The embedding problem over a PAC field and a generalization of Shafarevich's Cyclotomic Galois group conjecture

The Embedding Problem Over An Hilbertian PAC field: Original paper GQpresentation.pdf

Table of Contents:

I. PRESENTATIONS OF G_Q:
II. COMBINING HURWITZ SPACES AND CHEBOTAREV'S FIELD CROSSING ARGUMENT:
III. WHAT IS SPECIAL ABOUT SHAFAREVICH'S CASE, AND OTHERS WITH SIMILAR VIRTUES:

I. PRESENTATIONS OF G_Q:

The absolute Galois group G_Q of the rationals is an extension of a known group by a known group: G_Q maps surjectively to the direct product of the symmetric groups S_n, n=1,2, ... with kernel a profree group on a countable number of generators. This is a special case of the main theorem of this paper.

To get such presentations you produce Galois extensions of Q, as explicitly as possible, for which all arithmetic homological obstructions to embedding problems disappear. The remainder comes to considering split embedding problems using geometric properties of Hurwitz spaces. The key geometric property: We know precisely the components of sufficiently many Hurwitz spaces related to each finite group, the Conway-Fried-Parker-Voelklein Theorem CFPV.html. That has many application examples going beyond the special case of [FrVo, App.], using all conjugacy classes in the given group.

The general result of the paper is that a P(seudo)A(lgebraically) C(losed) field F inside the algebraic numbers has G_F pro-free if and only if F is Hilbertian. A practical aspect of the paper is that it shows how to identify points on Hurwitz spaces as giving solutions of embeddings problems as extensions of regular realizations of Galois groups.

II. COMBINING HURWITZ SPACES AND CHEBOTAREV'S FIELD CROSSING ARGUMENT:

The technical lemma that makes this work starts with a basic relation between inner Hurwitz spaces and absolute Hurwitz spaces in [FrVo, Main Thm]. You may view this as a geometric realization of Chebotarev's famous field crossing argument – used by Artin for his reciprocity law – writ large. That is, it here is applied far beyond the algebraic number theory cyclotomic field use of Chebotarev. Down-to-earth examples of the field crossing argument, a partly geometric and partly homological tool, are in [FrJ, pgs. 107, 130, 323, 429, 558, and a discourse in § 24.1]. Online, see RIGP-splitab.html for the split abelian case in the regular Inverse Galois Problem.

This paper makes obvious what is the right generalization:

Projectivity Conjecture: A subfield F of the algebraic numbers with G_F projective is Hilbertian if and only if G_Fis profree.

The paper suggests precisely why it is the best possible result that includes Shafarevich's famous conjecture: The cyclotomic closure (adjoin all roots of 1) of the rationals has pro-free absolute Galois group. So, the quotient in this case identifies with the profinite invertible integers ^{^}Z^*. Florian Pop, three years later (having watched four lectures on this topic, claimed an "independent" proof of our Main Theorem). His proof amounts to replacing Riemann's Existence Theorem in sufficiently many subcases of Nielsen classes by a lesser result, using Harbater Patching. It is also inexplict and less applicable than the techniques of this result.

III. WHAT IS SPECIAL ABOUT SHAFAREVICH'S CASE, AND OTHERS WITH SIMILAR VIRTUES:

Various formulas like the B(ranch) C(ycle) L(emma) suggest how to turn a presentation of G_Q into yet more precise data in Shafarevich's cyclotomic case. You construct geometric elements in the Galois group that give generators of that pro-free kernel, for which the action^{^}Z^* can be written explicitly. The minimum necessary here is that the G_Q quotient be defined by a Galois extension of Q requiring few artificial choices for its construction.

One candidate for this construction is the Alternating group closure of Q: The composite of all Galois extensions with an Alternating group as its Galois group. The paper attached to hf-can0611591.html doesn't exclude the possiblity for this, though it shows why an expected route doesn't work. To wit: We know the definition field of maximal families of 3-cycle covers is Q, so this is the natural case to consider. Indeed, following the technique of this paper, we would be done if every curve over Q has a Q map to P¹with odd order branching. What § 5.2 of hf-can0611591.pdf shows is that this is not the case; even though each curve has an odd order branching map to P¹, and other properties that make this a surprise.

Reason: The general curve of any genus would then be forced to have a half-canonical class also defined over Q, and the paper shows this is not possible. That is, half-canonical classes present an obstruction. The idea goes back to Riemann's generalization of Abel's Theorem. In considerint this hf-can0611591.pdf several problems that naturally challenge statements in Mumford's Curves on an Algebraic Curve. We make considerable application of Fay's Book, Theta functions on Riemann Surfaces that suggest non-abelian analogs of his variational formulas, like that due to Thomae as in yaac-cyc3thetaids04-08-07.html.

[FrJ] M. Fried and M. Jarden, Field Arithmetic, Springer Ergebnisse II Vol 11 (1986) 455 pgs., 2nd edition 2004, 780 pps. ISBN 3-540-22811-x. We quote the second ed., but all references are to statements also in the first.

[FrVo] M. Fried and H. Völklein, The inverse Galois problem and rational points on moduli spaces, Math. Ann. 290, (1991) 771–800.