The lifting problem for Galois representations

2025/01/31 by Alexander Merkurjev, Federico Scavia, Merkurjev, Alexander +1 · 1 citation
Computer Science · Engineering · Mathematics · #12F12 #12G05 (Primary) 11F80 #20J06 (Secondary) #Advanced Numerical Analysis Techniques #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2501.18906

openalex publication_date 2025/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We solve the lifting problem for Galois representations in every dimension and in every characteristic. That is, we determine all pairs (n,k), where n is a positive integer and k is a field of characteristic p>0, such that for every field F, every continuous homomorphism Γ_F→ GL_n(k) lifts to GL_n(W₂(k)), where Γ_F is the absolute Galois group of F and W₂(k) is the ring of p-typical length 2 Witt vectors of k.

Cited by

• Lifting Galois representations via Galois cohomology

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• Lifting Galois representations over arbitrary number fields
• Lifting N-dimensional Galois representations to characteristic zero
• Lifting Galois representations via Kummer flags
• Lifting low-dimensional local systems
• Lifting Galois representations via Galois cohomology