\mathscrD-elliptic sheaves and the Hasse principle

2024/09/28 by Keisuke Arai, Shin Hattori, Arai, Keisuke +5
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.2409.19268

openalex publication_date 2024/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be a rational prime, q>1 a power of p and F=\mathbbF_q(t). For an integer d≥ 2, let D be a central division algebra over F of dimension d² which is split at ∞ and has invariant inv_x(D)=1/d at any place x of F at which D ramifies. Let X^D be the Drinfeld--Stuhler variety, the coarse moduli scheme of the algebraic stack over F classifying \mathscrD-elliptic sheaves. In this paper, we establish various arithmetic properties of \mathscrD-elliptic sheaves to give an explicit criterion for the non-existence of rational points of X^D over a finite extension of F of degree d. As an application, for d=2, we present explicit infinite families of quadratic extensions of F over which the curve X^D violates the Hasse principle.

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