Mod \(p\) Galois representations of solvable image (Q2716093)

The authors investigate the following problem: Fix the dimension \(d\), a number field \(K\) and an ideal \(N\) of \(K\). Then, there should only exist finitely many \(d\)-dimensional representations of the Galois group of \(K\) with values in \(\overline{\mathbb F}_p\) (an algebraic closure of \(\mathbb F_p\), \(p\) a fixed prime) such that the prime-to-\(p\) part of the conductor divides \(N\).NEWLINENEWLINE Using class field theory and the Hermite-Minkowski theorem, they solve the problem in the case of solvable image.NEWLINENEWLINE In the general case, the problem is reduced to the case of image equal to a simple group of Lie type, via results of Larsen and Pink. The case of function fields is also considered.