Thin set (Serre)

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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).

[edit] 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.

[edit] 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 {p₁, ..., 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.