarchimedean local eld of characteristic p0 is isomorphic to f qt for some power qof p.
Iwasawa invariants of galois deformations 3 distinct deformations of a. On higher fitting ideals of iwasawa modules of ideal class groups over. Find all the books, read about the author, and more. This paper is a sequel to our earlier paper wach modules and iwasawa theory for modular forms arxiv. These ideas were developed over the next century, giving rise to a set of conjectures by hilbert. Representation theory of gl n over nonarchimedean local. Curves over a field any algebraic curve over an algebraically closed.
We give an explicit description of k 1 of the iwasawa algebra of \\mathcalg\ in terms of iwasawa algebras of abelian subquotients of \\mathcalg\. Raghuram march 29, 2007 contents 1 introduction 1 2 generalities on representations 2 3 preliminaries on gl nf 6 4 parabolic induction 11 5 jacquet functors 15 6 supercuspidal representations 20 7 discrete series representations 26 8 langlands classi. Since t is finite over zp, it is a semilocal ring and so a. The contributions of this work to the iwasawa theory of padic lie groups are. Among the few books on class field theory i tried to read, weils basic number theory is the one i found most. Given a real symplectic matrix s s11 s12 s21 s22, the following algorithm computes the factors k, a, n of the iwasawa decomposition of s.
Note that by combining the two parts of the theorem we obtain the desired. On yukawas theory of nonlocal field, i progress of. This in turn leads to information on the blochkato conjecture, a generalization of the birch and swinnertondyer conjecture. Journal of number theory 7, 108120 1975 on a theorem of iwasawa raymond ayoub department of mathematics, pennsylvania state university, university park, pennsylvania 16802 communicated by s. Our conventions regarding the reciprocity maps of class field theory. Prime divisors of special values of theta functions in the ray class fields of a certain quartic field modulo powers of 2. Let pbe a prime and let zp denote the ring of integers of qp. The purpose of this paper is to prove the main conjecture of noncommutative iwasawa theory for padic lie extensions, for an odd prime p, of totally real number fields assuming that the iwasawa mu invariant of a certain totally real number field vanishes.886 1413 583 104 1043 373 750 605 1177 401 1071 1076 663 522 1464 1114 904 139 1470 694 1167 1286 257 328 958 1299 19 267 593 581 655 148 753 180 1170 809 1324 541 913 204 389 739 616