Bogomolov conjecture

Last updated May 14, 2024

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 proven by Emmanuel Ullmo and Shou-Wu Zhang in 1998 using Arakelov theory. A further generalization to general abelian varieties was also proved by Zhang in 1998.

Statement

Let C be an algebraic curve of genus g at least two defined over a number field K, let ${\overline {K}}$ denote the algebraic closure of K, fix an embedding of C into its Jacobian variety J, and let ${\hat {h}}$ denote the Néron-Tate height on J associated to an ample symmetric divisor. Then there exists an $\epsilon >0$ such that the set

\{P\in C({\overline {K}}):{\hat {h}}(P)<\epsilon \}

is finite.

Since ${\hat {h}}(P)=0$ if and only if P is a torsion point, the Bogomolov conjecture generalises the Manin-Mumford conjecture.

Proof

The original Bogomolov conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang using Arakelov theory in 1998.^[1]^[2]

Generalization

In 1998, Zhang proved the following generalization:^[2]

Let A be an abelian variety defined over K, and let ${\hat {h}}$ be the Néron-Tate height on A associated to an ample symmetric divisor. A subvariety $X\subset A$ is called a torsion subvariety if it is the translate of an abelian subvariety of A by a torsion point. If X is not a torsion subvariety, then there is an $\epsilon >0$ such that the set

\{P\in X({\overline {K}}):{\hat {h}}(P)<\epsilon \}

is not Zariski dense in X.

Related Research Articles

Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field $of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and known as the Mordell conjecture until its 1983 proof by Gerd Faltings. The conjecture was later generalized by replacing by any number field.$

In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research on other topics in algebraic geometry and number theory.

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 Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds.

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.

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 representation on étale cohomology. The conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the Hodge conjecture.

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 algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence between line bundles and divisors, there is an equivalent notion of a nef divisor.

Shou-Wu Zhang is a Chinese-American mathematician known for his work in number theory and arithmetic geometry. He is currently a Professor of Mathematics at Princeton University.

In number theory, the Néron–Tate height is a quadratic form on the Mordell–Weil group of rational points of an abelian variety defined over a global field. It is named after André Néron and John Tate.

In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety.

In mathematics, Arakelov theory is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.

Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, $p$ -adic, 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, 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 theorem. The conjecture concerns itself with a characterization of the Zariski closure of sets of special points in Shimura varieties. A special case of the conjecture was stated by Yves André in 1989 and a more general statement was conjectured by Frans Oort in 1995. The modern version is a natural generalization of these two conjectures.

In mathematics, Vojta's conjecture is a conjecture introduced by Paul Vojta 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 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 respect to the embedding defined by the divisor. The concept is named after Rolf Nevanlinna.

Emmanuel Ullmo is a French mathematician, specialised in arithmetic geometry. Since 2013 he has served as director of the Institut des Hautes Études scientifiques.

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 tori and semiabelian varieties it was proposed by Boris Zilber and independently by Enrico Bombieri, David Masser, Umberto Zannier in the early 2000's. For semiabelian varieties the conjecture implies the Mordell–Lang and Manin–Mumford conjectures. Richard Pink proposed (again independently) a more general conjecture for Shimura varieties which also implies the André–Oort conjecture. In the case of algebraic tori, Zilber called it the Conjecture on Intersection with Tori (CIT). The general version is now known as the Zilber–Pink conjecture. It states roughly that atypical or unlikely intersections of an algebraic variety with certain special varieties are accounted for by finitely many special varieties.

References

↑ Ullmo, Emmanuel (1998), "Positivité et Discrétion des Points Algébriques des Courbes", Annals of Mathematics , 147 (1): 167–179, arXiv: alg-geom/9606017 , doi:10.2307/120987, Zbl 0934.14013 .
1 2 Zhang, S.-W. (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics , 147 (1): 159–165, doi:10.2307/120986

Other sources

Chambert-Loir, Antoine (2013). "Diophantine geometry and analytic spaces". In Amini, Omid; Baker, Matthew; Faber, Xander (eds.). Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011. Contemporary Mathematics. Vol. 605. Providence, RI: American Mathematical Society. pp. 161–179. ISBN 978-1-4704-1021-6. Zbl 1281.14002.