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: WV of normal varieties, defined over a number field K, we say pW (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 pW 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  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 NeronÃ¢â‚¬â„¢s high rank curves," TAMS 281 (1984), 615Ã¢â‚¬â€œ631. 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.