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In algebraic geometry and commutative algebra, the Zariski topology is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring.
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In mathematics, Diophantine geometry is the study of points of algebraic varieties with coordinates in the integers, rational numbers, and their generalizations. These generalizations typically are fields that are not algebraically closed, such as number fields, finite fields, function fields, and p-adic fields. It is a sub-branch of arithmetic geometry and is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry.
Serge Lang was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He received the Frank Nelson Cole Prize in 1960 and was a member of the Bourbaki group.
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
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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 variety of general type.
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Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.
In mathematics, in the field of algebraic number theory, an S-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for S-units.
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In arithmetic geometry, the Tate–Shafarevich groupШ(A/K), introduced by Serge Lang and John Tate (1958) and Igor Shafarevich (1959), of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group WC(A/K) = H1(GK, A) that become trivial in all of the completions of K (i.e. the p-adic fields obtained from K, as well as its real and complex completions). Thus, in terms of Galois cohomology, it can be written as
In mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyperbolic manifolds are an important class of complex manifolds, defined by the property that the Kobayashi pseudometric is a metric. Kobayashi hyperbolicity of a complex manifold X implies that every holomorphic map from the complex line C to X is constant.
In mathematics, Vojta's conjecture is a conjecture introduced by Paul Vojta (1987) about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory in complex analysis. It implies many other conjectures in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic.
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.
References
- Lang, Serge (1986). "Hyperbolic and Diophantine analysis" (PDF). Bulletin of the American Mathematical Society . 14 (2): 159–205. doi:10.1090/s0273-0979-1986-15426-1. Zbl 0602.14019.CS1 maint: discouraged parameter (link)
- Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8.CS1 maint: discouraged parameter (link)