Finiteness results on certain mod \(p\) Galois representations (Q1587424)

Let \(\mathbb F\) be the algebraic closure of the field with two elements. The author shows that there are only finitely many Galois extensions \(K\) of \(\mathbb Q\) which are unramified outside \(2\) and whose Galois group over \(\mathbb Q\) can be embedded into \(\text{GL}(4,\mathbb F).\) In order to achieve this result, the author first derives an upper bound on the discriminant of \(K\), coming from group theory (\(2\)-length of \(\text{Gal}(K/\mathbb Q)\)). This is then compared to the asymptotic lower bound on the discriminant from the work of \textit{H. M. Stark} and \textit{A. Odlyzko} [cf. \textit{G. Poitou}, Sémin. Bourbaki 1975/76, Exp. No. 479, Lect. Notes Math. 567 (1977; Zbl 0359.12010)] and the improved version by \textit{J.-P. Serre} [Œuvres. Collected papers, Vol. III: 1972--1984, pp. 240--243, Berlin: Springer-Verlag (1986; Zbl 0849.01049)] which uses Weil's explicit formulas. These bounds contradict when the degree of \(K\) goes to infinity. The result implies that the set of all isomorphism classes of continuous semisimple representations of the absolute Galois group of \(\mathbb Q\) into \(\text{GL}(4,\mathbb F)\) which are unramified outside 2 is finite. As Serre's estimates become better when assuming the generalized Riemann hypothesis, the author gets some more finiteness results as above under this assumption.