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

TL;DR

This paper proves that for specific small quadratic fields, certain types of 2-dimensional mod 2 Galois representations cannot exist if they are unramified outside 2.

Contribution

It establishes the non-existence of particular irreducible mod 2 Galois representations for some small quadratic fields, advancing understanding of Galois representation constraints.

Findings

Non-existence of certain mod 2 Galois representations for specific quadratic fields

Results apply to representations unramified outside 2

Contributes to the classification of Galois representations over quadratic fields

Abstract

For a few quadratic fields, the non-existence is proved of continuous irreducible mod 2 Galois representations of degree 2 unramified outside 2.