Class field theory is the description of abelian extensions of global fields and local. That approach was dramatically surpassed by the ideas of frey, serre, ribet and wiles. Class field theory describes the abelian extensions of a local or global field in. Class field theory, second edition ams chelsea publishing. Unramified class field theory of arithmetical surfaces created date. Class field theory proofs and applications download. Ams transactions of the american mathematical society. Click download or read online button to get class field theory proofs and applications book now. Unramified extension and class field theory mathoverflow. Thus local class field theory says that there is an isomorphism.
Local class field theory is a theory of abelian extensions of socalled local fields. Class field theory, diophantine analysis and the asymptotic. Note on galois descent of a normal integral basis of acyclic extension of degree p ichimura, humio, proceedings of the japan academy. Instead, he would have tried to find a good criterion for a cyclic extension of odd prime degree over an algebraic number field to be a. This classic book, originally published in 1968, is based on notes of a yearlong seminar the authors ran at princeton university. Unramified class field theory of arithmetical surfaces by kazuya kato and shuji saito let x be a connected scheme of finite type over the ring z of rational integers and let gbx be the abelian fundamental group of x which classifies unramified abelian coverings of x. Unramified class field theory 255 the given section of f gives rise to a splitting of the map h, so that the map g is surjective. We construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Unramified abelian extensions of number fields robert j. Let x be a regular proper arithmetic surface and let y be the support of divisor on x.
We remark that class field theory, and theory of cyclotomic fields, was once considered the key to fermats last theorem. This paper corrects and generalizes my paper relative k theory and class field theory for arithmetic surfaces math. We pass to the field extension f of fgenerated by all ppower roots of unity. Syl clis an algebraic number field with pclass group. By class field theory, k corresponds to a subgroup h of cl. A global function eld is a nite extension of f pt for some prime p, or equivalently is the function eld of a geometrically integral curve over a nite eld f q called the constant eld, where qis a power of some prime p. If lk is an abelian extension of number fields and p is unramified in lk. The ramification of infinite primes needs special care in this formulation. Intuitively, the hilbert class field of a number field k is the maximal unramified extension. Ga22 m pp k kk, the socalled second pclass group of k 5 6.
Seminar on higher dimensional class field theory fall 2014 wednesdays 67. Recall that the hilbert class field is the largest abelian everywhere unramified extension of k. Then we generalize the notions appearing in the proof to prove also the tamely ramified case relying on the unramified one. In part i we learn the new approach to higher dimensional cft based on the ideas of wiesend. We give a selfcontained proof of local class field theory, via lubintate theory and the hassearf theorem, refining the arguments of iwasawas book. Local class field theory university of british columbia. Geometry of bun g and hecke eigensheaves the gbundles bun g on xform a smooth algebraic stack locally of nite type cf. A note on unramified quadratic extensions over algebraic number fields ichimura, humio, proceedings of the japan academy, series a, mathematical sciences, 2000. Notes on class field theory notes from a onesemester course on class field theory uc berkeley, spring 2002. The primary goal of the book was to give a rather complete presentation of algebraic aspects of global class field theory, and the authors accomplished this goal spectacularly. Number theory 124 2007 4276 and, at this occasion, we bring into focus some classical. Distribution of galois groups of maximal unramified 2extensions over imaginary quadratic fields volume 237 sosuke sasaki. Class field theory of websites universiteit leiden.
It was also artins idea 11 to leave the abelian setting of class field theory and to consider the second hilbert pclass field f ff2 11 p pp kk, that is the maximal unramified metabelian pextension of k, and its galois group. Hence a compositum of unramified extensions is unramified. Fsep of f, fabutf the maximal abelian subextension in futf, ff the. Keywords hilbert pclass field tower, maximal unramified prop extension, pcapitulation of class groups, real quadratic fields 3, 3 1. The unramified characters form a subgroup of the group of all characters. The ray class field is the maximal abelian extension unramified outside the primes dividing the modulus and satisfying a particular ramification condition at the primes. It was also artins idea 11 to leave the abelian setting of class field theory and to consider the second hilbert p class field f ff2 11 p pp kk, that is the maximal unramified metabelian pextension of k, and its galois group. This albanese property also specializes to the cartier selfduality of the picard stack and is in fact equivalent to it. Our class field theory for curves over local fields corresponds to the case n 1 and dim x 1, and one of the most notable differences of the theory for n 2 1 from that for n 0 is the existence of nontrivial etale coverings of x in which all closed point split completely.
If the arithmetical invariants of k are known, in other words if class field theory over k is explicit, the situation for a finite extension l is a priori completely different, and one usually studies the corresponding invariants of l using several means. Apr 29, 2002 in this paper we extend the unramified class field theory for arithmetic surfaces of k. Unramified extensions and totally ramified extensions 36. Construction of unramified extensions with a prescribed galois group kim, kwangseob, osaka journal of mathematics, 2015 a cup product in the galois cohomology of number fields mccallum, william g. A concise general reference about kummer theory of elds is birchs. Nonabelian class field theory and fundamental groups. History of class field theory 3 in addition to the construction of abelian extensions, kronecker set o another path to class eld theory in an 1880 paper on densities of primes and factorization of polynomials.
From a different perspective, it describes the local components of the global artin map. Feb 08, 2010 unique factorization into primes, hence understanding the unit group and class group of the ring of integers of kwell. The theory had its origins in the proof of quadratic reciprocity by gauss at the end of 18th century. Unramified class field theory over function fields in. Source class field theory its centenary and prospect, k.
We first write in detail delignes proof to the unramified case of class field theory, including defining the required objects for the proof. Pdf in this paper we prove global class field theory using a purely geometric result. Hilbert conjectured in 1902 and furtwangler proved four years later that the maximal unramified abelian extension of a number field is isomorphic to the class group of the number field. Unramified class field theory of arithmetical schemes. Class field theory, the study of abelian extensions of algebraic number fields, is one of the largest branches of algebraic number theory. A character of the multiplicative group of a local field that is trivial on the group of units of is also called unramified. Let fut be the maximal unramified extension of f in the fixed separable closure. To analyse a f, the pprimary component of the minus part of the ideal class group of f, we use iwasawa theory. The hilbert class field is then the ray class field with respect to the trivial modulus 1. This is a oneyear course on class field theory one huge piece of intellectual work in the 20th century. Class field theory gives a description of this everywhere unramified abelian extension in terms of class groupsideles.
Unramified abelian extensions of number fields sciencedirect. Relative ktheory and class field theory for arithmetic. Unramified class field theory of arithmetical surfaces. The multiplicative group of a \\mathfrakp\adic number field 4. A local field is either a finite extension of characteristic 0 or. The ray class field is the maximal abelian extension unramified outside the primes dividing the modulus and satisfying a particular ramification condition at the primes dividing the modulus. Eventually fn is an abelian unramified extension of q, so fn q since q. Historically, local class field theory branched off from global, or classical.
A brief summary of the main statements of class field theory. For a ring a, the group glma will be the grothendieck group of the. 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. What i can find in my literature is just the correspondence between unramified extensions and norm groups. First, as a special case of hiberts irreducibility theorm, we construct. We prove a generalized albanese property for the picard stack of a smooth projective curve, which in particular implies delignes unramified geometric class field theory. Jossey, galois 2extensions unramified outside 2, j. Class field theory more or less solves this problem for lk abelian. This paper introduces basic theorems of class field theory and. First to give a detailed treatment of unramified geometric abelian class field theory concentrating on delignes geometric proof in order to remedy the unfortunate situation that the literature on this topic is very deficient, partial and sketchy written1.
This paper demonstrates that class field theory still has a role to play in the beautiful story of fermat. By local class field theory, and the inertia group maps to the units, hence an unramified character corresponds to a langlands parameter which is trivial on the inertia group. The maximal unramified extensions of the imaginary. Infact, bun gis equidimensional of pure dimension g 1dimg for g1, and the group of connected components. One is an extension of perret s result about the ideal class group problem. On a rigidity of some modular galois deformations shimada, yuichi, kodai mathematical journal, 2019. Journal of number theory 30, 110 1988 unramified abelian extensions of number fields robert j. Class field theory and algebraic ktheory springerlink. This is a graduated student seminar on higher dimensional class field theory held in harvard. Let k be an imaginary quadratic number field with class number two. Class field theory publish your masters thesis, bachelors. According to the class field theory, the ideal class group of a number field is closely related to the maximal unramified abelian extension of the field. The origins of class field theory lie in the quadratic reciprocity law proved by gauss. Ga22 m pp k kk, the socalled second p class group of k 5 6.
When you sort it out in this context, you will find that. Unramified geometric class field theory and cartier. In class field theory, one studies the ray class field with respect to a given modulus, which is a formal product of prime ideals including, possibly, archimedean ones. Mathematical society of japan, 2001, 307 312 hodge cycles and unramified class fields hiromichi yanai. The theory that gives a description of all abelian extensions finite galois extensions having abelian galois groups of a field that belongs to one of the following types. It brings together the quadratic and higher reciprocity laws of gauss, legendre, and others, and vastly generalizes them. We shall prove that the maximal unramified extension of k is k 1 or k 2, where k 1 is the hilbert class field of k and k 2 is that of k 1. The generalization took place as a longterm historical project, involving quadratic forms and their genus theory, work of ernst kummer and leopold kroneckerkurt hensel on ideals and completions, the theory of cyclotomic and kummer extensions. The main theorem of unramified class field theory, which states that the class group of the ring of integers of a number field, is canonically isomorphic to the galois group of the maximal totally unramified abelian extension of over, is generalized and proved for all infinite commutative rings with unit that, like rings of integers. Class field theory makes sense for arbitrary number fields, but for simplicity in this section and because it is all that is needed for our application to the bsd conjecture, we assume henceforth that is a totally imaginary number field, i.
In this talk we will use ideas from algebraic geometry to reformulate furtwanglers classical result. Main theorems throughout the present paper, we fix a prime number p and a finite field \mathrmf of characteristic p. Bond department of mathematics, boston college, chestnut hill, massachusetts 02167 communicated by o. We give a novel proof of the selfduality, by showing that any family of multiplicative line bundles on. You should have a vague understanding of the use of complex multiplication to generate abelian extensions of imaginary quadratic fields first, in order. Taussky todd received july 29, 1985 let k be an unramified abelian extension of a number field f with galois group g. This site is like a library, use search box in the widget to get ebook that you want. In the revised version, i positive characteristic case is included, ii the proof of base change is simplified, iii the construction of artin map is simplified, and strengthened so that we start with an arbitrary lubintate group. Relative ktheory and class field theory for arithmetic surfaces. In this paper we extend the unramified class field theory for arithmetic surfaces of k. A global function eld is a nite extension of f pt for some prime p, or equivalently is the function eld of a geometrically integral curve over a nite eld f q called the constant eld, where qis. For a polynomial fx 2zx, kronecker considered the number n pof roots of fx mod p in f p as pvaries.
On local class field theory we have the survey by hochschild 43 also see samuels report 94 on hochschilds results. In part ii we study other types of unramified galois extensions of quadratic number fields. Let cl, and cl, be the ideal class groups of f and k respectively. Instead, he would have tried to find a good criterion for a cyclic extension of odd prime degree over an algebraic number field to be a congruence class field. Recall that a global field is either a finite extension of characteristic 0 or a field of rational functions on a projective curve over a field of characteristic i. When the nth roots of unity are not in k, the situation is much more complicated, and is answered by class field theory. Local class field theory is a theory of abelian extensions of socalled local fields, typical examples of which are the padic number fields. The narrow hilbert class field of k is the maximal abelian extension of k unramified away from the infinite places. This definition is compatible with the preceding one, because by. We first write in detail delignes proof to the unramified case. Then we are interested in when ker jl is divisible by ici. Definition 4 we say a langlands parameter is unramified if its trivial on the inertia group. Course notes and supplementary material pdf format.
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