The non-existence of certain mod 2 Galois representations of some small quadratic fields (Q952014)

The authors show that for a quadratic number field \(F={\mathbb Q}(\sqrt d)\) with \(| d| \leq 6\) there does not exist a continuous irreducible 2-dimensional representation of the absolute Galois group of \(F\) over the algebraic closure of \({\mathbb F}_2\) which is unramified outside \(\{2,\infty\}\). The proof uses improvements on discriminant bounds at the prime 2 which are provided by the authors.