Thin set (Serre)

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 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 ${\displaystyle O\left({H^{1/2}}\right)}$, and this result is best possible.[1]

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]

${\displaystyle N(H)=O\left({H^{n-1/2}\log H}\right).}$

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][6]

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.[5][7] More generally a finite degree extension of a Hilbertian field is Hilbertian[8] and any finitely generated infinite field is Hilbertian.[6]

There are several results on the permanence criteria of Hilbertian fields. Notably Hilbertianity is preserved under finite separable extensions[9] 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.[5]

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.[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]

• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.
• Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051.
• Serre, Jean-Pierre (1989). Lectures on the Mordell-Weil Theorem. Aspects of Mathematics. Vol. E15. Translated and edited by Martin Brown from notes by Michel Waldschmidt. Braunschweig etc.: Friedr. Vieweg & Sohn. Zbl 0676.14005.
• Serre, Jean-Pierre (1992). Topics in Galois Theory. Research Notes in Mathematics. Vol. 1. Jones and Bartlett. ISBN 0-86720-210-6. Zbl 0746.12001.
• Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics and Its Applications. Vol. 77. Cambridge: Cambridge University Press. ISBN 0-521-66225-7. Zbl 0956.12001.