The hilbert class field is then the ray class field with respect to the trivial modulus 1. Journal of number theory 30, 110 1988 unramified abelian extensions of number fields robert j. 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. Seminar on higher dimensional class field theory fall 2014 wednesdays 67. Nonabelian class field theory and fundamental groups.
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. For a ring a, the group glma will be the grothendieck group of the. By class field theory, k corresponds to a subgroup h of cl. This paper corrects and generalizes my paper relative k theory and class field theory for arithmetic surfaces math. Eventually fn is an abelian unramified extension of q, so fn q since q. Unramified class field theory of arithmetical surfaces. 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. When you sort it out in this context, you will find that. Then we generalize the notions appearing in the proof to prove also the tamely ramified case relying on the unramified one. Main theorems throughout the present paper, we fix a prime number p and a finite field \mathrmf of characteristic p. This is a graduated student seminar on higher dimensional class field theory held in harvard.
Relative ktheory and class field theory for arithmetic surfaces. 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. This classic book, originally published in 1968, is based on notes of a yearlong seminar the authors ran at princeton university. 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. Hence a compositum of unramified extensions is unramified. 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. Feb 08, 2010 unique factorization into primes, hence understanding the unit group and class group of the ring of integers of kwell. Definition 4 we say a langlands parameter is unramified if its trivial on the inertia group. On local class field theory we have the survey by hochschild 43 also see samuels report 94 on hochschilds results. In this paper we extend the unramified class field theory for arithmetic surfaces of k. This paper introduces basic theorems of class field theory and.
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. 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. Jossey, galois 2extensions unramified outside 2, j. One is an extension of perret s result about the ideal class group problem. Fsep of f, fabutf the maximal abelian subextension in futf, ff the.
This definition is compatible with the preceding one, because by. It brings together the quadratic and higher reciprocity laws of gauss, legendre, and others, and vastly generalizes them. Let fut be the maximal unramified extension of f in the fixed separable closure. 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. Unramified extensions and totally ramified extensions 36. Class field theory more or less solves this problem for lk abelian. This albanese property also specializes to the cartier selfduality of the picard stack and is in fact equivalent to it. 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 note on unramified quadratic extensions over algebraic number fields ichimura, humio, proceedings of the japan academy, series a, mathematical sciences, 2000. Let cl, and cl, be the ideal class groups of f and k respectively.
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 character of the multiplicative group of a local field that is trivial on the group of units of is also called unramified. A local field is either a finite extension of characteristic 0 or. The theory had its origins in the proof of quadratic reciprocity by gauss at the end of 18th century.
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. If lk is an abelian extension of number fields and p is unramified in lk. 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. Class field theory, the study of abelian extensions of algebraic number fields, is one of the largest branches of algebraic number theory. Relative ktheory and class field theory for arithmetic. Unramified geometric class field theory and cartier. Historically, local class field theory branched off from global, or classical. Note on galois descent of a normal integral basis of acyclic extension of degree p ichimura, humio, proceedings of the japan academy. The ramification of infinite primes needs special care in this formulation. Unramified abelian extensions of number fields sciencedirect. Ga22 m pp k kk, the socalled second p class group of k 5 6.
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. Class field theory and algebraic ktheory springerlink. Unramified extension and class field theory mathoverflow. 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. On a rigidity of some modular galois deformations shimada, yuichi, kodai mathematical journal, 2019. 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. Distribution of galois groups of maximal unramified 2extensions over imaginary quadratic fields volume 237 sosuke sasaki. In part i we learn the new approach to higher dimensional cft based on the ideas of wiesend. Click download or read online button to get class field theory proofs and applications book now.
Local class field theory is a theory of abelian extensions of socalled local fields, typical examples of which are the padic number fields. Keywords hilbert pclass field tower, maximal unramified prop extension, pcapitulation of class groups, real quadratic fields 3, 3 1. Ams transactions of the american mathematical society. 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. We first write in detail delignes proof to the unramified case of class field theory, including defining the required objects for the proof. Taussky todd received july 29, 1985 let k be an unramified abelian extension of a number field f with galois group g. Ga22 m pp k kk, the socalled second pclass group of k 5 6. Class field theory, second edition ams chelsea publishing. 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. 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. Recall that the hilbert class field is the largest abelian everywhere unramified extension of k. Intuitively, the hilbert class field of a number field k is the maximal unramified extension. 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. Then we are interested in when ker jl is divisible by ici.
The narrow hilbert class field of k is the maximal abelian extension of k unramified away from the infinite places. 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. 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. 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. Thus local class field theory says that there is an isomorphism. This site is like a library, use search box in the widget to get ebook that you want. 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. You should have a vague understanding of the use of complex multiplication to generate abelian extensions of imaginary quadratic fields first, in order. 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. In part ii we study other types of unramified galois extensions of quadratic number fields. Source class field theory its centenary and prospect, k. 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. Unramified abelian extensions of number fields robert j. Syl clis an algebraic number field with pclass group.
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. Unramified class field theory over function fields in. The maximal unramified extensions of the imaginary. To analyse a f, the pprimary component of the minus part of the ideal class group of f, we use iwasawa theory. From a different perspective, it describes the local components of the global artin map. 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. Class field theory of websites universiteit leiden.
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. Unramified class field theory of arithmetical schemes. 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. Apr 29, 2002 in this paper we extend the unramified class field theory for arithmetic surfaces of k. We first write in detail delignes proof to the unramified case. Bond department of mathematics, boston college, chestnut hill, massachusetts 02167 communicated by o. Class field theory gives a description of this everywhere unramified abelian extension in terms of class groupsideles. We remark that class field theory, and theory of cyclotomic fields, was once considered the key to fermats last theorem. Local class field theory is a theory of abelian extensions of socalled local fields. Number theory 124 2007 4276 and, at this occasion, we bring into focus some classical. The unramified characters form a subgroup of the group of all characters. Unramified class field theory of arithmetical surfaces created date. We give a selfcontained proof of local class field theory, via lubintate theory and the hassearf theorem, refining the arguments of iwasawas book. Class field theory is the description of abelian extensions of global fields and local.
A brief summary of the main statements of class field theory. The multiplicative group of a \\mathfrakp\adic number field 4. Pdf in this paper we prove global class field theory using a purely geometric result. Notes on class field theory notes from a onesemester course on class field theory uc berkeley, spring 2002.
Class field theory publish your masters thesis, bachelors. What i can find in my literature is just the correspondence between unramified extensions and norm groups. For a polynomial fx 2zx, kronecker considered the number n pof roots of fx mod p in f p as pvaries. 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. We pass to the field extension f of fgenerated by all ppower roots of unity. Let x be a regular proper arithmetic surface and let y be the support of divisor on x. Class field theory describes the abelian extensions of a local or global field in. A concise general reference about kummer theory of elds is birchs. First, as a special case of hiberts irreducibility theorm, we construct. This paper demonstrates that class field theory still has a role to play in the beautiful story of fermat. The origins of class field theory lie in the quadratic reciprocity law proved by gauss. When the nth roots of unity are not in k, the situation is much more complicated, and is answered by class field theory. Let k be an imaginary quadratic number field with class number two. Class field theory, diophantine analysis and the asymptotic.
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