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 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 V(K) 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
O\left({H1/2
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[2]
N(H)=O\left({Hn-1/2logH}\right).
A Hilbertian variety V over K is one for which V(K) is not thin: this is a birational invariant of V.[3] A Hilbertian field K is one for which there exists a Hilbertian variety of positive dimension over K:[3] the term was introduced by Lang in 1962.[4] If K is Hilbertian then the projective line over K is Hilbertian, so this may be taken as the definition.[5]
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.[6] More generally a finite degree extension of a Hilbertian field is Hilbertian[7] and any finitely generated infinite field is Hilbertian.[5]
There are several results on the permanence criteria of Hilbertian fields. Notably Hilbertianity is preserved under finite separable extensions[8] and abelian extensions. If N is a Galois extension of a Hilbertian field, then although N need not be Hilbertian itself, Weissauer'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 numbers have all sets thin, for example. They, with the other local fields (real numbers, p-adic numbers) are not Hilbertian.[9]
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(Qp)
for all products over finite sets of prime numbers p, not including any of some set given once and for all. Ekedahl has proved that WWA for V implies V is Hilbertian.[10] 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.[10]
. Serge Lang . Survey of Diophantine Geometry . . 1997 . 3-540-61223-8 . 0869.11051 .
. Jean-Pierre Serre . Lectures on the Mordell-Weil Theorem . 1989 . Translated and edited by Martin Brown from notes by Michel Waldschmidt . 0676.14005 . Aspects of Mathematics . E15 . Braunschweig etc. . Friedr. Vieweg & Sohn .
. Jean-Pierre Serre . Topics in Galois Theory . Research Notes in Mathematics . 1 . Jones and Bartlett . 1992 . 0-86720-210-6 . 0746.12001 .
. Andrzej Schinzel . Polynomials with special regard to reducibility . 0956.12001 . Encyclopedia of Mathematics and Its Applications . 77 . Cambridge . . 2000 . 0-521-66225-7 . registration .