Galois representations modulo $p$ that do not lift modulo $p^2$

TL;DR

This paper constructs specific Galois representations modulo p that cannot be lifted to modulo p^2, providing counterexamples to previous conjectures and answering open questions in number theory.

Contribution

It explicitly demonstrates the existence of mod p Galois representations that do not lift to mod p^2, resolving a question of Khare and Serre and disproving Florence's conjecture.

Findings

Existence of non-liftable mod p Galois representations over certain fields.

Counterexamples to conjectures about lifting properties.

Clarification of the structure of negligible classes in cohomology.

Abstract

For every finite group $H$ and every finite $H$-module $A$, we determine the subgroup of negligible classes in $H^2(H,A)$, in the sense of Serre, over fields with enough roots of unity. As a consequence, we show that for every odd prime $p$, every integer $n\geq 3$, and every field $F$ containing a primitive $p$-th root of unity, there exists a continuous $n$-dimensional mod $p$ representation of the absolute Galois group of $F(x_1,\dots,x_p)$ which does not lift modulo $p^2$. This answers a question of Khare and Serre, and disproves a conjecture of Florence.