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

Alexander Merkurjev, Federico Scavia

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.

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

Abstract

For every finite group and every finite -module , we determine the subgroup of negligible classes in , in the sense of Serre, over fields with enough roots of unity. As a consequence, we show that for every odd prime , every integer , and every field containing a primitive -th root of unity, there exists a continuous -dimensional mod representation of the absolute Galois group of which does not lift modulo . This answers a question of Khare and Serre, and disproves a conjecture of Florence.

Paper Structure

This paper contains 11 sections, 15 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. The profinite inverse Galois problem
  3. The embedding problem with abelian kernel
  4. Lifting Galois representations
  5. The main theorem
  6. Non-liftable Galois representations
  7. Contents
  8. Negligible cohomology
  9. The map gamma
  10. Proof of Theorem \ref{['thm-negligible']}
  11. Proof of Theorem \ref{['thm-galois-rep']}

Key Result

Theorem 1.3

Let $H$ be a finite group, let $A$ be a finite $H$-module, and let $F$ be a field containing a primitive root of unity of order $e(A) e(H)$. Then the subgroup of negligible classes in $H^2(H,A)$ is generated by the images of the maps where $H'$ ranges over all subgroups of $H$.

Theorems & Definitions (43)

  • Conjecture 1.2: Florence
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Example 2.2: Kummer theory
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 33 more