Nonexistence of certain Galois representations for quadratic fields

Abstract

We establish new cases of quadratic number fields [Formula: see text] unramified away from a prime [Formula: see text] and [Formula: see text] whose absolute Galois group has no irreducible two-dimensional continuous Galois representations in [Formula: see text]. Our work builds on methods of Moon–Taguchi and Şengün and the usual analytic techniques of Odlyzko and Poitou where we note one of the new conditional cases arises via a correction of Poitou’s estimate. The results here seem optimal in that it seems these methods alone will yield no further cases either due to prohibitive computational issues or a failure of the analytic obstructions.