Thin set (Serre)

Thin set (Serre)

In mathematics, a thin set in the sense of Serre is a certain kind of subset constructed in algebraic geometry over a given field "K", by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or may not be the case; solving within "K" a polynomial that does not always factorise. One is also allowed to take finite unions.

More precisely, let "V" be an algebraic variety over "K" (assumptions here are: "V" is an irreducible set, a quasi-projective variety, and "K" has characteristic zero). A type I thin set is a subset of "V"("K") that is not Zariski-dense. That means it lies in an algebraic set that is a finite union of algebraic varieties of dimension lower than "d", the dimension of "V". A type II thin set is an image of an algebraic morphism (essentially a polynomial mapping) φ, applied to the "K"-points of some other "d"-dimensional algebraic variety "V"′, that maps essentially onto "V" as a ramified covering with degree "e" > 1. Saying this more technically, a thin set of type II is any subset of

:φ("V"′("K"))

where "V"′ satisfies the same assumptions as "V" and φ is generically surjective from the geometer's point of view. At the level of function fields we therefore have

: ["K"("V"): "K"("V"′)] = "e" > 1.

While a typical point "v" of "V" is φ("u") with "u" in "V"′, from "v" lying in "K"("V") we can conclude typically only that the coordinates of "u" come from solving a degree "e" equation over "K". The whole object of the theory of thin sets is then to understand that the solubility in question is a rare event. This reformulates in more geometric terms the classical Hilbert irreducibility theorem.

A thin set, in general, is a finite union of thin sets of types I and II. A Hilbertian variety "V" over "K" is one for which "V"("K") is "not" thin. A field "K" is Hilbertian if any Hilbertian variety "V" exists over it. The rational number field Q is Hilbertian, because the Hilbert irreducibility theorem has as a corollary that the projective line over Q is Hilbertian. Being Hilbertian is at the other end of the scale from being algebraically closed: the complex numbers have all sets thin, for example. They, with the other local fields (real numbers, p-adic numbers) are "not" Hilbertian. Any algebraic number field is Hilbertian.

A result of S. D. Cohen, based on the large sieve method, justifies the "thin" terminology by counting points by height function and showing, in a strong sense, that a thin set contains a low proportion of them (this is discussed at length in Serre's "Lectures on the Mordell-Weil theorem").

Colliot-Thélène conjecture

A conjecture of Jean-Louis Colliot-Thélène is that any smooth "K"-unirational variety over a number field "K" is Hilbertian. It is known that this would have the consequence that the inverse Galois problem over Q can be solved for any finite group "G".

WWA property

The WWA property (weak 'weak approximation', "sic") for a variety "V" over a number field is weak approximation (cf. approximation in algebraic groups), for finite sets of places of "K" avoiding some given finite set. For example take "K" = Q: it is required that "V"(Q) be dense in

:Π "V"(Q"p")

for all products over finite sets of prime numbers "p", not including any of some set {"p"1, ..., "p""M"} given once and for all. Ekedahl has proved that WWA for "V" implies "V" is Hilbertian. In fact Colliot-Thélène conjectures WWA, which is therefore a stronger statement.

References

*J.-P. Serre, "Lectures on the Mordell-Weil Theorem" (1989)
*J.-P. Serre, "Topics in Galois Theory" (1992)


Wikimedia Foundation. 2010.

См. также в других словарях:

  • List of exceptional set concepts — This is a list of exceptional set concepts. In mathematics, and in particular in mathematical analysis, it is very useful to be able to characterise subsets of a given set X as small , in some definite sense, or large if their complement in X is… …   Wikipedia

  • Jean-Pierre Serre — Infobox Scientist name = Jean Pierre Serre birth date = birth date and age|1926|09|15 birth place = Bages, Pyrénées Orientales, France residence = Paris, France nationality = France field = Mathematics work institutions = Centre National de la… …   Wikipedia

  • List of mathematics articles (T) — NOTOC T T duality T group T group (mathematics) T integration T norm T norm fuzzy logics T schema T square (fractal) T symmetry T table T theory T.C. Mits T1 space Table of bases Table of Clebsch Gordan coefficients Table of divisors Table of Lie …   Wikipedia

  • Hilbert's irreducibility theorem — In mathematics, Hilbert s irreducibility theorem, conceived by David Hilbert, states that an irreducible polynomial in two variables and having rational number coefficients will remain irreducible as a polynomial in one variable, when a rational… …   Wikipedia

  • literature — /lit euhr euh cheuhr, choor , li treuh /, n. 1. writings in which expression and form, in connection with ideas of permanent and universal interest, are characteristic or essential features, as poetry, novels, history, biography, and essays. 2.… …   Universalium

  • An Inconvenient Truth — Une vérité qui dérange Une vérité qui dérange (An Inconvenient Truth, titre en anglais) est un film documentaire américain traitant du changement climatique, spécialement du réchauffement planétaire, réalisé par Davis Guggenheim. Al Gore, ancien… …   Wikipédia en Français

  • An inconvenient truth — Une vérité qui dérange Une vérité qui dérange (An Inconvenient Truth, titre en anglais) est un film documentaire américain traitant du changement climatique, spécialement du réchauffement planétaire, réalisé par Davis Guggenheim. Al Gore, ancien… …   Wikipédia en Français

  • Une Vérité qui dérange — (An Inconvenient Truth, titre en anglais) est un film documentaire américain traitant du changement climatique, spécialement du réchauffement planétaire, réalisé par Davis Guggenheim. Al Gore, ancien vice président des États Unis d Amérique et… …   Wikipédia en Français

  • Une verite qui derange — Une vérité qui dérange Une vérité qui dérange (An Inconvenient Truth, titre en anglais) est un film documentaire américain traitant du changement climatique, spécialement du réchauffement planétaire, réalisé par Davis Guggenheim. Al Gore, ancien… …   Wikipédia en Français

  • Une vérité qui dérange — (An Inconvenient Truth, titre en anglais) est un documentaire américain traitant du changement climatique, spécialement du réchauffement planétaire, réalisé par Davis Guggenheim. Al Gore, ancien vice président des États Unis et prix Nobel de la… …   Wikipédia en Français


Поделиться ссылкой на выделенное

Прямая ссылка:
Нажмите правой клавишей мыши и выберите «Копировать ссылку»