Zig-zag for Galois Representations

2022/11/22 by Eknath Ghate, Ghate, Eknath · 1 citation
Mathematics · #11F80 (Primary) #14G22 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2211.12114

openalex publication_date 2022/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The zig-zag conjecture says that the reductions of two-dimensional crystalline representations of the Galois group of \mathbb Q_p of large exceptional weights and half-integral slopes up to (p-1)/(2) vary through an alternating sequence of irreducible and reducible mod p representations. We prove this conjecture in smoothly varying families of such representations for p ≥ 5. The proof uses a limiting argument due to Chitrao-Ghate-Yasuda to reduce to the case of semi-stable representations of weights at most p+1, and then appeals to the work of Breuil-Mézard, Guerberoff-Park and Chitrao-Ghate.

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