The group ck plays the role in global class field theory played by k. There were still a few things to be worked out, such as the local and global duality theories, and the cohomology of algebraic tori, but it was time for new directions. Class field theory describes the abelian extensions of a local or global field in terms of the. A brief summary of the main statements of class field theory. 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. 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.
On different notions of tameness in arithmetic geometry math. Using the higher tame symbol and kawada and satakes witt vector method, a. Class field theory says that every central division algebra over k is cyclic. Parshin developed class field theory for positive characteristic higher local fields, defining reciprocity maps separately for the tamely ramified and wildly ramified cases. 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. Class formations and higher dimensional local class field. Some aspects of eld theory are popular in algebra courses at the undergraduate or graduate levels, especially the theory of nite eld extensions and galois theory. Chow group of 0cycles with modulus and higher dimensional class field theory article pdf available in duke mathematical journal 16515.
Trans dimensional unified field theory natural philosophy. Forschungsseminar higher dimensional class field theory and. Chow group of 0cycles with modulus and higherdimensional class field theory kerz, moritz and saito, shuji, duke mathematical journal, 2016. Among other things, artin reciprocity showed that the unrami ed extensions of a global eld can be described by an abelian object only depending on intrinsic data of the eld. Local class field theory is concerned with abelian extensions of a local field f, i. Morrow, grothendiecks trace map for arithmetic surfaces via residues and higher adeles, algebra and number theory j. Pdf higher dimensional global monopole in brans dicke theory. Fesenkolocal fields, local class field theory, higher local class field theory via algebraick. We will survey some results from global class eld theory with examples, before discussing any proofs. In higher class eld theory, the domain of the reciprocity map is a milnor kgroup, in particular the group ktop n f for an n dimensional local eld f. A reader familiar with galois theory will easily recognize that the. The main content of the thesis is divided into two parts. Higher local and global class field theory uses algebraic ktheory and appropriate milnor kgroups replace which is in use in onedimensional class field theory.
Mathematics theory of high dimensional expanders has emerged. Milnor ktheory of local rings with finite residue fields j. Class field theory is one of the major achievements in the number theory of the rst half of the 20h century. Also the gravitational properties of the monopole solution are discussed. This is a graduated student seminar on higher dimensional class field theory held in harvard. Twodimensional localglobal class field theory in positive. In this note we present a new selfcontained approach to the class field theory of arithmetic schemes in the sense of wiesend. The theory is referred to as higher local class field theory and higher global class field theory.
The aim of class field theory is to understand abelian extensions of k in terms of information. Vostokov, local fields and their extensions, second extended edition. Higher dimensional class field theory rims, kyoto university. Covering data and higher dimensional global class field theory.
In this thesis, we study various aspects of physics in higher dimensional. On the usual local fields typically completions of number fields or the quotient fields of local rings of algebraic curves there is a unique surjective discrete valuation of rank 1 associated to a choice of. The solution of this problem in the case dimx1 was one of the major achievements of number theory in the. Higher class field theory and the connected component. Higher dimensional class field theory from a topological point of view. Pdf the gravitational field of a higher dimensional global monopole in the context of bransdicke theory of gravity is investigated. 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. Using adelic analytic duality and a two dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. Localglobal principles for 1motives harari, david and szamuely, tamas, duke mathematical journal, 2008. Higher local and global class field theory uses algebraic k theory and appropriate milnor kgroups replace which is in use in one dimensional class field theory. We show how one can deduce the more classical version of higher global class field theory due to kato and saito from wiesends version. What does tate mean when he wrote higher dimensional. The question for a higher dimensional analogue of artinreciprocity. Pdf theory of binet formulas for fibonacci and lucas p.
Covering data and higher dimensional global class field. Blochs formula for 0cycles with modulus and higher. Syder, kirsty 2014 twodimensional localglobal class field. In higher class eld theory, the domain of the reciprocity map is a milnor kgroup, in particular the group ktop n f for an ndimensional local eld f.
For the sake of formal simplicity we have used the notion of a. Keeton, eric rains, nicholas shepherdbarron, and john t. Syder, two dimensional local global class field theory in positive characteristic, arxiv. Kato defined complexes of gerstenblochogus type involving the galois cohomology groups of all residue fields of x. In mathematics, a higher dimensional local field is an important example of a complete discrete valuation field. Finally, the geodesic of a test particle due to the gravitational field of the monopole is. Higher dimensional class field theory penn math university of. Let l be a prime not equal to the characteristic of k. We give two proofs, both of which are algebrogeometric and related to higher regulators. The arithmetic of curves over two dimensional local fields draouil, belgacem, bulletin of the belgian mathematical society simon stevin, 2009. For a proper regular model of an elliptic curve over a global field, a two dimensional version of the theory of tate and iwasawa is derived.
Along the way we prove new results on space filling curves on arithmetic schemes and on the class field theory of local rings. An approach to time series analysis parzen, emanuel. We have put artins and tates theory of class formations at the beginning. As such, geometric class field theory has become part of the geometric langlands program and of higher dimensional class field theory. For one dimensional fields we cover the cases of local fields and global fields of positive characteristic. In this volume, we discuss some relations between this problem and cusp forms of weight 1. One of the outstanding problems in arithmetic is a generalization of class field theory to nonabelian galois extension of number fields. Higher local and global class field theory was developed by a. What does tate mean when he wrote higher dimensional class. Let kbe a number eld and sa nite set of places containing all. We investigate the spacetime of a global monopole in a five dimensional spacetime in the presence of the cosmological term. An overview of class field theory california institute of. Syder, twodimensional localglobal class field theory in positive characteristic, arxiv. Higher local class field theory is compatible with class field theory at the residue field level.
The theory had its origins in the proof of quadratic reciprocity by gauss at the end of the 18th century. The gravitational field of a higher dimensional global monopole in the context of bransdicke theory of gravity is investigated. Main results of the theory belong to the first half of the 20th century hilbert, artin, tate, hasse and are quite different for the local field from the global field case. The reciprocity law of higher dimensional local class field theory is proved with the help of class formations. Note that if n 1, then ktop n f f so this theory is compatible with the usual one. Class field theory cft is the main achievement of algebraic number. For a connected regular scheme x, flat and of finite type over specz, we construct a reciprocity homomorphism. Nagoya mathematical journal latest issue cambridge core.
Higher dimensional global monopole in bransdicke theory. Parshin, on the arithmetic of two dimensional schemes. Mtheory and loop quantization of higher dimensional sugra. Parshin, kazuya kato, ivan fesenko, spencer bloch, shiji saito and many other mathematicians. Higher dimensional class field theory and ramification. This thesis investigates class field theory for one dimensional fields and higher dimensional fields.
Class field theory, its three main generalisations, and applications ivan fesenko class field theory cftis the main achievement of algebraic number theory of the 20th century. Field theory 3 about these notes the purpose of these notes is to give a treatment of the theory of elds. Investigate higher dimensional extensions of this theorem to invariant. Saito, global class field theory of arithmetic schemes, contemp. Theory with gravity is dual to theory without gravity like gauge symmetry, general covariance is a derived concept precise definition of string theory as least in negatively curved spacetime new insights into gauge theory and string theory, in particular, holography and issues in the black hole information puzzle. It explains the matrix and periodic table of a multidimensional universe. Now we explain how higher dimensional unramified class field theory can be deduced from our main results of the last section.
For higher dimensional fields we study the case of higher local fields of positive characteristic. A similar but weaker statement holds for smooth varieties over. Then classical global class field theory deals with the question. Wiesend developed a new approach to higher dimensional class. Class field theory studies finitedimensional abelian field extensions of number fields and of function fields, hence of global fields by relating them to the idele class group class field theory clarifies the origin of various reciprocity laws in number theory. Im not sure which tate is actually referring to, since i dont have a copy of the new preface. Ramification filtration by moduli in higherdimensional. The theory just described is called global class field theory because it deals with a number field. Higher local reciprocity homomorphism from this quotient of the nth milnor kgroup to the galois group of the maximal abelian extension of the higher local field has many features similar to those of the one dimensional local class field theory. Vi can be reformulated as theorems about one dimensional arithmetical schemes, whose function fields are precisely the. The basic one dimensional class field theory stems from the ideas of kronecker and weber, and results of hilbert soon after them.
In part i we learn the new approach to higher dimensional cft based on the ideas of wiesend. Chow group of 0cycles with modulus and higherdimensional class field theory, duke math. Class formations and higher dimensional local class field theory. 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. Newtons theory was presented as a theory of 2body interaction. Jordan using zeta functions to factor polynomials over finite fields.
Syder, kirsty 2014 twodimensional localglobal class. In this thesis, a pitchfork bifurcation of an m1 dimensional invariant pdf pda ra submani. This series of talks gives a brief account of this theory a suitable preparation for those interested in attending the threeday workshop on high dimensional expanders at the simons foundation in new york city, oct. Application of bifurcation theory to a variety pdf in wordpress post of problems in physics and applied. An overview of class field theory the statements of class eld theory become easier to understand once placed in their historical contexts. Field theoretic aspects of newtons theory of gravitation. Vostokov, local fields and their extensions, second extended edition, ams 2002, 341 pp. Introduction our universe is a multidimensional universe where processes and procedures involving natural physics relate and exist simultaneously in multiple dimensions. These ideas were developed over the next century, giving rise to a set of conjectures by hilbert. Covering data and higher dimensional global class field theory j. On a duality theorem of abelian varieties over higher. Pdf higher dimensional global monopole in bransdicke theory.