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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. In 1983 it was proved by Gerd Faltings, and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field.
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 space for a group G is a non-empty set X on which G acts freely and transitively . An analogous definition holds in other categories, where, for example,
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 are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety A over a number field K; or more generally.
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer.
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. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.
Igor Rostislavovich Shafarevich was a Russian mathematician who contributed to algebraic number theory and algebraic geometry. He wrote books and articles that criticised socialism, and he was an important dissident during the Soviet regime.
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. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.
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 simple algebras in such a way that the algebra splits over K if and only if the variety has a point rational over K. Francesco Severi (1932) studied these varieties, and they are also named after Richard Brauer because of their close relation to the Brauer group.
Yuri Ivanovich Manin is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Moreover, Manin was one of the first to propose the idea of a quantum computer in 1980 with his book Computable and Uncomputable.
In arithmetic geometry, the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. John Tate (1958) named it for François Châtelet (1946) who introduced it for elliptic curves, and André Weil (1955), who introduced it for more general groups. It plays a basic role in the arithmetic of abelian varieties, in particular for elliptic curves, because of its connection with infinite descent.
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 generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point.
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.
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 typically functions from a set of points on algebraic varieties to the real numbers.
In mathematics, Arakelov theory is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
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 geometry. The conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang in 1998. A further generalization to general abelian varieties was also proved by Zhang in 1998.
Alexei Nikolaievich Skorobogatov is a British-Russian mathematician and Professor in Pure Mathematics at Imperial College London specialising in algebraic geometry. His work has focused on rational points, the Hasse principle, the Manin obstruction, exponential sums, and error-correcting codes.
In arithmetic geometry, the Tate–Shafarevich groupШ(A/K) 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, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by Emil Artin as an expression appearing in the functional equation of an Artin L-function.
In mathematics, the Néron–Ogg–Shafarevich criterion states that if A is an elliptic curve or abelian variety over a local field K and ℓ is a prime not dividing the characteristic of the residue field of K then A has good reduction if and only if the ℓ-adic Tate module Tℓ of A is unramified. Andrew Ogg (1967) introduced the criterion for elliptic curves. Serre and Tate (1968) used the results of André Néron (1964) to extend it to abelian varieties, and named the criterion after Ogg, Néron and Igor Shafarevich.
In mathematics, the Hasse invariant of an algebra is an invariant attached to a Brauer class of algebras over a field. The concept is named after Helmut Hasse. The invariant plays a role in local class field theory.
References
- Serge Lang (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 250–258. ISBN 3-540-61223-8. Zbl 0869.11051.
- Alexei N. Skorobogatov (1999). Appendix A by S. Siksek: 4-descent. "Beyond the Manin obstruction". Inventiones Mathematicae . 135 (2): 399–424. arXiv: alg-geom/9711006 . Bibcode:1999InMat.135..399S. doi:10.1007/s002220050291. Zbl 0951.14013.
- Alexei Skorobogatov (2001). Torsors and rational points. Cambridge Tracts in Mathematics. 144. Cambridge: Cambridge University Press. pp. 1–7, 112. ISBN 0-521-80237-7. Zbl 0972.14015.
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