View source for Shafarevich conjecture

''in inverse Galois theory''

The absolute Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s1302602.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s1302603.png" /> (cf. also [[Galois group|Galois group]]) is a free [[Profinite group|profinite group]] of countable rank. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s1302604.png" /> is the maximal Abelian extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s1302605.png" />, or, equivalently (by the [[Kronecker–Weber theorem]]), the maximal cyclotomic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s1302606.png" />.

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|Tower of fields]]; [[Class field theory|Class field theory]]). The conjecture would imply an affirmative answer to the inverse Galois problem over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s1302608.png" />, i.e. that every [[Finite group|finite group]] is a Galois group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s1302609.png" /> (cf. also [[Galois theory, inverse problem of|Galois theory, inverse problem of]]). By the Iwasawa theorem [[#References|[a7]]], p. 567, (see also [[#References|[a1]]], Cor. 24.2), a profinite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026010.png" /> of countable rank is free (as a profinite group) if and only if every finite embedding problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026011.png" /> has a proper solution. Thus, the Shafarevich conjecture is equivalent to the assertion that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026012.png" /> is a quotient of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026013.png" />, then every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026014.png" />-Galois field extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026015.png" /> is dominated by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026016.png" />-Galois field extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026017.png" />.

A weakening of this assertion is known: that the profinite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026018.png" /> is projective, i.e. every finite embedding problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026019.png" /> has a weak solution (cf. also [[Projective group|Projective group]]). Projectivity is equivalent to the condition of [[Cohomological dimension|cohomological dimension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026020.png" /> [[#References|[a12]]], Chap. 1; Props. 16, 45, and this holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026021.png" /> by [[#References|[a12]]], Chap. 2; Prop. 9. On the other hand, the absolute Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026023.png" /> is not projective, since the surjection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026024.png" /> corresponding to the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026025.png" /> does not factor through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026026.png" />. Thus, the analogue of the Shafarevich conjecture does not hold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026027.png" />.

==Evidence for the conjecture.==
Many finite groups, including  "most"  simple groups, have been realized as Galois groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026028.png" /> [[#References|[a9]]], Chap. II, Sec. 10. These realizations provide evidence for the inverse Galois problem over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026029.png" /> and hence for the Shafarevich conjecture. Typically, these realizations have been achieved by constructing Galois branched covers of the projective line over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026030.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026031.png" /> is Hilbertian [[#References|[a13]]], Cor. 1.28, such a realization implies that the covering group is a Galois group of a field extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026032.png" />. Most of these branched covers have been constructed by means of rigidity; cf. [[#References|[a9]]] and [[#References|[a13]]] for a discussion of this approach. (Some of these covers are actually defined over the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026033.png" />-line, and their covering groups are thus Galois groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026034.png" />.)

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. [[#References|[a9]]], Chap. 4, Sec. 3.1). Many simple groups have been GAR-realized over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026035.png" />, and the Shafarevich conjecture would follow if it were shown that every finite simple group has a GAR-realization over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026036.png" />. See [[#References|[a9]]], Chap. 4; Sec. 3, 4.

The solvable case of the Shafarevich conjecture has been proven: K. Iwasawa [[#References|[a7]]] showed that the maximal pro-solvable quotient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026037.png" /> is a free pro-solvable group of countable rank. In particular, every finite solvable group is a Galois group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026038.png" />, and every embedding problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026039.png" /> with finite solvable kernel has a proper solution. Iwasawa's result also holds for the maximal Abelian extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026040.png" /> of any global field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026041.png" />, and for the maximal cyclotomic extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026042.png" /> of any global field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026043.png" /> [[#References|[a7]]], Thm. 6, 7.

==Generalizations.==
The Shafarevich conjecture can be posed with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026044.png" /> replaced by any [[Global field|global field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026045.png" />. In this generalized form, it asserts that the absolute Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026046.png" /> 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 [[#References|[a6]]], Cor. 4.2, and F. Pop [[#References|[a10]]] in the case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026047.png" /> is the function field of a curve over a [[Finite field|finite field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026048.png" />. (See also [[#References|[a5]]], Cor. 4.7, and [[#References|[a9]]], Sec. V.2.4.) Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026049.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026050.png" /> is a finite field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026051.png" />, this assertion is equivalent to stating that the absolute Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026052.png" /> is free of countable rank if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026053.png" /> is the function field of a curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026054.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026055.png" /> have a proper solution — i.e. that every connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026056.png" />-Galois branched cover of the curve is dominated by a connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026057.png" />-Galois branched cover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026058.png" /> is a quotient of the finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026059.png" />. By Iwasawa's theorem [[#References|[a7]]], p. 567, the result follows. The proof also shows that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026060.png" /> is a curve over an arbitrary [[Algebraically closed field|algebraically closed field]] of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026061.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026062.png" /> is the function field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026063.png" />, then every finite embedding problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026064.png" /> has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026065.png" /> proper solutions. By the Mel'nikov–Chatzidakis theorem [[#References|[a8]]], Lemma 2.1, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026066.png" /> is free profinite of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026067.png" />, generalizing the geometric case of the Shafarevich conjecture (see [[#References|[a6]]], Thm. 4.4, [[#References|[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 [[#References|[a2]]], p. 470, that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026068.png" /> is a countable Hilbertian field whose absolute Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026069.png" /> is projective, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026070.png" /> is free of countable rank. They proved a special case of this [[#References|[a2]]], Thm. A, viz. that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026071.png" /> is free of countable rank if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026072.png" /> is a countable Hilbertian pseudo-algebraically closed field (a PAC field) of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026073.png" />. For example, this applies to the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026074.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026075.png" /> is the field of totally real algebraic numbers, by results of R. Weissauer and Pop; see [[#References|[a13]]], p. 151, [[#References|[a9]]], p. 286. Later, Pop [[#References|[a11]]], Thm. 1, removed the characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026076.png" /> hypothesis from the above result. This solves a problem in [[#References|[a1]]], Problem 24.41. (See also [[#References|[a4]]].) Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026077.png" /> is not PAC (as proven by G. Frey [[#References|[a1]]], Cor.10.15), this result does not prove the Shafarevich conjecture itself. But it does imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026078.png" /> has a free normal subgroup of countable rank for which the quotient is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026079.png" /> [[#References|[a2]]] (instead of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026080.png" /> as in the Shafarevich conjecture). The above Fried–Völklein conjecture holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026081.png" /> is Galois over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026082.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026083.png" /> an algebraically closed field ([[#References|[a8]]], Prop. 4.4, using the geometric case of the Shafarevich conjecture [[#References|[a6]]], [[#References|[a10]]]). More generally, it holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026084.png" /> is large in the sense of Pop [[#References|[a11]]], Thm. 2.1; cf. also [[#References|[a9]]], Sec. V.4. A solvable case of the conjecture holds, extending Iwasawa's result: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026085.png" /> Hilbertian with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026086.png" /> projective, every embedding problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026087.png" /> with finite solvable kernel has a proper solution [[#References|[a13]]], Cor. 8.25.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Fried,   M. Jarden,   "Field arithmetic" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Fried,   H. Völklein,   "The embedding problem over a Hilbertian PAC field"  ''Ann. of Math.'' , '''135'''  (1992)  pp. 469–481</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  "Recent developments in the inverse Galois problem"  M. Fried (ed.) , ''Contemp. Math.'' , '''186''' , Amer. Math. Soc.  (1995)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Haran,   M. Jarden,   "Regular split embedding problems over complete valued fields"  ''Forum Math.'' , '''10'''  (1998)  pp. 329–351</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Haran,   H. Völklein,   "Galois groups over complete valued fields"  ''Israel J. Math.'' , '''93'''  (1996)  pp. 9–27</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D. Harbater,   "Fundamental groups and embedding problems in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130260/s13026088.png" />"  M. Fried (ed.) , ''Recent Developments in the Inverse Galois Problem'' , ''Contemp. Math.'' , '''186''' , Amer. Math. Soc.  (1995)  pp. 353–370</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  K. Iwasawa,   "On solvable extensions of algebraic number fields"  ''Ann. of Math.'' , '''58'''  (1953)  pp. 548–572</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  G. Malle,   B.H. Matzat,   "Inverse Galois theory" , Springer  (1999)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  F. Pop,   "Étale Galois covers over smooth affine curves"  ''Invent. Math.'' , '''120'''  (1995)  pp. 555–578</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  F. Pop,   "Embedding problems over large fields"  ''Ann. of Math.'' , '''144'''  (1996)  pp. 1–34</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J.-P. Serre,   "Cohomologie Galoisienne" , ''Lecture Notes in Mathematics'' , '''5''' , Springer  (1964)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  H. Völklein,   "Groups as Galois groups" , ''Studies in Adv. Math.'' , '''53''' , Cambridge Univ. Press  (1996)</TD></TR></table>