Points of bounded height on curves and the dimension growth conjecture over \mathbbF_q[t]

2020/03/24 by Floris Vermeulen, Vermeulen, Floris
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2003.10988

openalex publication_date 2020/03/24 · openalex created_date 2020/04/03 · openalex updated_date 2026/07/28

Abstract

In this article we prove several new uniform upper bounds on the number of points of bounded height on varieties over \mathbbF_q[t]. For projective curves, we prove the analogue of Walsh' result with polynomial dependence on q and the degree d of the curve. For affine curves, this yields an improvement to bounds by Sedunova, and Cluckers, Forey and Loeser. In higher dimensions, we prove a version of dimension growth for hypersurfaces of degree d≥ 64, building on work by Castryck, Cluckers, Dittmann and Nguyen in characteristic zero. These bounds depend polynomially on q and d, and it is this dependence which simplifies the treatment of the dimension growth conjecture.

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