Chow group of 0cycles with modulus and higher dimensional. Covering data and higher dimensional global class field theory. String theory describes how strings propagate through space and interact with each other. This thesis investigates class field theory for one dimensional fields and higher dimensional fields. One of the outstanding problems in arithmetic is a generalization of class field theory to nonabelian galois extension of number fields. We show how to extract from this set the algebraic structure of a vertex algebra. In this thesis, a pitchfork bifurcation of an m1 dimensional invariant pdf pda ra submani.
In which sense is summing two numbers a 2dimensional process. Class field theory describes the abelian extensions of a local or global field in terms of the arithmetic of the. The theory is referred to as higher local class field theory and higher global class field theory. Apr 16, 20 chow group of 0cycles with modulus and higher dimensional class field theory article pdf available in duke mathematical journal 16515 april 20 with 28 reads how we measure reads.
On a duality theorem of abelian varieties over higher. Fugroup arithmetic geometry ws 14 introduction let kbe a 1 dimensional global eld, i. Msri moved online tensor categories and topological. An invitation to higher dimensional mathematics and physics. Another natural generalization is higher class field theory. Algebraic number theory and related topics, surikaisekikenkyusho kokyuroku rims proceedings 10, 168178 2005. 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. Field theoretic notions appeared, even though still implicitly, in the modern theory of solvability of polynomial equations, as introduced by abel and galois in the early nineteenth century. We propose tensor type chernsimons theory as the effective field theory for membranes in aclass topological insulators. Wiesend developeda new approach to higher dimensional class. 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. Parshin developed class field theory for positive characteristic higher local fields, defining reciprocity maps separately for the tamely ramified and wildly ramified cases.
Class formations ans higher dimensional local class field. On different notions of tameness in arithmetic geometry math. We also consider the k 1 of a product of curves over a number field. An introduction to the theory of higherdimensional. A key ingredient is the construction of a cycletheoretic avatar of a refined artin conductor in ramification theory originally studied by kazuya kato. Class formations ans higher dimensional local class field theory.
Yoon, generalised kawadasatake method for mackey functors in class field theory, europ. Some remarks concerning the origins and nature of this material. In which sense is summing two numbers a 2 dimensional process. In this thesis, we study various aspects of physics in higherdimensional. Forschungsseminar higher dimensional class field theory. Chow group of 0cycles with modulus and higher dimensional class field theory article pdf available in duke mathematical journal 16515. 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.
As well as the foundational theory, we summarise the theory of topologies on higher dimensional local. For one dimensional fields we cover the cases of local fields and global fields of positive characteristic. Nagoya mathematical journal latest issue cambridge core. 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. Application of bifurcation theory to a variety pdf in wordpress post of problems in physics and applied. Localglobal principles for 1motives harari, david and szamuely, tamas, duke mathematical journal, 2008. Higherdimensional algebra and planckscale physics john c. Class field theory clarifies the origin of various reciprocity laws in number theory. The problem is translated into the language of higher dimensional class field theory over finite fields, which describes the abelian fundamental group of u by chow groups of 0cycles with moduli. The starting point for string theory is the idea that the pointlike particles of particle physics can also be modeled as onedimensional objects called strings. Forschungsseminar higher dimensional class field theory and rami cation mentor. Higher local class field theory is compatible with class field theory at the residue field level. What does tate mean when he wrote higher dimensional class.
Higher dimensional quantum hall effect as aclass topological. Baez department of mathematics, university of california riverside, california 92521, usa january 28, 1999. We propose tensor type chernsimons theory as the effective field theory for membranes in a class topological insulators. The focus will be on the interplay among topics such as higher category theory, as illustrated by the cobordism hypothesis, conformal field theory, tensor categories describing the quantum symmetries, and the relation to topological phases of matter. The problem is translated into the language of higherdimensional class field theory over finite fields, which describes the abelian fundamental group of u by chow groups of 0cycles with moduli. Class formations and higher dimensional local class field theory. Now we explain how higher dimensional unramified class field theory can be deduced from our main results of the last section. Then classical global class field theory deals with the question. Some consequences of wiesends higher dimensional class field.
B class field theories, onedimensional and higher dimensional b16 class field theory, its three main generalisations, and applications pdf b15 i. Forschungsseminar higher dimensional class field theory and. Field is turned on and craft zips off to interstellar destination, never locally breaking the speed of light, but covering the distance d in an arbitrarily short period of time. Experiential learning theory as a guide for experiential. Let kbe a number eld and sa nite set of places containing all. Statistics of kgroups modulo p for the ring of integers of a varying quadratic number field. Pdf higher dimensional quantum hall effect as aclass. The connection between two types of monopole is applied to generalize the concept of flux attachment in quantum hall effect to aclass topological insulator. Higher dimensional class field theory for regular arithmetic schemes after wiesend 4.
Higher dimensional class field theory and ramification. Experiential learning theory as a guide for experiential educators in higher education alice y. Class field theory studies finite dimensional abelian field extensions of number fields and of function fields, hence of global fields by relating them to the idele class group. In this volume, we discuss some relations between this problem and cusp forms of weight 1.
For a polynomial fx 2zx, kronecker considered the number n. For a regular arithmetic scheme xde ne the class group c x and the reciprocity homomorphism cx abx. In this thesis, a pitchfork bifurcation of an m1dimensional invariant pdf pda ra submani. Schmidt, a higher dimensional class field theory from a topological point of view. Covering data and higher dimensional global class field theory j. I learned early on in my undergraduate education that while it is instructive to read, and to attend to the words of informed speakers, i cannot gain the feeling that i understand a subject until i have done my best to write about it. It is a careful and detailed introduction to the higherdimensional theory of quasiconformal mappings from the geometric viewpoint, based primarily on the technique of the conformal modulus of a curve family.
The basic one dimensional class field theory stems from the ideas of kronecker. 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. Florian pop, advisor let kbe a nite eld, and suppose that the arithmetical variety x. Chow groups of 0cycles with modulus and higher dimensional class field theory following kerzsaito, we introduce the chow group of 0cycles with modulus, relate it to the wiesend idele class group, and show the existence of a canonical reciprocity map on the chow group with modulus, with values in the abelianization of the fundamental group. The aim of class field theory is the description of the abelian etale coverings. Pdf blochs formula for 0cycles with modulus and higher. This is a graduated student seminar on higher dimensional class field theory held in harvard. As well as class field theory, higher dimensional fields have found applications in the development of explicit approaches to grothendieck duality and trace maps. Twodimensional localglobal class field theory in positive. We prove that the kernel of the reciprocity map for a product of curves over a padic field with split semistable reduction is divisible. Chow group of 0cycles with modulus and higher dimensional class field theory, pdf. Galois structure of class formations let k be a number field and let l be a finite galois extension of k, with galois group g gallk. Chow group of 0cycles with modulus and higherdimensional class field theory kerz, moritz and saito, shuji, duke mathematical journal, 2016. Higher dimensional class field theory from a topological point of view.
Fugroup arithmetic geometry ws 14 introduction let kbe a 1dimensional global eld, i. From a different perspective, it describes the local components of the global artin map. In part i we learn the new approach to higher dimensional cft based on the ideas of wiesend. 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. Seminar on higher dimensional class field theory fall 2014 wednesdays 67. 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. The starting point for string theory is the idea that the pointlike particles of particle physics can also be modeled as one dimensional objects called strings. Higher dimensional class field theory penn math university of.
Mtheory and loop quantization of higher dimensional sugra. Artin reciprocity is a particular case of onedimensional class field theory. Class field theory for a product of curves over a local field. String theory is a theoretical framework that attempts to address these questions and many others. Investigate higher dimensional extensions of this theorem to invariant. Higher dimensional class field theory, using the chow group with modulus, is developed in. Higher dimensional local fields are defined by induction. B class field theories, one dimensional and higher dimensional b16 class field theory, its three main generalisations, and applications pdf b15 i. From a first analysis of correlation functions of a quantum field theory, we extract an infinite set of meromorphic sections of line bundles on the riemann sphere. The connection between two types of monopole is applied to generalize the concept of flux attachment in quantum hall effect to a class topological insulator. In this section we sketch a proof of the isomorphism theorem for a higher dimensional local field as an analogue of the above argument. Using the higher tame symbol and kawada and satakes witt vector method, a. Milnor ktheory of local rings with finite residue fields j. Jordan using zeta functions to factor polynomials over finite fields.
An introduction to nonabelian class field theory series on. Published in physics meets philosophy at the planck scale, eds. Field is turned off at standoff distance dfrom the destination, and craft finishes journey conventionally. Algebraic number theory and related topics, surikaisekikenkyusho kokyuroku rims. Suppose that c x is the wiesend id ele class group of x. Higher dimensional class field theory rims, kyoto university. Roquette received december 9, 1995 the reciprocity law of higher dimensional local class field theory is proved with the help of class. Core concepts of experiential learning theorythe learning cycle, learning style, and learning spacehave been. It is a careful and detailed introduction to the higher dimensional theory of quasiconformal mappings from the geometric viewpoint, based primarily on the technique of the conformal modulus of a curve family. The reciprocity law of higher dimensional local class field theory is proved with the help of class formations. In this thesis, we study various aspects of physics in higher dimensional. Another generalization of class field theory is anabelian geometry which studies algorithms to restore the original object e.