Glossary of arithmetic and Diophantine geometry

This is a glossary of arithmetic and Diophantine geometry in mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, areas growing out of the traditional study of Diophantine equation

Diophantine equation

In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...

s to encompass large parts of number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

and algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

. Much of the theory is in the form of proposed conjecture

Conjecture

A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

s, which can be related at various levels of generality.

Diophantine geometry

Diophantine geometry

In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general...

in general is the study of algebraic varieties V over fields K that are finitely generated over their prime fields—including as of special interest number fields and finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

s—and over local field

Local field

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

s. Of those, only the complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s are algebraically closed; over any other K the existence of points of V with co-ordinates in K is something to be proved and studied as an extra topic, even knowing the geometry of V.

Arithmetical or arithmetic (algebraic) geometry is a field with a less elementary definition. After the advent of scheme theory it could reasonably be defined as the study of Alexander Grothendieck

Alexander Grothendieck

Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...

's schemes of finite type over the spectrum

Spectrum of a ring

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...

of the ring of integers

Ring of integers

In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...

Z. This point of view has been very influential for around half a century; it has very widely been regarded as fulfilling Leopold Kronecker

Leopold Kronecker

Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...

's ambition to have number theory operate only with rings that are quotients of polynomial ring

Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...

s over the integers (to use the current language of commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

). In fact scheme theory uses all sorts of auxiliary constructions that do not appear at all 'finitistic', so that there is little connection with 'constructivist' ideas as such. That scheme theory may not be the last word appears from continuing interest in the 'infinite primes' (the real and complex local fields), which do not come from prime ideal

Prime ideal

In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

s as the p-adic number

P-adic number

In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

s do.

A

abc conjecture

The abc conjecture

Abc conjecture

The abc conjecture is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser in 1985. The conjecture is stated in terms of three positive integers, a, b and c , which have no common factor and satisfy a + b = c...

of Masser

David Masser

David William Masser is Professor of Mathematics at the University of Basel, in Basel, Switzerland. He obtained his Ph.D. from University of Cambridge in 1974 on the topic of Elliptic Functions and Transcendence....

and Oesterlé

Joseph Oesterlé

Joseph Oesterlé is a French mathematician who, along with David Masser, formulated the abc conjecture in 1985. It is perhaps the most important unsolved problem in diophantine analysis.-External links:*...

attempts to state as much as possible about repeated prime factors in an equation a + b = c. For example 3 + 125 = 128 but the prime powers here are exceptional.

Arakelov theory

Arakelov theory is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.-Background:...

is an approach to arithmetic geometry that explicitly includes the 'infinite primes'.

Arithmetic of abelian varieties

See main article arithmetic of abelian varieties

Arithmetic of abelian varieties

In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those. It goes back to the studies of Fermat on what are now recognised as elliptic curves; and has become a very substantial area both in terms of results and conjectures...

Artin L-functions

Artin L-function

In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin...

s are defined for quite general Galois representations. The introduction of étale cohomology

Étale cohomology

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...

in the 1960s meant that Hasse-Weil L-functions (q.v.) could be regarded as Artin L-functions for the Galois representations on l-adic cohomology groups.

B

Bad reduction

See good reduction.

Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture

Birch and Swinnerton-Dyer conjecture

In mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. Its status as one of the most challenging mathematical questions has become widely recognized; the conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay...

on elliptic curves postulates a connection between the rank of an elliptic curve and the order of pole of its Hasse-Weil L-function. It has been an important landmark in Diophantine geometry since the mid-1960s, with important results such as the Coates–Wiles theorem, Gross–Zagier theorem and Kolyvagin's theorem

Bombieri–Lang conjecture

Enrico Bombieri

Enrico Bombieri is a mathematician who has been working at the Institute for Advanced Study in Princeton, New Jersey. Bombieri's research in number theory, algebraic geometry, and mathematical analysis have earned him many international prizes --- a Fields Medal in 1974 and the Balzan Prize in 1980...

, Serge Lang

Serge Lang

Serge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra...

and Paul Vojta

Paul Vojta

Paul Alan Vojta is an American mathematician, known for his work in number theory on diophantine geometry and diophantine approximation....

have conjectured that algebraic varieties of general type do not have Zariski dense subsets of K-rational points, for K a finitely-generated field. This circle of ideas includes the understanding of analytic hyperbolicity and the Lang conjectures on that, and the Vojta conjectures. An analytically holomorphic algebraic variety V over the complex numbers is one such that no holomorphic mapping from the whole complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

to it exists, that is not constant. Examples include compact Riemann surface

Compact Riemann surface

In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact and the open .A compact Riemann surface C that is a...

s of genus g > 1. Lang conjectured that V is analytically holomorphic if and only if all subvarieties are of general type.

C

Canonical height

The canonical height on an abelian variety

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...

is a height function that is a distinguished quadratic form

Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

. See Néron-Tate height.

Chabauty's method

Chabauty's method, based on p-adic analytic functions, is a special application but capable of proving cases of the Mordell conjecture for curves whose Jacobian's rank is less than its dimension. It developed ideas from Thoralf Skolem

Thoralf Skolem

Thoralf Albert Skolem was a Norwegian mathematician known mainly for his work on mathematical logic and set theory.-Life:...

's method for an algebraic torus

Algebraic torus

In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of tori in Lie group theory...

. (Other older methods for Diophantine problems include Runge's method.)

Crystalline cohomology

In mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by and developed by . Its values are modules over rings of Witt vectors over the base field....

is a p-adic cohomology theory in characteristic p, introduced by Alexander Grothendieck

Alexander Grothendieck

to fill the gap left by étale cohomology

Étale cohomology

which is deficient in using mod p coefficients in this case. It is one of a number of theories deriving in some way from Dwork's method (q.v.), and has applications outside purely arithmetical questions.

D

Diagonal forms

Diagonal forms are some of the simplest projective varieties to study from an arithmetic point of view (including the Fermat varieties). Their local zeta-function

Local zeta-function

In number theory, a local zeta-functionis a function whose logarithmic derivative is a generating functionfor the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F.-Formulation:...

s are computed in terms of Jacobi sum

Jacobi sum

In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J for Dirichlet characters \chi, \psi modulo a prime number p, defined by...

s. Waring's problem

Waring's problem

In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers...

is the most classical case.

Dwork's method

Bernard Dwork

Bernard Morris Dwork was an American mathematician, known for his application of p-adic analysis to local zeta functions, and in particular for the first general results on the Weil conjectures. Together with Kenkichi Iwasawa he received the Cole Prize in 1962.Dwork received his Ph.D. at Columbia...

used distinctive methods of p-adic analysis

P-adic analysis

In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers....

, p-adic algebraic differential equation

Algebraic differential equation

In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used....

s, Koszul complex

Koszul complex

In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul...

es and other techniques that have not all been absorbed into general theories such as crystalline cohomology (q.v.). He first proved the rationality

Rational function

In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

of local zeta-functions, the initial advance in the direction of the Weil conjectures (q.v.)

E

Étale cohomology

The search for a Weil cohomology (q.v.) was at least partially fulfilled in the étale cohomology

Étale cohomology

theory of Alexander Grothendieck

Alexander Grothendieck

and Michael Artin

Michael Artin

Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry. and also generally recognized as one of the outstanding professors in his field.Artin was born in Hamburg,...

. It provided a proof of the functional equation

Functional equation (L-function)

In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural...

for the local zeta-function

Local zeta-function

s, and was basic in the formulation of the Tate conjecture (q.v.) and numerous other theories.

F

Fermat's last theorem

Fermat's Last Theorem

In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....

, the most celebrated conjecture of Diophantine geometry, was proved by Andrew Wiles

Andrew Wiles

Sir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at Oxford University, specializing in number theory...

and Richard Taylor

Richard Taylor (mathematician)

-External links:**...

Flat cohomology

Flat cohomology is, for the school of Grothendieck, one terminal point of development. It has the disadvantage of being quite hard to compute with. The reason that the flat topology

Flat topology

In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also has played a fundamental role in the theory of descent...

has been considered the 'right' foundational topos

Topos

In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...

for scheme theory goes back to the fact of faithfully-flat descent, the discovery of Grothendieck that the representable functor

Representable functor

In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...

s are sheaves for it (i.e. a very general gluing axiom

Gluing axiom

In mathematics, the gluing axiom is introduced to define what a sheaf F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor...

holds).

Function field analogy

It was realised in the nineteenth century that the ring of integers

Ring of integers

In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...

of a number field has analogies with the affine coordinate ring of an algebraic curve or compact Riemann surface, with a point or more removed corresponding to the 'infinite places' of a number field. This idea is more precisely encoded in the theory that global field

Global field

In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...

s should all be treated on the same basis. The idea goes further. Thus elliptic surface

Elliptic surface

In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism to an algebraic curve, almost all of whose fibers are elliptic curves....

s over the complex numbers, also, have some quite strict analogies with elliptic curve

Elliptic curve

In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

s over number fields.

G

Geometric class field theory

The extension of class field theory

Class field theory

In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...

-style results on abelian coverings to varieties of dimension at least two is often called geometric class field theory.

Good reduction

Fundamental to local analysis

Local analysis

In mathematics, the term local analysis has at least two meanings - both derived from the idea of looking at a problem relative to each prime number p first, and then later trying to integrate the information gained at each prime into a 'global' picture...

in arithmetic problems is to reduce modulo

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

all prime numbers p. In the typical situation this presents little difficulty for almost all

Almost all

In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....

p; for example denominators of fractions are tricky, in that reduction modulo a prime in the denominator looks like division by zero

Division by zero

In mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...

, but that rules out only finitely many p per fraction. With a little extra sophistication, homogeneous coordinates

Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...

allow clearing of denominators by multiplying by a common scalar. For a given, single point one can do this and not leave a common factor p. However singularity theory

Singularity theory

-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...

enters: a non-singular point may become a singular point

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

on reduction modulo p, because the Zariski tangent space

Zariski tangent space

In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V...

can become larger when linear terms reduce to 0 (the geometric formulation shows it is not the fault of a single set of coordinates). Good reduction therefore excludes a finite set S of primes for a given variety V, assumed smooth, such that there is otherwise a smooth reduced V_p over Z/pZ. The theory is subtle, in the sense that the freedom to change variables to try to improve matters is rather unobvious: see Néron model

Néron model

In algebraic geometry, a Néron model for an abelian variety AK defined over a local field K is the "best possible" group scheme AO defined over the ring of integers R of the local field K that becomes isomorphic to AK after base change from R to K.They were introduced by André Néron...

, potential good reduction, Tate curve, semistable abelian variety, semistable elliptic curve

Semistable elliptic curve

In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field....

, Ogg-Néron-Shafarevich criterion, Serre-Tate theorem.

Grothendieck-Katz conjecture

The Grothendieck-Katz p-curvature conjecture applies reduction modulo primes to algebraic differential equation

Algebraic differential equation

s, to derive information on algebraic function

Algebraic function

In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...

solutions. It is an open problem . The initial result of this type was Eisenstein's theorem

Eisenstein's theorem

In mathematics, Eisenstein's theorem, named after the German mathematician Gotthold Eisenstein, applies to the coefficients of any power series which is an algebraic function with rational number coefficients...

H

Hasse principle

The Hasse principle

Hasse principle

In mathematics, Helmut Hasse's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number...

states that solubility for a global field

Global field

is the same as solubility in all relevant local field

Local field

s. One of the main objectives of Diophantine geometry is to classify cases where the Hasse principle holds. Generally that is for a large number of variables, when the degree of an equation is held fixed. The Hasse principle is often associated with the success of the Hardy-Littlewood circle method

Hardy-Littlewood circle method

In mathematics, the Hardy–Littlewood circle method is one of the most frequently used techniques of analytic number theory. It is named for G. H. Hardy and J. E...

. When the circle method works, it can provide extra, quantitative information such as asymptotic number of solutions. Reducing the number of variables makes the circle method harder; therefore failures of the Hasse principle, for example for cubic form

Cubic form

In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form.In , Boris Delone and Dmitriĭ Faddeev showed that binary cubic forms with integer coefficients can be used to parametrize orders in cubic fields...

s in small numbers of variables (and in particular for elliptic curve

Elliptic curve

s as cubic curves) are at a general level connected with the limitations of the analytic approach.

Hasse-Weil L-function

A Hasse-Weil L-function, sometimes called a global L-function, is an Euler product

Euler product

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.-Definition:...

formed from local zeta-functions. The properties of such L-function

L-function

The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...

s remain largely in the realm of conjecture, with the proof of the Taniyama-Shimura conjecture being a breakthrough. The Langlands philosophy is largely complementary to the theory of global L-functions.

Height function

A height function in Diophantine geometry quantifies the size of solutions to Diophantine equations. It is standard to take a logarithmic scale

Logarithmic scale

A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...

: that is, the height is proportional to the number of bits a computer needs to store a point in homogeneous coordinates

Homogeneous coordinates

. Heights were initially developed by André Weil

André Weil

André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...

and D. G. Northcott. Innovations around 1960 were the Néron-Tate height (q.v.) and the realisation that heights were linked to projective representations in much the same way that ample line bundle

Ample line bundle

In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold M into projective space. An ample line bundle is one such that some positive power is very ample...

s are in pure geometry.

Hilbertian fields

A Hilbertian field K is one for which the projective space

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

s over K are not thin set

Thin set (Serre)

In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field K, by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or...

s in the sense of Jean-Pierre Serre

Jean-Pierre Serre

Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...

. This is a geometric take on Hilbert's irreducibility theorem

Hilbert's irreducibility theorem

In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers...

which shows the rational numbers are Hilbertian. Results are applied to the inverse Galois problem

Inverse Galois problem

In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers Q. This problem, first posed in the 19th century, is unsolved....

. Thin sets (the French word is mince) are in some sense analogous to the meagre set

Meagre set

In the mathematical fields of general topology and descriptive set theory, a meagre set is a set that, considered as a subset of a topological space, is in a precise sense small or negligible...

s (French maigre) of the Baire category theorem

Baire category theorem

The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....

I

Igusa zeta-function

An Igusa zeta-function

Igusa zeta-function

In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p2, p3, and so on.- Definition :For a prime number p let K be a p-adic field, i.e...

, named for Jun-ichi Igusa

Jun-Ichi Igusa

is a Japanese mathematician, known for his contributions to algebraic geometry and number theory. Igusa zeta-function were named after him.-External links:...

, is a generating function

Generating function

In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...

counting numbers of points on an algebraic variety modulo high powers pⁿ of a fixed prime number p. General rationality theorems

Rational function

are now known, drawing on methods of mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

Infinite descent

In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the fact that the natural numbers are well ordered. One typical application is to show that a given equation has no solutions. Assuming a solution exists, one shows that another exists, that...

was Pierre de Fermat

Pierre de Fermat

Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...

's classical method for Diophantine equations. It became one half of the standard proof of the Mordell-Weil theorem, with the other being an argument with height functions (q.v.). Descent is something like division by two in a group of principal homogeneous space

Principal homogeneous space

In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial...

s (often called 'descents', when written out by equations); in more modern terms in a Galois cohomology

Galois cohomology

In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...

group which is to be proved finite. See Selmer group.

Iwasawa theory

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur...

builds up from the analytic number theory

Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

and Stickelberger's theorem

Stickelberger's theorem

In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields...

as a theory of ideal class group

Ideal class group

In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...

s as Galois module

Galois module

In mathematics, a Galois module is a G-module where G is the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module...

s and p-adic L-function

P-adic L-function

In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic...

s (with roots in Kummer congruence on Bernoulli number

Bernoulli number

In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....

s). In its early days in the late 1960s it was called Iwasawa's

Kenkichi Iwasawa

Kenkichi Iwasawa was a Japanese mathematician who is known for his influence on algebraic number theory.Iwasawa was born in Shinshuku-mura, a town near Kiryū, in Gunma Prefecture...

analogue of the Jacobian. The analogy was with the Jacobian variety

Jacobian variety

In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...

J of a curve C over a finite field F (qua Picard variety), where the finite field has roots of unity added to make finite field extensions F′ The local zeta-function (q.v.) of C can be recovered from the points J(F′) as Galois module. In the same way, Iwasawa added pⁿ-power roots of unity for fixed p and with n → ∞, for his analogue, to a number field K, and considered the inverse limit

Inverse limit

In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...

of class groups, finding a p-adic L-function earlier introduced by Kubota and Leopoldt.

K

K-theory

Algebraic K-theory

In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....

is on one hand a quite general theory with an abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

flavour, and, on the other hand, implicated in some formulations of arithmetic conjectures. See for example Birch-Tate conjecture, Lichtenbaum conjecture.

L

Local zeta-function

A local zeta-function

Local zeta-function

is a generating function

Generating function

for the number of points on an algebraic variety V over a finite field

Finite field

F, over the finite field extension

Field extension

In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...

s of F. According to the Weil conjectures (q.v.) these functions, for non-singular varieties, exhibit properties closely analogous to the Riemann zeta-function, including the Riemann hypothesis

Riemann hypothesis

In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

M

Mordell conjecture

The Mordell conjecture is now the Faltings theorem, and states that a curve of genus at least two has only finitely many rational points.

Mordell-Lang conjecture

The Mordell-Lang conjecture is a complex of a number of conjectures of Serge Lang unifying the Mordell conjecture and Manin-Mumford conjecture in an abelian variety

Abelian variety

or semi-abelian variety.

Mordell-Weil theorem

The Mordell-Weil theorem is a foundational result stating that for an abelian variety A over a number field K the group A(K) is a finitely-generated abelian group. This was proved initially for number fields K, but extends to all finitely-generated fields.

N

Néron-Tate height

The Néron-Tate height (also often referred to as the canonical height) on an abelian variety

Abelian variety

A is a height function (q.v.) that is essentially intrinsic, and an exact quadratic form

Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

, rather than approximately quadratic with respect to the addition on A as provided by the general theory of heights. It can be defined from a general height by a limiting process; there are also formulae, in the sense that it is a sum of local field contributions.

Q

Quasi-algebraic closure

The topic of quasi-algebraic closure, i.e. solubility guaranteed by a number of variables polynomial in the degree of an equation, grew out of studies of the Brauer group

Brauer group

In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...

and the Chevalley–Warning theorem. It stalled in the face of counterexample

Counterexample

In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....

s; but see Ax-Kochen theorem from mathematical logic

Mathematical logic

S

Sato-Tate conjecture

The Sato-Tate conjecture

Sato-Tate conjecture

In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep over the finite field with p elements, with p a prime number, obtained from an elliptic curve E over the rational number field, by the process of reduction modulo a prime for almost all p...

on elliptic curves is a conjectural result on the distribution of Frobenius elements in the Tate module

Tate module

In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G...

. It is a prototype for Galois representations in general.

Skolem's method

See Chabauty's method.

T

Tamagawa numbers

The direct Tamagawa number definition works well only for linear algebraic group

Linear algebraic group

In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...

s. There the Weil conjecture on Tamagawa numbers

Weil conjecture on Tamagawa numbers

In mathematics, the Weil conjecture on Tamagawa numbers is a result about algebraic groups formulated by André Weil in the late 1950s and proved in 1989...

was eventually proved. For abelian varieties, and in particular the Birch-Swinnerton-Dyer conjecture (q.v.), the Tamagawa number approach to a local-global principle fails on a direct attempt, though it has had heuristic value over many years. Now a sophisticated equivariant Tamagawa number conjecture is a major research problem.

Tate conjecture

The Tate conjecture

Tate conjecture

In mathematics, the Tate conjecture is a 1963 conjecture of John Tate linking algebraic geometry, and more specifically the identification of algebraic cycles, with Galois modules coming from étale cohomology...

(John Tate

John Tate

John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...

, 1963) provided an analogue to the Hodge conjecture

Hodge conjecture

The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...

, also on algebraic cycle

Algebraic cycle

In mathematics, an algebraic cycle on an algebraic variety V is, roughly speaking, a homology class on V that is represented by a linear combination of subvarieties of V. Therefore the algebraic cycles on V are the part of the algebraic topology of V that is directly accessible in algebraic geometry...

s, but well within arithmetic geometry. It also gave, for elliptic surface

Elliptic surface

In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism to an algebraic curve, almost all of whose fibers are elliptic curves....

s, an analogue of the Birch-Swinnerton-Dyer conjecture (q.v.), leading quickly to a clarification of the latter and a recognition of its importance.

Tate curve

The Tate curve is a particular elliptic curve over the p-adic number

P-adic number

s introduced by John Tate to study bad reduction (see good reduction).

V

Vojta conjecture

The Vojta conjecture is a complex of conjectures by Paul Vojta

Paul Vojta

Paul Alan Vojta is an American mathematician, known for his work in number theory on diophantine geometry and diophantine approximation....

, making analogies between Diophantine approximation

Diophantine approximation

In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....

and Nevanlinna theory

Nevanlinna theory

Nevanlinna theory is a branch of complex analysis developed by Rolf Nevanlinna. It deals with the value distribution theory of holomorphic functions in one variable, usually denoted z....

W

Weights

The yoga of weights is a formulation by Alexander Grothendieck

Alexander Grothendieck

of analogies between Hodge theory

Hodge theory

In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...

and l-adic cohomology.

Weil cohomology

The initial idea, later somewhat modified, for proving the Weil conjectures (q.v.), was to construct a cohomology theory applying to algebraic varieties over finite field

Finite field

s that would both be as good as singular homology

Singular homology

In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....

at detecting topological structure, and have Frobenius mappings acting in such a way that the Lefschetz fixed-point theorem

Lefschetz fixed-point theorem

In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X...

could be applied to the counting in local zeta-function

Local zeta-function

s. For later history see motive (algebraic geometry)

Motive (algebraic geometry)

In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer...

, motivic cohomology

Motivic cohomology

Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s. At that time, it was conceived as a theory constructed on the basis of the so-called standard conjectures on algebraic cycles, in algebraic geometry...

Weil conjectures

The Weil conjectures

Weil conjectures

In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....

were three highly-influential conjectures of André Weil

André Weil

, made public around 1949, on local zeta-functions. The proof was completed in 1973. Those being proved, there remain extensions of the Chevalley–Warning theorem congruence, which comes from an elementary method, and improvements of Weil bounds, e.g. better estimates for curves of the number of points than come from Weil's basic theorem of 1940. The latter turn out to be of interest for Goppa codes.

Weil distributions on algebraic varieties

André Weil proposed a theory in the 1920s and 1930s on prime ideal

Prime ideal

In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

decomposition of algebraic numbers in co-ordinates of points on algebraic varieties. It has remained somewhat under-developed.

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Further reading