×

Some finite groups which appear as \(\mathrm{Gal } L/K\), where \(K\subseteq \mathbb{Q}(\mu_n)\). (English) Zbl 0552.12004

The topic of this paper is the famous ‘inverse problem’ of Galois theory: Which groups occur as Galois groups over a fixed ground field \(k\)? \(k=\mathbb{Q}\) or \(k\) a small extension of \(\mathbb{Q}\) are of special interest. The most striking consequence of this work is the realization of the Fischer-Griess monster as a Galois group over \(\mathbb{Q}\).
The author deals more general with non-abelian finite simple groups \(F\) that are ‘rigid’. This means: there are conjugacy classes \(C_1,\ldots,C_k\) in \(F\), such that
(i) \(F\) is generated by \(x_1,\ldots,x_k\) for every \((x_1,\ldots,x_k)\in C_1\times\cdots\times C_k\) with \(x_1\cdot\cdots\cdot x_k=1\), and
(ii) \(F\) operates (by conjugation) transitively on these generators \((x_1,\ldots,x_k)\).
This is the fundamental group theoretical concept. Examples of rigid simple groups given by the author are the Fischer-Griess monster and the groups \(\mathrm{PSL}(2,2^n)\).
The author then constructs Galois extensions \(L\,\vert\, K\) of some cyclotomic field \(K\) with a given rigid group \(F\) as Galois group, and more general: Galois extensions \(N \vert \mathbb{Q}\) with a Galois group built out of \(F\) in an explicit form.
The construction of these field extensions is based on the theory of Riemann surfaces and of discrete subgroups of \(\mathrm{PSL}_2(\mathbb{R})\). The proof is rather long and involved, containing a study of the congruence subgroup \(\Gamma_0(12)\) with its corresponding modular curve \(X_0(12)\), and explicit calculations in terms of Puiseux series.
To fix the ideas behind all this one should add: A group \(G\) generated by \(x_1,\ldots,x_k\) with \(x_1\cdot\cdots\cdot x_k=1\) can be realized as the group of covering transformations of some compact Riemann surface \(X\to\mathbb{P}^1(\mathbb{C})\), unramified outside \(k\) points in \(\mathbb{P}^1(\mathbb{C})\). Hence \(G\) is the Galois group of an algebraic function field \(L\) over the field of rational functions \(\mathbb{C}(T)\). Here \(\mathbb{C}\) may be replaced by \(\bar{\mathbb{Q}}\), the field of algebraic numbers. The main problem then is, to reduce \(\bar {\mathbb{Q}}\) further to some small subfield \(K\).
In the author’s exposition \(X_0(12)\) plays the role of \(\mathbb{P}^1(\mathbb{C})\), and the ramification points are the six cusps of \(X_0(12)\). To reduce the field \(K\) of definition for \(L \vert \mathbb{C}(T)\) he uses explicit calculations in terms of Puiseux series.
One should add that in general – by the work of B. H. Matzat – one can get information about \(K\) in purely group theoretical terms of \(F\), and at this point the rigidity of \(F\) comes in.

MSC:

11R32 Galois theory
12F12 Inverse Galois theory
20D08 Simple groups: sporadic groups
20F29 Representations of groups as automorphism groups of algebraic systems
11R58 Arithmetic theory of algebraic function fields
20D06 Simple groups: alternating groups and groups of Lie type
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
30F10 Compact Riemann surfaces and uniformization
14H05 Algebraic functions and function fields in algebraic geometry
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Artin, E., Galois Theory, Notre Dame Mathematical Lectures (1953)
[2] Bieberbach, L., (Lehrbuch der Funktionentheorie, Vol. 1 (1945), Chelsea: Chelsea New York) · Zbl 0060.19907
[3] Hilbert, D., Gesammelte Abhandlungen (1965), Chelsea: Chelsea New York · JFM 59.0037.06
[4] Queen, L., (Ph.D. thesis (1979), University of Cambridge)
[5] Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions (1971), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J · Zbl 0221.10029
[6] Siegel, C., Vorlesungen über ausgewählte Kapitel der Funktionentheorie (1964), Teil I, Göttingen
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.