Galois stability, integrality and realization fields for representations of finite Abelian groups (Q1811029)

Let \(E/F\) be a finite Galois extension of fields and \(G\) a finite Abelian subgroup of \(\text{GL}_n(E)\), which is \(G_{E/F}\)-stable. The paper resumes earlier work of the author and proves the existence of such \(G\) so that \(E\) is the field \(F(G)\) generated by the matrix entries of \(G\) over \(F\), provided that some natural conditions on \([E:F]\), \(n\), \(\exp(G)\) are satisfied. (1) Let \(\text{char}(F)=0\), \([E:F]>1\), \(t>1\), \(n\geq\phi_E(t)[E:F]\), where \(\phi_E(t)=[E(\zeta_t):E]\). Then there exists a \(G_{E/F}\)-stable Abelian subgroup \(G\) of exponent \(t\) in \(\text{GL}_n(E)\) such that \(E=F(G)\). If \(F\) is a number field, then \(n=\phi(t)[E:F]\) is the minimal possible \(n\) and \(G\) is irreducible under conjugation in \(\text{GL}_n(F)\); moreover, the smallest possible order of \(G\) only occurs when \(G\) is elementary Abelian. Toward an integral analogue the author shows the following. (2) Let \(K=\mathbb{Q}\) or imaginary quadratic, \(N\in\mathbb{N}\), \({\mathfrak o}(N)=\{\alpha\in{\mathfrak o}_K:|N_{K/\mathbb{Q}}(\alpha)|\leq N\}\), \(\nu(N)\) the number of polynomials of degree \(m\) with coefficients in \({\mathfrak o}(N)\), \(\psi(N)\) the number of those such polynomials whose splitting fields do not contain a field \(K(G)\neq K\) for \(G\subset\text{GL}_n({\mathfrak o}_E)\), \(E\supset K\), \(n\) fixed. Then \(\lim_{N\to\infty}\tfrac{\psi(N)}{\nu(N)}=1\). (3) To a number field \(F\) and integers \(n\), \(t\) there exist only finitely many Galois extensions \(E=F(G)\) with \(G\leq\text{GL}_n({\mathfrak o}_E)\) a finite Abelian \(G_{E/F}\)-stable group of exponent \(t\). This selection of results is not complete. The proofs (of which not every detail has been checked by the reviewer) are quite involved and combine ramification theory, representation theory and, among work of others, \textit{S. D. Cohen}'s paper [Ill. J. Math. 23, 135-152 (1979; Zbl 0402.12005)].