Lifting Galois representations via Kummer flags

2024/03/13 by Conti, Andrea, Demarche, Cyril, Florence, Mathieu · 2 citations
#11F80 #Algebraic Topology (math.AT) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2403.08888

Abstract

Let Γ be either i) the absolute Galois group of a local field F, or ii) the topological fundamental group of a closed connected orientable surface of genus g. In case i), assume that μ_p² ⊂ F. We give an elementary and unified proof that every representation ρ₁: Γ→ GL_d(\mathbbF_p) lifts to a representation ρ₂: Γ→ GL_d(ℤ/p²). [In case i), it is understood these are continuous.] The actual statement is much stronger: for all r ≥ 1, under "suitable" assumptions, triangular representations ρ_r: Γ→ B_d(ℤ/p^r) lift to ρ_r+1: Γ→ B_d(ℤ/p^r+1), in the strongest possible step-by-step sense. Here "suitable" is made precise by the concept of Kummer flag. An essential aspect of this work, is to identify the common properties of groups i) and ii), that suffice to ensure the existence of such lifts.

Cited by

• Lifting Galois representations via Galois cohomology
• Massey products and unipotent extensions with restricted ramification

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