The number of monomial mod \(p\) Galois representations with bounded conductor (Q1422328)

This article gives an upper bound for the number of \(n\)-dimensional monomial representations of the absolute Galois group of \(\mathbb{Q}\) with values in a finite field of characteristic \(p\) and bounded conductor, and also a bound for the order of the image of such representations, in terms of \(n\), \(p\) and the conductor. Observe that in the non-monomial case (arbitrary \(n\)-dimensional mod \(p\) Galois representations with bounded conductor) the finiteness of the number of Galois representations is not known. Let us stress that to give an explicit bound is a much harder problem than to just prove finiteness (in the case of monomial representations, finiteness follows from the Hermite-Minkowski theorem and the finiteness of ray class groups). The proof uses bounds for discriminants of number fields, geometry of numbers and bounds for class numbers.