On the existence of large degree Galois representations for fields of small discriminant
Let \(K\) be a number field, let \(\mathrm{Disc}(K)\) be its absolute discriminant and let \(\mathrm{rd}_K=(|\mathrm{Disc}(K)|)^{1/[K:\mathbb Q]}\) be the root discriminant of \(K\). It is known from Minkowski's work on the geometry of numbers that \(\mathrm{rd}_K>e^2\pi/4-o(1)\), so that the discriminant of a number field cannot be too small. A lot of work has been done in recent years in the study of fields with small discriminant. As a motivation for the paper under review, we cite [\textit{J. Leshin}, Proc. Am. Math. Soc. 141, No. 10, 3341--3352, (2013; Zbl 1307.11114)], in which, for any \(N>0\) and any positive integer \(n\), the set of length \(n\) solvable extensions of a number field \(K\) with root discriminant less than \(N\) is shown to be finite. So one should expect that if \(L/K\) is a Galois extension of number fields and \(\mathrm{rd}_L\) is small, then \(\mathrm{Gal}(L/K)\) has to be far away from being abelian. In the paper under review this intuition is made concrete by giving two lower bounds for the maximum \(r\) of the degrees of the irreducible complex representations of \(\mathrm{Gal}(L/K)\). These bounds are non-trivial when \([K:\mathbb Q]\) and \(\mathrm{rd}_L\) are small compared to \([L:\mathbb Q]\). The first inequality holds unconditionally and its proof relies on the existence of an intermediate field \(F\) of \(L/K\) such that \(L/F\) is abelian and \([F:K]\leq (r!)^2\). The second lower bound of the paper is stronger than the first one, but it is conditional on the Artin conjecture and the generalized Riemann hypothesis. As a corollary the authors prove an upper bound for the number of linearly disjoint unramified Galois extensions of a number field \(K\), such that their Galois groups over \(K\) have an irreducible representation of some fixed degree \(r\).
scientific article; zbMATH DE number 6369636