The Nonexistence of Certain Representations of the Absolute Galois Group of Quadratic Fields

TL;DR

This paper proves the nonexistence of certain irreducible mod p Galois representations for some quadratic fields and explores their implications for elliptic curves with specific reduction properties.

Contribution

It establishes new nonexistence results for irreducible Galois representations over quadratic fields and examines their connection to elliptic curves with particular reduction behaviors.

Findings

No such irreducible representations exist for certain (K,p) pairs.

Lists of imaginary quadratic fields with existing representations.

Implications for elliptic curves with good reduction away from 2.

Abstract

For a quadratic field K, we investigate continuous mod p representations of the absolute Galois groups of K that are unramified away from p and infinity. We prove that for certain pairs (K,p), there are no such irreducible representations. We also list some imaginary quadratic fields for which such irreducible representations exist. As an application, we look at elliptic curves with good reduction away from 2 over quadratic fields.