Lectures on the approach to iwasawa theory for hasseweil. These are notes i took in class, taught by professor rankf calegari and george boxer. Motives and automorphic forms september 28 october 2, 2015 abstracts of talks massimo bertolini. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Preliminaries for iwasawa theory and the langlands program. Pdf available in bulletin of the american mathematical society 211989 january 1989 with 52 reads. Iwasawa theory for class groups of cm fields with p 2. Class eld theory gives an isomorphism between a dense subgroup of gkabjk, called the weil group, and the multiplicative group of k. Pdf available in bulletin of the american mathematical society 211989 january 1989 with.
Using adelic analytic duality and a twodimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. Journal of number theory 23, 238242 1986 c n t invariants of zpextensions of real quadratic fields takashi fukuda department of mathematics, tokyo, 152 japan and keiichi komatsu department of mathematics, tokyo university of agriculture and technology, fuchu, tokyo, 183 japan communicated by h. Class eld theory viewed as a langlands correspondence. Congruent numbers and elliptic curves 26 pages pdf. Click download or read online button to get class field theory proofs and applications book now. Like the title says, im looking for what i need to familiarize myself with before trying to learn iwasawa theory or the main conjectures of the langlands program.
Course announcement math 205ab, fall 2006 winter 2007. In the early 1970s, barry mazur considered generalizations of iwasawa theory to abelian varieties. Kenkichi iwasawa this book provides a readable introduction to local class field theory, a theory of algebraic extensions, in particular abelian extensions of socalled local fields. This theory describes the maximal abelian extension kab of kand its galois group gkabjk. Lectures on the approach to iwasawa theory for hasseweil lfunctions via b dr. Local class field theory the main goal of local class field theory is a precise description of the maximal abelian extension of an arbitrary local field. 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. This site is like a library, use search box in the widget to get ebook that you want. The theory had its origins in the proof of quadratic reciprocity by gauss at the end of 18th century. Included are the two sections on cyclic extensions and abelian extensions. One of the main questions to answer is to how many abelian extensions exists over a global or local field, and the numbertheoretic phenomena occurring in. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel.
Introductory iwasawa theory notes 102 pages pdf i am. Lubintate formal groups and local class field theory. More recently, ralph greenberg has proposed an iwasawa theory. More recent results are phrased in terms of main conjectures of iwasawa theory. One of the most powerful tools known at present attacking these conjectures is iwasawa theory. I claim no credit to the originality of the contents of these notes. Eudml local class field theory via lubintate theory. More recently, ralph greenberg has proposed an iwasawa theory for motives.
It began as a galois module theory of ideal class groups, initiated by kenkichi iwasawa, as part of the theory of cyclotomic fields. For a proper regular model of an elliptic curve over a global field, a twodimensional version of the theory of tate and iwasawa is derived. In order that senior thesis produced by harvard math students are easier for other undergrads to benefit from, we would like to exhibit more senior theses online while all theses are available through harvard university archives, it would be more convenient to have them online. Jurgen neukirch class field theory, and kenkichi iwasawa, local class field theory. An introductory lecture on euler systems barry mazur, harvard university notes by jungjo lee, ariel pacetti, and john voight the purpose of these notes is to describe the notion of an euler system, a col lection of compatible cohomology classes arising from a tower of. Before that he worked on lie groups and lie algebras, introducing the general iwasawa decomposition. Alex, i will take congruent numbers for one million dollars please 14 pages submitted pdf. Local class field theory university of british columbia. Class field theory proofs and applications download.
Backgroundmaterial is presented, though in moreof a fact gatheringframework. We give a selfcontained proof of local class field theory, via lubintate theory and the hassearf theorem, refining the arguments of iwasawa s book. Contributions in mathematical and computational sciences, vol 7. Historically, local class field theory branched off from global, or classical. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle. Numerous and frequentlyupdated resource results are available from this search. Iwasawa is perhaps best known for introducing what is now called iwasawa theory, which developed from researches on cyclotomic fields from the later parts of the 1950s. Other readers will always be interested in your opinion of the books youve read.
Nor do i claim that they are without errors, nor readable. The lifted root number conjecture and iwasawa theory. Classically iwasawa theory was concerned with the study of sizes of class groups of cyclotomic. Motives and automorphic forms september 28 october 2. This book provides a readable introduction to local class field theory, a theory of algebraic extensions. The main idea of iwasawa theory is to relate various arithmetic objects to complex \l\functions via a socalled \p\adic \l\function. It has been an object of intense study since the nineteenth. A note on the finiteness of certain cuspidal divisor class groups.
Course announcement math 205ab, fall 2006 winter 2007 local and global class field theory local class field theory is the precise description of the maximal abelian extension of an arbitrary local field. Local class field theory is a theory of abelian extensions of socalled local fields, typical examples of which are the padic number fields. Milne, class field theory, neukirch, class field theory, guillot, a gentle course in local class field theory elementary serre, local fields advanced students minor thesis on tates thesis, buzzard, tates thesis, recommended. Local class field theory oxford mathematical monographs by kenkichi iwasawa author visit amazons kenkichi iwasawa page. Find all the books, read about the author, and more. Using the iwasawa main conjecture, we study the 2component of the ideal class group of a cm field k of finite degree as a galois module. Local class field theory by kenkichi iwasawa, 9780195040302, available at book depository with free delivery worldwide. Lubintate formal groups and local class field theory submitted by emily riehl in partial ful llment of the requirements for the degree of bachelor of arts with honors department of mathematics harvard university april 3, 2006 1 introduction the goal of local class eld theory is to classify abelian galois extensions of a local eld k. In the late 1950s, iwasawa initiated a study of the growth of class groups in certain towers of number. This book offers a selfcontained exposition of local class field theory, serving as a second course on galois theory. This arithmetic object could be the ideal class group of a number field, the selmer group of an elliptic. From a different perspective, it describes the local components of the global artin map.
In the rst chapter of this thesis we consider class eld theory for a local eld k. A field is called local if it is complete with respect to a discrete valuation and has a finite residue field. In number theory, iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. Mathematical society of japan, 2001, 335 385 iwasawa theory past and present ralph greenberg. Explicit class field theory for rational function fields. The idele class group is a collection of multiplicative groups of local fields, giving a view of class field theory for global fields as a collection of class field theories for local fields. Right now im pretty good with modular forms, modular lfunctions, galois theory, algebraic number theory but not class field theory yet, and classical realcomplex analysis. Topics in iwasawa theory ralph greenberg december 15, 2006 1 ideal class groups. First of all, we show that the classical iwasawa main conjecture holds true even for p 2 over a totally real field k assuming. It opens with a discussion of several fundamental topics in algebra, such as profinite groups, padic fields, semisimple algebras and their modules, and homological algebra with the example of group cohomology. It covers abelian extensions in particular of socalled local fields, typical examples of which are the padic number fields. Local class field theory oxford mathematical monographs. Iwasawa, a note on the group of units of an algebraic number field, j.