This is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality. Euclid, detail from The School of Athens by Raphael. ... In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove. ...

Diophantine geometry 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 fields — and over local fields. Of those, only the complex numbers 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. In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. ... This article is about algebraic varieties. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ...

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's schemes of finite type over the spectrum of the ring of integers Z. This point of view has been very influential for around half a century; it has very widely been regarded as fulfilling Leopold Kronecker's ambition to have number theory operate only with rings that are quotients of polynomial rings over the integers (to use the current language of 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 ideals as the p-adic numbers do. In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form a0xn + a1xn−1 + ··· + an −1x + an = 0 where n is a positive integer called the degree... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... The p-adic number systems were first described by Kurt Hensel in 1897. ...

abc conjecture

The abc conjecture attempts to state as much as possible about repeated prime factors in an equation a + b = c. For example 8 + 1 = 9 but the prime powers here are exceptional. The abc conjecture in number theory was first formulated by Joseph Oesterlé and David Masser in 1985. ...

Arakelov theory

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

Arithmetic of abelian varieties

See main article 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. ...

Artin L-functions

Artin L-functions are defined for quite general Galois representations. The introduction of étale cohomology 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. 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 1920s by Emil Artin, in connection with his research into class field theory. ... In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group — equivalently for a Galois group G and a group ring R[G] of G with respect to some ring R, it is some R[G]-module M. In that general sense... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ...

Bad reduction

See good reduction.

Birch-Swinnerton-Dyer conjecture

The Birch–Swinnerton-Dyer conjecture 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 In mathematics, elliptic curves are defined by certain cubic (the superscript exponent is three, a. ...

Bombieri-Lang conjecture

Enrico Bombieri, Serge Lang and Paul Vojta 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 to it exists, that is not constant. Examples include compact Riemann surfaces of genus g > 1. Lang conjectured that V is analytically holomorphic if and only if all subvarieties are of general type. Enrico Bombieri (born November 26, 1940) is a Italian mathematician, born in Milan. ... Serge Lang (May 19, 1927–September 12, 2005) was a French-born American mathematician. ... Paul Vojta is an American mathematician, known for his work in number theory on diophantine geometry and diophantine approximation. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. ...

Canonical height

The canonical height on an abelian variety is a height function that is a distinguished quadratic form. See Néron-Tate height. In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

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's method for an algebraic torus. (Other older methods for Diophantine problems include Runge's method.) In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. ... Albert Thoralf Skolem (May 23, 1887 - March 23, 1963) was a Norwegian mathematician. ... In mathematics, an algebraic torus over a field K is an algebraic group which is isomorphic over the algebraic closure of K to (GL1)r for some integer r, the rank of the torus. ...

Crystalline cohomology

Crystalline cohomology is a p-adic cohomology theory in characteristic p, introduced by Alexander Grothendieck to fill the gap left by 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. (See also crystal (mathematics).) In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... Crystal in homological algebra is a diagram with points and arrows. ...

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-functions are computed in terms of Jacobi sums. Waring's problem is the most classical case. This article is about algebraic varieties. ... In number theory, a local zeta-function is a generating function Z(t) for the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F. The analogy with the Riemann zeta-function comes via consideration of the logarithmic derivative . Given... In mathematics, a Jacobi sum is a type of character sum formed with one or more Dirichlet characters. ... In number theory, Warings 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. ...

Dwork's method

Bernard Dwork used distinctive methods of p-adic analysis, p-adic algebraic differential equations, Koszul complexes and other techniques that have not all been absorbed into general theories such as crystalline cohomology (q.v.). He first proved the rationality of local zeta-functions, the initial advance in the direction of the Weil conjectures (q.v.) Bernard Morris Dwork (May 27, 1923 – May 9, 1998) 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. ... P-adic analysis (p-adic analysis) is a branch of mathematics that deals with functions of p-adic numbers. ... In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In number theory, a local zeta-function is a generating function Z(t) for the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F. The analogy with the Riemann zeta-function comes via consideration of the logarithmic derivative . Given...

Étale cohomology

The search for a Weil cohomology (q.v.) was at least partially fulfilled in the étale cohomology theory of Alexander Grothendieck and Michael Artin. It provided a proof of the functional equation for the local zeta-functions, and was basic in the formulation of the Tate conjecture (q.v.) and numerous other theories. In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. ... Michael Artin is an American mathematician, known for his contributions to algebraic geometry. ... In mathematics, the L-functions of number theory have certain functional equations, as one of their characteristic properties. ... In number theory, a local zeta-function is a generating function Z(t) for the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F. The analogy with the Riemann zeta-function comes via consideration of the logarithmic derivative . Given...

Fermat's last theorem

Fermat's last theorem, the most celebrated conjecture of Diophantine geometry, was proved by Andrew Wiles and Richard Taylor. Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats last theorem (edition of 1670). ... Sir Andrew John Wiles (April 11, 1953) is a British-American mathematician. ... Richard Taylor is a British mathematician working in the field of number theory. ...

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 has been considered the 'right' foundational topos for scheme theory goes back to the fact of faithfully-flat descent, the discovery of Grothendieck that the representable functors are sheaves for it (i.e. a very general gluing axiom holds). Sheaves were introduced into mathematics in the 1940s and, a major theme since then has been to study a space by studying sheaves on that space. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, especially in category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ... 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 F: Open(X) → C to a category C which initially one takes to be the category of...

Function field analogy

It was realised in the nineteenth century that the ring of integers 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 fields should all be treated on the same basis. The idea goes further. Thus elliptic surfaces over the complex numbers, also, have some quite strict analogies with elliptic curves over number fields. In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form a0xn + a1xn−1 + ··· + an −1x + an = 0 where n is a positive integer called the degree... In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... The term global field refers to either of the following: a number field, i. ... In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected smooth morphism to an algebraic curve, almost all of whose fibers are elliptic curves. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ...

Geometric class field theory

The extension of class field theory-style results on abelian coverings to varieties of dimension at least two is often called geometric class field theory. In mathematics, class field theory is a major branch of algebraic number theory. ...

Good reduction

Fundamental to local analysis in arithmetic problems is to reduce modulo all prime numbers p. In the typical situation this presents little difficulty for almost all p; for example denominators of fractions are tricky, in that reduction modulo a prime in the denominator looks like division by zero, but that rules out only finitely many p per fraction. With a little extra sophistication, homogeneous coordinates 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 enters: a non-singular point may become a singular point on reduction modulo p, because the Zariski tangent space 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, potential good reduction, Tate curve, semistable elliptic curve, Ogg-Néron-Shafarevich criterion, Serre-Tate theorem. 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. ... The word modulo is the Latin ablative of modulus. ... In mathematics, the phrase almost all has a number of specialised uses. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... In mathematics, a division is called a division by zero if the divisor is zero. ... In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. ... For non-mathematical singularity theories, see singularity. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space, at a point P on an algebraic variety V (and more generally). ... In mathematics, a semistable elliptic curve in diophantine geometry is an elliptic curve that has bad reduction only of multiplicative type. ...

Grothendieck-Katz conjecture

The Grothendieck-Katz p-curvature conjecture applies reduction modulo primes to algebraic differential equations, to derive information on algebraic function solutions. It is an open problem as of 2005. The initial result of this type was Eisenstein's theorem. In mathematics, an algebraic function of indeterminates X1, X2, ..., Xn, is a function F that satisfies some non-trivial equation P(F, X1, X2, ..., Xn) = 0, with P a polynomial in n + 1 variables over a given field K. That is, F is an implicit function that solves an algebraic... 2005 is a common year starting on Saturday of the Gregorian calendar. ...

Hasse principle

The Hasse principle states that solubility for a global field is the same as solubility in all relevant local fields. 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. 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 forms in small numbers of variables (and in particular for elliptic curves as cubic curves) are at a general level connected with the limitations of the analytic approach. In mathematics, Helmut Hasses local-global principle, also known as the Hasse principle, is the assertion that an equation can be solved over the rational numbers if and only if it can be solved over the real numbers and over the p-adic numbers for every prime p. ... The term global field refers to either of the following: a number field, i. ... In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ... In mathematics, the Hardy-Littlewood circle method is one of the most frequently used techniques of analytic number theory. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In mathematics, a cubic curve is a plane curve C defined by a cubic equation F(X,Y,Z) = 0 applied to homogeneous coordinates [X:Y:Z] for the projective plane; or the inhomogeneous version for the affine space determined by setting Z = 1 in such an equation. ...

Hasse-Weil L-function

A Hasse-Weil L-function, sometimes called a global L-function, is an Euler product formed from local zeta-functions. The properties of such L-functions 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. In mathematics, the Hasse-Weil zeta function attached to an algebraic variety V defined over a number field K is one of the most important types of L-function. ... In mathematics, an Euler product is an infinite product expansion, indexed by prime numbers p, of a Dirichlet series. ... The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. ... The Taniyama-Shimura theorem establishes an important connection between elliptic curves, which are objects from algebraic geometry, and modular forms, which are certain periodic holomorphic functions investigated in number theory. ... In mathematics, the Langlands program is a web of far-reaching and influential conjectures that connect number theory and the representation theory of certain groups. ...

Height function

A height function in Diophantine geometry quantifies the size of solutions to Diophantine equations. It is standard to take a logarithmic scale: that is, the height is proportional to the number of bits a computer needs to store a point in homogeneous coordinates. Heights were initially developed by André Weil 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 bundles are in pure geometry. A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself. ... In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... In mathematics, in algebraic geometry or the theory of complex manifolds, a very ample line bundle is one with enough sections to set up an embedding of its base variety or manifold into projective space. ...

Hilbertian fields

A Hilbertian field K is one for which the projective spaces over K are not thin sets in the sense of Jean-Pierre Serre. This is a geometric take on Hilbert's irreducibility theorem which shows the rational numbers are Hilbertian. Results are applied to the inverse Galois problem. Thin sets (the French word is mince) are in some sense analogous to the meagre sets (French maigre) of the Baire category theorem. In mathematics, a projective space is a fundamental construction from any vector space. ... In mathematics, a thin set in the sense of 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 may not be... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... In mathematics, Hilberts irreducibility theorem is a result of David Hilbert, stating that an irreducible polynomial in two variables and having rational number coefficients will remain irreducible as a polynomial in one variable, when a rational number is substituted for the other variable, in infinitely many ways. ... In mathematics, the inverse Galois problem concerns whether or not we can find a rational field extension with a given Galois group. ... In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ... The Baire category theorem is an important tool in general topology and functional analysis. ...

Igusa zeta-function

An Igusa zeta-function, named for Jun-ichi Igusa, is a generating function counting numbers of points on an algebraic variety modulo high powers pⁿ of a fixed prime number p. General rationality theorems are now known, drawing on methods of mathematical logic. In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...

Infinite descent

Infinite descent was Pierre de Fermat'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 spaces (often called 'descents', when written out by equations); in more modern terms in a Galois cohomology group which is to be proved finite. See Selmer group. In mathematics, a proof by infinite descent is a particular kind of proof by mathematical induction. ... Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, southern France, and a mathematician who is given credit for the development of modern calculus. ... In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ... 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. ... In mathematics, the Weil-Châtelet group of an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. It is named for André Weil, who introduced the general group operation in it, and F. Châtelet. ...

Iwasawa theory

Iwasawa theory builds up from the analytic number theory and Stickelberger's theorem as a theory of ideal class groups as Galois modules and p-adic L-functions (with roots in Kummer congruence on Bernoulli numbers). In its early days in the late 1960s it was called Iwasawa's analogue of the Jacobian. The analogy was with the Jacobian variety Jof 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 of class groups, finding a p-adic L-function earlier introduced by Kubota and Leopoldt. In number theory, Iwasawa theory is a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa as part of the theory of cyclotomic fields. ... Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ... In mathematics, Stickelbergers theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. ... In mathematics the theory of algebraic number fields gives rise to a finite abelian group constructed from each such field, its ideal class group. ... In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group — equivalently for a Galois group G and a group ring R[G] of G with respect to some ring R, it is some R[G]-module M. In that general sense... In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ... Kenkichi Iwasawa (岩澤 健吉 Iwasawa Kenkichi, September 11, 1917 - October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory. ... For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group — equivalently for a Galois group G and a group ring R[G] of G with respect to some ring R, it is some R[G]-module M. In that general sense... In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ...

K-theory

Algebraic K-theory is on one hand a quite general theory with an abstract algebra flavour, and, on the other hand, implicated in some formulations of arithmetic conjectures. See for example Birch-Tate conjecture, Lichtenbaum conjecture. In mathematics, algebraic K-theory is an advanced part of homological algebra concerned with defining and applying a sequence Kn(R) of functors from rings to abelian groups, for n = 0,1,2, ... . Here for traditional reasons the cases of K0 and K1 are thought of in somewhat different terms... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ...

Local zeta-function

A local zeta-function is a generating function for the number of points on an algebraic variety V over a finite field F, over the finite field extensions 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. In number theory, a local zeta-function is a generating function Z(t) for the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F. The analogy with the Riemann zeta-function comes via consideration of the logarithmic derivative . Given... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In abstract algebra, a subfield of a field L is a subset K of L which is closed under the addition and multiplication operations of L and itself forms a field with these operations. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. ... In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous of all unsolved problems. ...

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. In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. ...

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 or semi-abelian variety. In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. ... In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ...

Mordell-Weil theorem

The Mordell-Weil theorem is a foundational result stating that for an abelian variety A over a 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. In mathematics, the Mordell-Weil theorem states that for an abelian variety A over a number field K, the group A(K) of K-rational points of A is a finitely-generated abelian group. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ...

Néron-Tate height

The Néron-Tate height on an abelian variety A is a height function (q.v.) that is essentially intrinsic, and an exact quadratic form, 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. In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

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 and the Chevalley-Warning theorem. It stalled in the face of counterexamples; but see Ax-Kochen theorem from mathematical logic. In mathematics, quasi-algebraic closure of a field F is the property that for every homogeneous polynomial P over F in indeterminates X1, ..., XN, and of degree d satisfying d < N has a non-trivial zero over F; that is, for some xi in F, not all 0, we have... In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer. ... In mathematics, Chevalleys theorem on solutions of polynomial equations over a finite field F with q elements, q a power of the prime number p, states that for a polynomial P(X1, ..., XN) of total degree d, with d < N, the number M of solutions of P(X1, ..., XN... In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...

Reduction modulo a prime number or ideal

See good reduction.

Sato-Tate conjecture

The Sato-Tate conjecture on elliptic curves is a conjecture result on the distribution of Frobenius elements in the Tate module. It is a prototype for Galois representations in general. 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... In mathematics, the Frobenius automorphism is an automorphism induced by a prime power mapping defined for various extensions of fields. ... In mathematics, a Tate module is a Galois module constructed from an abelian variety A over a field K. It is denoted Tl(A) where l is a given prime number (the letter p is traditionally reserved for the characteristic of K; the case where K has characteristic p is... In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group — equivalently for a Galois group G and a group ring R[G] of G with respect to some ring R, it is some R[G]-module M. In that general sense...

Skolem's method

See Chabauty's method.

Tamagawa numbers

The direct Tamagawa number definition works well only for linear algebraic groups. There the Weil conjecture on Tamagawa numbers 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. In mathematics, an adelic algebraic group is a topological group defined by an algebraic group G over a number field K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only... In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations. ... In mathematics, the Weil conjecture on Tamagawa numbers was formulated by André Weil in the late 1950s. ... In mathematics, Helmut Hasses local-global principle, also known as the Hasse principle, is the assertion that an equation can be solved over the rational numbers if and only if it can be solved over the real numbers and over the p-adic numbers for every prime p. ...

Tate conjecture

The Tate conjecture (John Tate, 1963) provided an analogue to the Hodge conjecture, also on algebraic cycles, but well within arithmetic geometry. It also gave, for elliptic surfaces, an analogue of the Birch-Swinnerton-Dyer conjecture (q.v.), leading quickly to a clarification of the latter and a recognition of its importance. 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. ... You may be looking for John Tate (boxer) John Torrence Tate, born March 13, 1925 in Minneapolis, Minnesota, is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. ... The Hodge conjecture is a major unsolved problem of algebraic geometry. ... In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected smooth morphism to an algebraic curve, almost all of whose fibers are elliptic curves. ...

Tate curve

The Tate curve is a particular elliptic curve over the p-adic numbers introduced by John Tate to study bad reduction (see good reduction). The p-adic number systems were first described by Kurt Hensel in 1897. ... You may be looking for John Tate (boxer) John Torrence Tate, born March 13, 1925 in Minneapolis, Minnesota, is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. ...

Vojta conjecture

The Vojta conjecture is a complex of conjectures by Paul Vojta, making analogies between Diophantine approximation and Nevanlinna theory. Paul Vojta is an American mathematician, known for his work in number theory on diophantine geometry and 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. ... In mathematics, the value distribution theory of holomorphic functions is a division of mathematical analysis. ...

Weights

The yoga of weights is a formulation by Alexander Grothendieck of analogies between Hodge theory and l-adic cohomology. Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. ... In mathematics, Hodge theory 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 Laplacian operators associated to a Riemannian metric... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ...

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 fields that would both be as good as singular homology at detecting topological structure, and have Frobenius mappings acting in such a way that the Lefschetz fixed-point theorem could be applied to the counting in local zeta-functions. For later history see motive (algebraic geometry), motivic cohomology. In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ... In mathematics, the Lefschetz fixed-point theorem counts the number of fixed points of a mapping from a topological space X to itself (subject to some mild conditions on X), by means of traces of the induced mappings on the homology groups of X. The counting is subject to some... In number theory, a local zeta-function is a generating function Z(t) for the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F. The analogy with the Riemann zeta-function comes via consideration of the logarithmic derivative . Given... In algebraic geometry the idea of a motive intuitively refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ... Motivic cohomology is a homological theory in mathematics, the existence of which was first conjectured by Grothendieck during the 1960s. ...

Weil conjectures

The Weil conjectures were three highly-influential conjectures of 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. In mathematics, the Weil conjectures, which had become theorems by 1975, were some highly-influential proposals from the late 1940s by Andre Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... In number theory, a local zeta-function is a generating function Z(t) for the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F. The analogy with the Riemann zeta-function comes via consideration of the logarithmic derivative . Given... In mathematics, Chevalleys theorem on solutions of polynomial equations over a finite field F with q elements, q a power of the prime number p, states that for a polynomial P(X1, ..., XN) of total degree d, with d < N, the number M of solutions of P(X1, ..., XN... In mathematics, a Goppa code is a general type of linear code constructed by using an algebraic curve X over a finite field . ...

Weil distributions on algebraic varieties

André Weil proposed a theory in the 1920s and 1930s on prime ideal decomposition of algebraic numbers in co-ordinates of points on algebraic varieties. It has remained somewhat under-developed. In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...