# The non-existence of some Galois representations of moderate dimension in small characteristic

## TL;DR

This paper proves the non-existence of certain low-dimensional irreducible mod p Galois representations unramified outside p, under GRH, refining previous arguments and establishing new non-existence results in small characteristic.

## Contribution

It establishes new non-existence results for low-dimensional irreducible mod p Galois representations unramified outside p, under GRH, and proves unconditional largeness of certain symplectic representations.

## Key findings

- Non-existence of irreducible mod 2 Galois representations of dimension ≤4 unramified outside 2.
- Non-existence of totally real mod 2 representations of dimension ≤8 unramified outside 2.
- Unconditional largeness of irreducible mod 2 symplectic 4-dimensional Galois representations unramified outside 2.

## Abstract

Refining arguments of Hyunsuk Moon, under the assumption of the Generalized Riemann Hypothesis, we prove the non-existence of irreducible mod 2 Galois representations unramified outside 2 of dimensions $\leq 4$, and of totally real such representations of dimensions $\leq 8$. We also prove the non-existence of irreducible totally real mod 3 representations unramified outside 3 of dimensions $\leq 4$. We show unconditionally that the image of an irreducible mod 2 symplectic 4-dimensional Galois representation that is unramified outside 2 must be large. Under GRH, we then deduce the non-existence of such representations.