Domain of Hilbert's
The phrase HIT result will refer to any statement over a field that
captures or enhances the original statement:
(*) Given an irreducible polynomial f(z,w)
with coefficients in Q in two
variables of degree n in w, then for infinitely many z0 in Q (even in the integers Z) f(z0,w) is irreducible in w of degree n over Q.
I use Q- for the
algebraic closure of Q.
To understand the domain of the result, we notice first that it
translates to a statement about projective normal (nonsingular in this
case) covers. I explain that. The set in the complex numbers defined by
maps to the z-line P1z by φ': (z,w) → z. The degree of this map – number
of points in Q- in
the fiber over all but
finitely many z0
in Q- is also n.
A singular point (z0,w0)∈ X' is one at which the partials
of f in both z and w are zero. Denote X' with its singular points
removed by X''. There is
then a unique non-singular projective algebraic variety X containing X'' as a natural subset and having
a map (function) φ: X → P1z with these properties:
Since only a finite number of points have changed, only the wording of statement (*)
changes to say that there are infinitely many z0 in Z for which the fiber is irreducible over Q. Denote the set of exceptions to HIT for (φ, X) by Rφ, its complement in Z by Iφ. There are many
advantages to changing (φ', X')
to (φ, X). For one, putting a
polynomial f in front of
someone gives them nothing but the coefficients of the polynomial.
There is no useful "name" or "type" attached to the polynomial.
- φ extends φ' on X'.
- φ is a finite flat morphism – of degree n in every fiber.
There are, however, nice pieces of data attached to the type of (φ, X). In asking to enhance (*) we
can make use of their "type," called a Nielsen class. The simplest
observation on a cover is that it comes with two groups, its arithmetic (resp. geometric) Galois closure
group ˆGφ (resp. Gφ).
Hilbert's application of his HIT was to show for a given (φ, X), for infinitely many z0 in Q, the specialized polynomial f(z0,w) also has ˆGφ
closure group. Call the set of such z0
in Z the Galois closure set GCφ.
When we want to consider values of z0
in Q, or over another domain,
we add corresponding indicative decoration. Now I list a collection of
enhancements; most are still centers of active research now.
After a listing of our major topics, there is a URL on that topic.
First there are results in the classical domain, over any number field,
The accompanying files on #3 and #4 start with an archetype that will
reader accustom to covers. The distinction between
ˆGφ and Gφ
in #4 show how to use a geometric expression for describing GCφ. #3 also raises a practical consideration of
there is an HIT result by replacing P1z by other spaces. There is!
Still, the depth of it would be to specialize fibers over certain
special rational points when the base of the cover is group scheme.
- HIT applies to specialize general objects defined by covers to specific cases. Applications like specializations of the Galois closure
of a cover and Néron's production of elliptic curves of
high rank over Q simultaneously applies HIT to many covers. HITcovers.html
- For infinite collections of polynomials f(z,w) based on their type, we can
describe the set Rφ,
up to a finite set: A near
description of Rφ.
Example: Applying HIT to the Inverse Galois Problem,
have simplifications using Nielsen classes, allowing a near description
of GCφ. HITsiegel.html
- For any Hilbertian set Iφ finding an explicit representative in it is a tough problem. Subtler questions arise for (infinite) Universal
Hilbert Subsets: sets nearly
contained, meaning up to a finite set, in each Rφ. HITUHS.html
- There are infinite sequences of covers … → Xk → Xk-1 → … → X0 → P1z, a tower, for which an
enhancement of statement (*) holds: (**) For infinitely many z0 in Q the fiber of Xk → P1z over z0 is irreducible
for each k. A most
significant phrase, Frattini cover.
All methods of transparently finding a representative of Iφ, such as finding an explicit arithmetic progression in it, run into nontrivial points about explicit primes. Debes and Zannier, and their cowriters, have explored the connections of diophantine geometry conjectures to this topic. There is no hope of classifying Universal
Hilbert Subsets in #5, but their existence first started as a general
argument, and then led to making them explicit. Given one such, finding the last member not in a given Iφ must be a very subtle problem.
The archetype for #6 is a tower of modular curves; there the
result is called Serre's Open Image
Theorem. By thinking of HIT in this
way, we point out that there are myriad circumstances
theoretically suiting #6. It is a major research project to
prove results for them. A subfield L
of Q- is
if a version of (*) holds for all two-variable polynomials over L.
Versions of HIT are compatible with various versions of other
diophantine properties. Our archetype is from §3 of HIT74.pdf, giving an HIT version as a consequence of the
Chebotarev density theorem.
By considering the HIT property simultaneously with other properties of
fields, in varying the field, #7 and #8 open relations among
various diophantine properties, including HIT. Various analogs of
Hilbertianity, arise – our 2nd application of Frattini covers –. As an example, #8 compares the
Fried-Jarden Σ-Hilbertian fields with various
- It is known that in the lattice of fields, the Hilbertian
property jumps around all over the place. Yet, Weiesauer's Statement
plays a special role in identifying Hilbertian fields. HITWeissauer.html
- In the class of
P(seudo)A(lgebraically)C(losed), the Galois group of the field
interprets HIT, and many distinct variants on it. HITPAC.html
- Those PAC fields that are Galois extensions of Q provide a way to describe the
absolute Galois group of Q, GQ. HITGQ.html
#9 is the territory of Fried-Voelklein:
fields, Hilbertianity is equivalent to pro-free absolute Galois
group. (It thereby produced what are still substantially the only
known presentations of the absolute Galois group of Q, but it also
shows how close is Fried-Voelklein to proving generalizations that
include Shafarevich's conjecture.) PAC fields and, more
generally, fields of projective dimension 0 (e.g., cyclotomic numbers)
are tools for dissecting GQ.
A theme in the Fried-Jarden book Field Arithmetic, is to collect
subfields of the algebraic numbers according to various possible of
their diophantine properties. By looking at fields satisfying subsets
of properties, one sees the intrinsic relation between the properties.
With the Fried-Voelklein result describing presentations of the
absolute Galois group of Q, we see that properties of large fields can
reveal structure in classical objects. The main tool in this is a
variant of Hilbert's Theorem. Likewise, a view of Serre's Open Image
Theorem – on modular curves – is that towers of covers can exhibit a
Hilbertian property. Those two topics are deeper results. Yet, in this
file we see that variant's on Hilbert's Theorem make clever tools for
investigating many problems.
Moshe Jarden didn't
use my writing on the bigger themes that relate to the Hilbertian
property. They may have been too abstract for his tastes. Still,
relating HIT to
other diophantine properties was underdone in Fried-Jarden.
Especially since that was what attracted Jarden to the book idea when I
first proposed the title "Field
Arithmetic" and its major topics in 1975. It was
I who refereed Jarden's thesis (James Ax gave it to me) and solved the
problems his thesis had left unsolved (in my Math Reviews of the same
January 12, 2009 Michael D. Fried