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)