In mathematics, a higher (-dimensional) local field is an important example of a complete discrete valuation field. Such fields are also sometimes called multi-dimensional local fields. On the usual

local field In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...

s (typically completions of

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

s or the quotient fields of

local ring In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...

s of

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

s) there is a unique surjective discrete valuation (of rank 1) associated to a choice of a local parameter of the fields, unless they are archimedean local fields such as the

real numbers In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...

and

complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

. Similarly, there is a discrete valuation of rank ''n'' on almost all ''n''-dimensional local fields, associated to a choice of ''n'' local parameters of the field. In contrast to one-dimensional local fields, higher local fields have a sequence of

residue field In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and \mathfrak is a maximal ideal, then the residue field is the quotient ring k=R/\mathfrak, which is a field. Frequently, R is a local ri ...

s.Fesenko, I., Kurihara, M. (eds.) ''Invitation to Higher Local Fields''. Geometry and Topology Monographs, 2000, section 1 (Zhukov). There are different integral structures on higher local fields, depending how many residue fields information one wants to take into account. Geometrically, higher local fields appear via a process of localization and completion of local rings of higher dimensional schemes. Higher local fields are an important part of the subject of higher dimensional number theory, forming the appropriate collection of objects for local considerations.

Definition

Finite fields have dimension 0 and complete discrete valuation fields with finite residue field have dimension one (it is natural to also define archimedean local fields such as R or C to have dimension 1), then we say a complete discrete valuation field has dimension ''n'' if its residue field has dimension ''n''−1. Higher local fields are those of dimension greater than one, while one-dimensional local fields are the traditional local fields. We call the residue field of a finite-dimensional higher local field the 'first' residue field, its residue field is then the second residue field, and the pattern continues until we reach a finite field.

Examples

Two-dimensional local fields are divided into the following classes: *Fields of positive characteristic, they are

formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...

in variable ''t'' over a one-dimensional local field, i.e. F_''q''((''u''))((''t'')). *Equicharacteristic fields of characteristic zero, they are formal

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

''F''((''t'')) over a one-dimensional local field ''F'' of characteristic zero. * Mixed-characteristic fields, they are finite extensions of fields of type ''F'', ''F'' is a one-dimensional local field of characteristic zero. This field is defined as the set of formal power series, infinite in both directions, with coefficients from ''F'' such that the minimum of the valuation of the coefficients is an integer, and such that the valuation of the coefficients tend to zero as their index goes to minus infinity. *Archimedean two-dimensional local fields, which are formal power series over the

R or the

Constructions

Higher local fields appear in a variety of contexts. A geometric example is as follows. Given a surface over a finite field of characteristic p, a curve on the surface and a point on the curve, take the local ring at the point. Then, complete this ring, localise it at the curve and complete the resulting ring. Finally, take the quotient field. The result is a two-dimensional local field over a finite field. There is also a construction using commutative algebra, which becomes technical for non-regular rings. The starting point is a Noetherian, regular, ''n''-dimensional ring and a full

flag A flag is a piece of textile, fabric (most often rectangular) with distinctive colours and design. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design employed, and fla ...

of prime ideals such that their corresponding quotient ring is regular. A series of completions and localisations take place as above until an ''n''-dimensional local field is reached.

Topologies on higher local fields

One-dimensional local fields are usually considered in the valuation topology, in which the discrete valuation is used to define open sets. This will not suffice for higher dimensional local fields, since one needs to take into account the topology at the residue level too. Higher local fields can be endowed with appropriate topologies (not uniquely defined) which address this issue. Such topologies are not the topologies associated with discrete valuations of rank ''n'', if ''n'' > 1. In dimension two and higher the additive group of the field becomes a topological group which is not

locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...

and the base of the topology is not countable. The most surprising thing is that the multiplication is not continuous; however, it is

sequentially continuous In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...

, which suffices for all reasonable arithmetic purposes. There are also iterated Ind–Pro approaches to replace topological considerations by more formal ones.Fesenko, I., Kurihara, M. (eds.) ''Invitation to Higher Local Fields''. Geometry and Topology Monographs, 2000, several sections.

Measure, integration and harmonic analysis on higher local fields

There is no translation

invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mappin ...

on two-dimensional local fields. Instead, there is a

finitely additive In mathematics, an additive set function is a function \mu mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this ad ...

translation invariant measure defined on the ring of sets generated by closed balls with respect to two-dimensional discrete valuations on the field, and taking values in formal power series R((''X'')) over reals. This measure is also countably additive in a certain refined sense. It can be viewed as higher

Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfr� ...

on higher local fields. The additive group of every higher local field is non-canonically self-dual, and one can define a higher

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

on appropriate spaces of functions. This leads to higher

harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...

Higher local class field theory

Local class field theory In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite re ...

in dimension one has its analogues in higher dimensions. The appropriate replacement for the multiplicative group becomes the nth Milnor K-group, where ''n'' is the dimension of the field, which then appears as the domain of a reciprocity map to the

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

of the maximal

abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian group, abelian. When the Galois group is also cyclic group, cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galoi ...

over the field. Even better is to work with the quotient of the nth Milnor K-group by its subgroup of elements divisible by every positive integer. Due to Fesenko theorem, this quotient can also be viewed as the maximal separated topological quotient of the K-group endowed with appropriate higher dimensional topology. Higher local reciprocity homomorphism from this quotient of the nth Milnor K-group 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. Higher local class field theory is compatible with class field theory at the residue field level, using the border map of Milnor K-theory to create a commutative diagram involving the reciprocity map on the level of the field and the residue field.Fesenko, I., Kurihara, M. (eds.) ''Invitation to Higher Local Fields''. Geometry and Topology Monographs, 2000, section 5 (Kurihara). General higher local class field theory was developed by

Kazuya Kato is a Japanese mathematician who works at the University of Chicago and specializes in number theory and arithmetic geometry. Early life and education Kazuya Kato grew up in the prefecture of Wakayama in Japan. He attended college at the Univer ...

and by Ivan Fesenko. Higher local class field theory in positive characteristic was proposed by Aleksei Parshin. Translation (1991), ''Proc. Steklov Inst. Math.'' 4: 191–201.

Notes

References

* * {{Citation , editor-last=Fesenko , editor-first=Ivan B. , editor-link=Ivan Fesenko , editor2-last=Kurihara , editor2-first=Masato , title=Invitation to Higher Local Fields. Extended version of talks given at the conference on higher local fields, Münster, Germany, August 29–September 5, 1999 , publisher= Mathematical Sciences Publishers , location= University of Warwick , year=2000 , series=Geometry and Topology Monographs , volume=3 , edition=First , doi=10.2140/gtm.2000.3 , issn=1464-8989 , zbl=0954.00026 , url=https://www.maths.nottingham.ac.uk/personal/ibf/volume.html Field (mathematics) Algebraic number theory Harmonic analysis