Covers
and Hilbert's Irreducibility Theorem
We start with the application of H(ilbert's) I(rreducibility) T(heorem)
to groups as Galois groups. HIT is really about specializing covers.
One easily forgots that attached to any cover (even in positive
characteristic if the cover is separable), essentially canonically one
has a Galois closure. This automatically gives many covers attached to
the initial cover. In recognition of the complication of going to
a Galois closure, §II considers a precise way to view the covers
that arise in applying it, especially for distinguishing the arithmetic
from the geometric part of the Galois closure. The precise methods
continue in HITsiegel.html. Finally,
§III considers HIT results over group schemes. Without this we've
left out the possibility of producing groups as a Galois groups (and
other algebraic constructions) by specialization that potentially could
be very advantageous.
I. Galois Group Application of HIT: For any cover φ: W → V of normal varieties, defined over
a number field K, we say p∈W (an algebraic point) has full Galois closure over V, if the degree of the Galois
closure Gφ,p
of K(p)/K(φ(p)) is the same
as the degree of the Galois closure ^W of φ. Since Gφ,p
is a decomposition subgroup of Gφ, the full Galois
closure condition implies the groups are equal.
Thm. From HIT conclude the following [FJ: 16.1.5].
(*) If V is any open subset
of Pu, then there is a dense set
of t ∈Pu(K) so that any p∈W over t has full
Galois closure.
Proof: Avoid t in the branch
locus of φ. Then, any ^p∈^W
over t has
a decomposition group Dp, a subgroup of Gφ,
and the image of ^p in
the quotient ^W/Dp
is a K point over t. Let H run over the maximal proper
subgroups of Gφ and choose t ∈Pu(K) so that none of the ^W/H
has a K point over t. Any of the
versions of HIT say that there is an infinite set of t that
avoid being the image of K
points from any finite set of Pu covers of degree at least
2.
Question to consider: Why would it be easier to realize a finite group
as the Galois group of a geometric
cover over Q, and only
then apply the Theorem above to realize it as the group of a finite
extension of Q?
Answer 1: Maybe it isn't "easier," though that has been essentially the
only way to realize any perfect groups: Geometrically first, then
specialize. To understand this, realize that one must tie the groups to
equations.
II.
Distinguishing the Geometric and Arithmetic Galois Closures:
Since One of the basic corollaries of Its arithmetic cover \hat
G_{\phi} has an associated collection of conjugacy classes of subgroups
I_\phi. T
o [H]\in I_\phi get (g_[H],K_{[H]}) the genus and the definition field.
Geometric cover defined by K equivalent to =\hat G_\p
hi. Get branch cycle description by applying the permutation
representation for H to the branch cycles for \phi.
III. Specializing over rational
points in group schemes:
In my HIT stage I played with getting results for fibers of covers
over
the points generated by a non-torsion point on an Abelian variety,
thinking that these might have been versions of universal Hilbert
subsets, so long as the cover didn't factor through an abelian variety
by translate of an isogeny, pretty much like your Theorem 4.2, but I
was speculating. I need to read your proof in detail to see what
exactly is the problem with going beyond products of elliptic curves.
This was around the time I wrote on "Constructions arising from
Neron’s high rank curves," TAMS
281 (1984), 615–631.
UMBERTO RESPONSE: As I remark somewhere in the paper, the method in
principle adapts to any abelian variety, PROVIDED one has sufficiently
good information on the Galois structure of torsion points. I succeed
for powers of elliptic curves just because of Serre's theorem. For the
rest, there is nothing, it seems to me, special of that situation. For
instance, it should go through in dimension $2$. I have however not
worked out other special cases, because this would not introduce
anything new. That paper happened because Serre didn't believe the
N\'eron paper you reference. UMBERTO RESPONSE: Very interesting! I did
not suspect this, also because he quotes the paper.
He wrote to me doubting it, and I said I would show that N\'eron was
basically correct. In the end I certainly was using N\'eron's idea, but
was using HIT much differently than did he, based on a different
construction. Indeed I had trouble with his version of HIT, and
replaced the setup so I could use the traditional version. Still, it
was by thinking about his version that the topic of speciallization
over group schemes arose. Serre and I had an interchange of letters
over it. I still don't have my N\'eron paper on my web site, and
haven't looked at the topic since then. UMBERTO RESPONSE: I have read a
review of N\'eron's paper by B. Segre. From what he writes, N\'eron
worked with a cover $W ---->B$ where $B$ is an abelian variety AND
$W$ MAY BE EMBEDDED IN ANOTHER ABELIAN VARIETY $A$. Now, this is
extremely strong, and e.g. nowadays we can use Faltings' proof of the
lang conjecture to locate the rational points of $W$ in an abelian
subvariety of $A$, so N\'eron's result becomes automatic. PS: I do have
one immediate suggestion, and that is your title. It is your paper,
though if it were mine, my first stab at a title would be something
like "Hilbertian specializations over rational points on group
schemes." One example where I didn't use the title well was my own HIT
paper in 1974, but that paper was written within two years of my
thesis, and I was still taking advice from people like Schinzel, who
think much less philosophically or telegraphically than do I.