\(S_4\) and \(\widetilde{S}_4\) extensions of \(\mathbb{Q}\) ramified at only one prime
Extensions of \(\mathbb Q\) with Galois group \(S_4\) that can be embedded in a \(\text{ GL}_2(\mathbb F_3)\)-extension are interesting in connection with the Serre conjectures and have been studied e.g. by \textit{P. Cassou-Nogues} and \textit{A. Jehanne} [J. Number Theory 57, 366-384 (1996; Zbl 0858.11058) and Exp. Math. 5, 275-290 (1997)]. In this note, the author considers odd primes of the form \(p^* = (256n^3 - k^2)/27\), with \(27 \mid k\) if \(3 \mid n\) and shows that the splitting field \(K\) of \(f(x) = (x+n)^4 - p^*x\) is an \(S_4\)-extension of \(\mathbb Q\) unramified outside \(p^*\infty\) unless \(p^* = 1 + 4n\). Write \(p = | p^*| \); if \(p \equiv 3 \bmod 8\), then \(K\) can be embedded in a \(\text{ GL}_2(\mathbb F_3)\)-extension by results of \textit{P. Bayer} and \textit{G. Frey} [Math. Z. 207, No.~3, 395-408 (1991; Zbl 0726.11037)]. The decomposition of primes in this extension is then related to a certain \(2\)-dimensional Galois representation, and this in turn via Serre's ``epsilon conjecture'' to a cusp form of weight \(1 + (p-1)^2/8\) and level \(1\) which is a simultaneous eigenform for the Hecke operators. The last section contains an explicit computation of such an eigenform for the primes \(p = 59\), \(107\) and \(283\).
scientific article; zbMATH DE number 1275042