A non-solvable extension of \(\mathbb Q\) unramified outside 7 (Q2894199)

In [Int. Math. Res. Not. 1998, No. 16, 865--875 (1998; Zbl 0978.11018)], \textit{B. H. Gross} proposed the following conjecture: For any prime number \(p\), there is a non-solvable Galois extension of \(\mathbb{Q}\) ramifying only at \(p\). It is known that this conjecture is true for \(p\neq 7\). In the present paper, the author proved that this conjecture for \(p=7\) is true.NEWLINENEWLINEIn [Lect. Notes Math. 317, 319--338 (1973; Zbl 0276.14013)], \textit{J.-P. Serre} proved that the conjecture for \(p> 7\) is true using Galois representations attached to elliptic modular forms. For \(p \leq 7\), one cannot apply his method to the conjecture. It seems almost impossible to show the conjecture by using Galois representation attached to automorphic forms of \(\mathrm{GL}_2\) over \(\mathbb{Q}\). However, one believes that we can attack the conjecture by similar methods with Galois representations attached to some other algebraic groups or other base fields.NEWLINENEWLINEIndeed, changing the base field to some totally real fields, i.e., using Hilbert modular forms, \textit{L. Dembélé} [C. R., Math., Acad. Sci. Paris 347, No. 3--4, 111--116 (2009; Zbl 1166.11038)] proved the conjecture for \(p=2\) and \textit{L. Dembélé} et al. [Compos. Math. 147, No. 3, 716--734 (2011; Zbl 1269.11112)] proved the conjecture for \(p=3,5\). As mentioned in [loc. cit.] \S.3, their methods don't work for \(p=7\) case.NEWLINENEWLINEThe author proves this conjecture for \(p=7\) using a Siegel modular form. More precisely, using a genus \(2\) Siegel cusp form of level \(1\) and weight 28, which is not a Maass spezialform, he proved the existence of a non-solvable Galois extension \(F/\mathbb{Q}\) ramifying only at \(7\). He also proved that the root discriminant of \(F/\mathbb{Q}\) is smaller than \(471.22\).