Algebraic geometry
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Number theory
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Diophantine equation
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History of number theory
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Rational point
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...
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 ...
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 th...
Conductor of an abelian variety - Wikipedia
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...
Severi–Brauer variety
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 varie...
Chevalley–Warning theorem
In algebra, 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 (1936) a...
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.
They give conditions for the group G(k) to be dense ...
Siegel's lemma
In transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence o...
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...
Glossary of arithmetic and Diophantine geometry - Wikipedia
Rational point
In number theory, a rational point is a point in space each of whose coordinates are rational; that is, the coordinates of the point are elements of the field of rational numbers, as well as being ele...
Birch and Swinnerton-Dyer conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. It is widely recognized as one of the most challenging mathematical problems; the conjecture ...
Mordellic variety
Mordell–Weil theorem
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...
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 Galoi...
Weil conjecture on Tamagawa numbers
In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number τ(G) of a simply connected simple algebraic group defined over a number field is 1. Weil (1959) ...
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 equationTherefore in terms of the affine plane its equa...
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 ell...
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...
Bogomolov conjecture
In mathematics, the Bogomolov conjecture, named for Fedor Bogomolov, is the following statement:Let C be an algebraic curve of genus g at least two defined over a number field K, let denote the algeb...
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 p...
Faltings' theorem
In number theory, the Mordell conjecture is the conjecture made by Mordell (1922) that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. T...
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.
Arakelov 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.
If S is a set with height function H, s...
Mazur's torsion theorem
In algebraic geometry, Mazur's torsion theorem, due to Barry Mazur, classifies the possible torsion subgroups of the group of rational points on an elliptic curve defined over the rational numbers. If...
Mazur's torsion theorem - Wikipedia
Local zeta-function
Suppose that V is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements. In the number theory, the local zeta function Z(V, s) of V (or, sometimes called the...
Local zeta-function - Wikipedia
Manin obstruction
In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction (named after Yuri Manin) is attached to a geometric object X which measures the failure of the Hasse principle for ...
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, p, p, and so on.
For a prime number p let K be a p-adic field, i...
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. Equivalently, a principal homogeneous ...
Integer lattice
In mathematics, the n-dimensional integer lattice (or cubic lattice), denoted Z, is the lattice in the Euclidean space R whose lattice points are n-tuples of integers. The two-dimensional integer latt...