A zig-zag conjecture and local constancy for Galois representations

2019/03/20 by Eknath Ghate, Ghate, Eknath · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1903.08996

openalex publication_date 2019/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We make a zig-zag conjecture describing the reductions of irreducible crystalline two-dimensional representations of G_ℚp of half-integral slopes and exceptional weights. Such weights are two more than twice the slope mod (p-1). We show that zig-zag holds for half-integral slopes at most (3)/(2). We then explore the connection between zig-zag and local constancy results in the weight. First we show that known cases of zig-zag force local constancy to fail for small weights. Conversely, we explain how local constancy forces zig-zag to fail for some small weights and half-integral slopes at least 2. However, we expect zig-zag to be qualitatively true in general. We end with some compatibility results between zig-zag and other results.

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