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2011-11-04T21:54:12Z

58.145.148.113: Removed the statement that the Strand 500 is the most current console, it's now been surpassed by the Pallete and I can't verify it's compatibility with D54

{{Other uses|Thin set (disambiguation){{!}}Thin set}}

In [[mathematics]], a '''thin set in the sense of Serre''', named after [[Jean-Pierre Serre]], is a certain kind of subset constructed in [[algebraic geometry]] over a given [[field (mathematics)|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.

==Formulation==

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 an algebraic variety|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 field of an algebraic variety|function field]]s 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 subset of a finite union of thin sets of types I and II .

The terminology ''thin'' may be justified by the fact that if ''A'' is a thin subset of the line over '''Q''' then the number of points of ''A'' of height at most ''H'' is ≪ ''H'': the number of integral points of height at most ''H'' is <math>O\left({N^{1/2}}\right)</math>, and this result is best possible.<ref name=SB26>Serre (1992) p.26</ref>

A result of S. D. Cohen, based on the [[large sieve method]], extends this result, 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''). Let ''A'' be a thin set in affine ''n''-space over '''Q''' and let ''N''(''H'') denote the number of integral points of naive height at most ''H''. Then<ref name=SB27>Serre (1992) p.27</ref>

:<math> N(H) = O\left({H^{n-1/2} \log H}\right) . </math>

==Hilbertian fields==
A '''Hilbertian variety''' ''V'' over ''K'' is one for which ''V''(''K'') is ''not'' thin: this is a [[birational invariant]] of ''V''.<ref name=SB19>Serre (1992) p.19</ref> A '''Hilbertian field''' ''K'' is one for which there exists a Hilbertian variety of positive dimension over ''K''.<ref name=SB19/> If ''K'' is Hilbertian then the [[projective line]] over ''K'' is Hilbertian, so this may be taken as the definition.<ref name=SB20/>

The rational number field '''Q''' is Hilbertian, because [[Hilbert's irreducibility theorem]] has as a corollary that the [[projective line]] over '''Q''' is Hilbertian: indeed, any [[algebraic number field]] is Hilbertian, again by the Hilbert irreducibility theorem.<ref name=SB20/><ref name=L41>Lang (1997) p.41</ref> More generally a finite degree extension of a Hilbertian field is Hilbertian<ref name=SB21>Serre (1992) p.21</ref> and any finitely generated infinite field is Hilbertian.

There are several results on the permanence criteria of Hilbertian fields. Notably Hilbertianity is preserved under finite separable extensions<ref name=FJ224>Fried & Jarden (2008) p.224</ref> and abelian extensions. If ''N'' is a Galois extension of a Hilbertian field, then although ''N'' need not be Hilbertian itself, Weisseauer's results asserts that any proper finite extension of ''N'' is Hilbertian. The most general result in this direction is [[Haran's diamond theorem]]. A discussion on these results and more appears in Fried-Jarden's ''Field Arithmetic''.

Being Hilbertian is at the other end of the scale from being [[algebraically closed]]: the [[complex number]]s have all sets thin, for example. They, with the other [[local field]]s ([[real number]]s, [[p-adic number]]s) are ''not'' Hilbertian.<ref name=SB20>Serre (1992) p.20</ref>

==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.<ref name=SB29>Serre (1992) p.29</ref> In fact Colliot-Thélène conjectures WWA holds for any [[unirational variety]], which is therefore a stronger statement. This conjecture would imply a positive answer to the [[inverse Galois problem]].<ref name=SB29/>

==References==
{{reflist}}
* {{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=3rd revised | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=[[Springer-Verlag]] | year=2008 | isbn=978-3-540-77269-9 | zbl=1145.12001 }}
* {{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Survey of Diophantine Geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | zbl=0869.11051 }}
* {{cite book | first=Jean-Pierre | last=Serre | authorlink=Jean-Pierre Serre | title=Lectures on the Mordell-Weil Theorem | year=1989 }}
* {{cite book | first=Jean-Pierre | last=Serre | authorlink=Jean-Pierre Serre | title=Topics in Galois Theory | series=Research Notes in Mathematics | volume=1 | publisher=Jones and Bartlett | year=1992 | isbn=0-86720-210-6 | zbl=0746.12001 }}

[[Category:Diophantine geometry]]
[[Category:Field theory]]

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