Shafarevich conjecture

in inverse Galois theory

The absolute Galois group of (cf. also Galois group) is a free profinite group of countable rank. Here, is the maximal Abelian extension of , or, equivalently (by the Kronecker–Weber theorem), the maximal cyclotomic extension of .

I.R. Shafarevich posed this assertion as an important problem during a 1964 series of talks at Oberwolfach on the solution to the class field tower problem (cf. Tower of fields; Class field theory). The conjecture would imply an affirmative answer to the inverse Galois problem over , i.e. that every finite group is a Galois group over (cf. also Galois theory, inverse problem of). By the Iwasawa theorem [a7], p. 567, (see also [a1], Cor. 24.2), a profinite group of countable rank is free (as a profinite group) if and only if every finite embedding problem for has a proper solution. Thus, the Shafarevich conjecture is equivalent to the assertion that if is a quotient of a finite group , then every -Galois field extension of is dominated by a -Galois field extension of .

A weakening of this assertion is known: that the profinite group is projective, i.e. every finite embedding problem for has a weak solution (cf. also Projective group). Projectivity is equivalent to the condition of cohomological dimension [a12], Chap. 1; Props. 16, 45, and this holds for by [a12], Chap. 2; Prop. 9. On the other hand, the absolute Galois group is not projective, since the surjection corresponding to the extension does not factor through . Thus, the analogue of the Shafarevich conjecture does not hold for .

Evidence for the conjecture.

Many finite groups, including "most" simple groups, have been realized as Galois groups over [a9], Chap. II, Sec. 10. These realizations provide evidence for the inverse Galois problem over and hence for the Shafarevich conjecture. Typically, these realizations have been achieved by constructing Galois branched covers of the projective line over . Since is Hilbertian [a13], Cor. 1.28, such a realization implies that the covering group is a Galois group of a field extension of . Most of these branched covers have been constructed by means of rigidity; cf. [a9] and [a13] for a discussion of this approach. (Some of these covers are actually defined over the -line, and their covering groups are thus Galois groups over .)

The rigidity approach also suggests a possible way of proving the Shafarevich conjecture. B.H. Matzat introduced the notion of GAR-realizability of a group, this being realizability as the Galois group of a branched cover with certain additional properties (cf. [a9], Chap. 4, Sec. 3.1). Many simple groups have been GAR-realized over , and the Shafarevich conjecture would follow if it were shown that every finite simple group has a GAR-realization over . See [a9], Chap. 4; Sec. 3, 4.

The solvable case of the Shafarevich conjecture has been proven: K. Iwasawa [a7] showed that the maximal pro-solvable quotient of is a free pro-solvable group of countable rank. In particular, every finite solvable group is a Galois group over , and every embedding problem for with finite solvable kernel has a proper solution. Iwasawa's result also holds for the maximal Abelian extension of any global field , and for the maximal cyclotomic extension of any global field [a7], Thm. 6, 7.

Generalizations.

The Shafarevich conjecture can be posed with replaced by any global field . In this generalized form, it asserts that the absolute Galois group of is free of countable rank (as a profinite group). This conjecture remains open (as of 2001) in the number field case, but has been proven by D. Harbater [a6], Cor. 4.2, and F. Pop [a10] in the case that is the function field of a curve over a finite field . (See also [a5], Cor. 4.7, and [a9], Sec. V.2.4.) Since if is a finite field of characteristic , this assertion is equivalent to stating that the absolute Galois group of is free of countable rank if is the function field of a curve over . This result is shown by using patching methods involving formal schemes or rigid analytic spaces, in order to show that all finite embedding problems for have a proper solution — i.e. that every connected -Galois branched cover of the curve is dominated by a connected -Galois branched cover, if is a quotient of the finite group . By Iwasawa's theorem [a7], p. 567, the result follows. The proof also shows that if is a curve over an arbitrary algebraically closed field of cardinality , and if is the function field of , then every finite embedding problem for has exactly proper solutions. By the Mel'nikov–Chatzidakis theorem [a8], Lemma 2.1, it follows that is free profinite of rank , generalizing the geometric case of the Shafarevich conjecture (see [a6], Thm. 4.4, [a10], Cor. to Thm. A).

As another proposed generalization of the Shafarevich conjecture (which would subsume the above case of global fields), M. Fried and H. Völklein conjectured [a2], p. 470, that if is a countable Hilbertian field whose absolute Galois group is projective, then is free of countable rank. They proved a special case of this [a2], Thm. A, viz. that is free of countable rank if is a countable Hilbertian pseudo algebraically closed field (a PAC field) of characteristic . For example, this applies to the field , where is the field of totally real algebraic numbers, by results of R. Weissauer and Pop; see [a13], p. 151, [a9], p. 286. Later, Pop [a11], Thm. 1, removed the characteristic hypothesis from the above result. This solves a problem in [a1], Problem 24.41. (See also [a4].) Since is not PAC (as proven by G. Frey [a1], Cor.10.15), this result does not prove the Shafarevich conjecture itself. But it does imply that has a free normal subgroup of countable rank for which the quotient is of the form [a2] (instead of the form as in the Shafarevich conjecture). The above Fried–Völklein conjecture holds if is Galois over , for an algebraically closed field ([a8], Prop. 4.4, using the geometric case of the Shafarevich conjecture [a6], [a10]). More generally, it holds if is large in the sense of Pop [a11], Thm. 2.1; cf. also [a9], Sec. V.4. A solvable case of the conjecture holds, extending Iwasawa's result: For Hilbertian with projective, every embedding problem for with finite solvable kernel has a proper solution [a13], Cor. 8.25.

References

[a1]	M. Fried, M. Jarden, "Field arithmetic" , Springer (1986)
[a2]	M. Fried, H. Völklein, "The embedding problem over a Hilbertian PAC field" Ann. of Math. , 135 (1992) pp. 469–481
[a3]	"Recent developments in the inverse Galois problem" M. Fried (ed.) , Contemp. Math. , 186 , Amer. Math. Soc. (1995)
[a4]	D. Haran, M. Jarden, "Regular split embedding problems over complete valued fields" Forum Math. , 10 (1998) pp. 329–351
[a5]	D. Haran, H. Völklein, "Galois groups over complete valued fields" Israel J. Math. , 93 (1996) pp. 9–27
[a6]	D. Harbater, "Fundamental groups and embedding problems in characteristic " M. Fried (ed.) , Recent Developments in the Inverse Galois Problem , Contemp. Math. , 186 , Amer. Math. Soc. (1995) pp. 353–370
[a7]	K. Iwasawa, "On solvable extensions of algebraic number fields" Ann. of Math. , 58 (1953) pp. 548–572
[a8]	M. Jarden, "On free profinite groups of uncountable rank" M. Fried (ed.) , Recent Developments in the Inverse Galois Problem , Contemp. Math. , 186 , Amer. Math. Soc. (1995) pp. 371–383
[a9]	G. Malle, B.H. Matzat, "Inverse Galois theory" , Springer (1999)
[a10]	F. Pop, "Étale Galois covers over smooth affine curves" Invent. Math. , 120 (1995) pp. 555–578
[a11]	F. Pop, "Embedding problems over large fields" Ann. of Math. , 144 (1996) pp. 1–34
[a12]	J.-P. Serre, "Cohomologie Galoisienne" , Lecture Notes in Mathematics , 5 , Springer (1964)
[a13]	H. Völklein, "Groups as Galois groups" , Studies in Adv. Math. , 53 , Cambridge Univ. Press (1996)