Thin set (Serre)
Encyclopedia
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.
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
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
s we therefore have
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 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. Any algebraic number field
is Hilbertian. More generally any finitely generated infinite field is Hilbertian.
There are several results on the permanence criteria of Hilbertian fields. Notably Hilbertianity is preserved under finite extensions 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.
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).
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.
), 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
for all products over finite sets of prime numbers p, not including any of some set {p1, ..., pM} 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a thin set in the sense of Serre, named after Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
, is a certain kind of subset constructed in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
over a given field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
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 varietyAlgebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
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
Algebraic set
In mathematics, an algebraic set over an algebraically closed field K is the set of solutions in Kn of a set of simultaneous equationsand so on up to...
that is a finite union of algebraic varieties of dimension lower than d, the dimension
Dimension of an algebraic variety
In mathematics, the dimension of an algebraic variety V in algebraic geometry is defined, informally speaking, as the number of independent rational functions that exist on V.For example, an algebraic curve has by definition dimension 1...
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
Function field
Function field may refer to:*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 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
Projective line
In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
over Q is Hilbertian. Being Hilbertian is at the other end of the scale from being algebraically closed: the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s have all sets thin, for example. They, with the other local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
s (real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, p-adic number
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
s) are not Hilbertian. Any algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
is Hilbertian. More generally any finitely generated infinite field is Hilbertian.
There are several results on the permanence criteria of Hilbertian fields. Notably Hilbertianity is preserved under finite extensions 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
Haran's diamond theorem
In mathematics, the Haran diamond theorem gives a general sufficient condition for a separable extension of a Hilbertian field to be Hilbertian.- Statement of the diamond theorem :Let K be a Hilbertian field and L a separable extension of K...
. A discussion on these results and more appears in Fried-Jarden's Field Arithmetic.
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-ThélèneJean-Louis Colliot-Thélène
Jean-Louis Colliot-Thélène is a French mathematician, born on 2 December 1947. He is a Directeur de Recherches at CNRS at the Université Paris-Sud in Orsay.He studies mainly number theory and algebraic geometry with an arithmetic flavor.-Awards:...
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
Inverse Galois problem
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers Q. This problem, first posed in the 19th century, is unsolved....
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 groupsApproximation in algebraic groups
In mathematics, strong approximation in linear algebraic groups is an important arithmetic property of matrix groups. In rough terms, it explains to what extent there can be an extension of the Chinese remainder theorem to various kinds of matrices...
), 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 {p1, ..., pM} 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.