# Category: Diophantine geometry

Manin obstruction
In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction (named after Yuri Manin) is attached to a variety X over a global field, which measures the failure of the Hasse pr
Field of definition
In mathematics, the field of definition of an algebraic variety V is essentially the smallest field to which the coefficients of the polynomials defining V can belong. Given polynomials, with coeffici
Mordell–Weil theorem
In mathematics, the Mordell–Weil theorem states that for an abelian variety over a number field , the group of K-rational points of is a finitely-generated abelian group, called the Mordell–Weil group
Chevalley–Warning theorem
In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by Ewald Warning and a slight
Severi–Brauer variety
In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central
Bombieri–Lang conjecture
In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variet
Height function
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typica
Tsen rank
In mathematics, the Tsen rank of a field describes conditions under which a system of polynomial equations must have a solution in the field. The concept is named for C. C. Tsen, who introduced their
Bogomolov conjecture
In mathematics, the Bogomolov conjecture is a conjecture, named after Fedor Bogomolov, in arithmetic geometry about algebraic curves that generalizes the Manin-Mumford conjecture in arithmetic geometr
Torsion conjecture
In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that the order of the torsion group of an abelian vari
Mordell–Weil group
In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety defined over a number field , it is an arithmetic invariant of the Abelian variety. It is simply th
Nevanlinna invariant
In mathematics, the Nevanlinna invariant of an ample divisor D on a normal projective variety X is a real number connected with the rate of growth of the number of rational points on the variety with
Weil's conjecture on Tamagawa numbers
In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number of a simply connected simple algebraic group defined over a number field is 1. In this case, simply co
Conductor of an abelian variety
In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the
Semistable abelian variety
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. For an abelian vari
Diophantine geometry
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods
Approximation in algebraic groups
In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraic groups G over global fields k.
Local zeta function
In number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as where V is a non-singular n-dimensional projective algebr
Arithmetic of abelian varieties
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what a
Faltings's theorem
In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points.
Arithmetic surface
In mathematics, an arithmetic surface over a Dedekind domain R with fraction field is a geometric object having one conventional dimension, and one other dimension provided by the infinitude of the pr
Manin conjecture
In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his
A Pythagorean quadruple is a tuple of integers a, b, c, and d, such that a2 + b2 + c2 = d2. They are solutions of a Diophantine equation and often only positive integer values are considered. However,
André–Oort conjecture
In mathematics, the André–Oort conjecture is a problem in Diophantine geometry, a branch of number theory, that can be seen as a non-abelian analogue of the Manin–Mumford conjecture, which is now a th
Rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is
Birch and Swinnerton-Dyer conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an o
Integer lattice
In mathematics, the n-dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are n-tuples of integers. The two-dimensional integer lattice
Principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous
Quasi-algebraically closed field
In mathematics, a field F is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than it
Arakelov theory
In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
Fermat curve
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation Therefore, in terms of the affine plane its eq
Tate conjecture
In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois
Glossary of arithmetic and diophantine geometry
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 ge
Height zeta function
In mathematics, the height zeta function of an algebraic variety or more generally a subset of a variety encodes the distribution of points of given height.
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.
Zilber-Pink conjecture
In mathematics, the Zilber–Pink conjecture is a far-reaching generalisation of many famous Diophantine conjectures and statements, such as André-Oort, Manin–Mumford, and Mordell-Lang. For algebraic to
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
Siegel's lemma
In mathematics, specifically in transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxilia
Mordellic variety
In mathematics, a Mordellic variety is an algebraic variety which has only finitely many points in any finitely generated field. The terminology was introduced by Serge Lang to enunciate a range of co