Refinements of Katz-Sarnak theory for the number of points on curves over finite fields

2023/03/31 by Jonas Bergström, Bergström, Jonas, Everett W. Howe +5
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2303.17825

openalex publication_date 2023/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

This paper goes beyond Katz-Sarnak theory on the distribution of curves over finite fields according to their number of rational points, theoretically, experimentally and conjecturally. In particular, we give a formula for the limits of the moments measuring the asymmetry of this distribution for (non-hyperelliptic) curves of genus g ≥ 3. The experiments point to a stronger notion of convergence than the one provided by the Katz-Sarnak framework for all curves of genus ≥ 3. However, for elliptic curves and for hyperelliptic curves of every genus we prove that this stronger convergence cannot occur.

Citations

• Torelli Groups and Geometry of Moduli Spaces of Curves
• On the distribution of the trace in the unitary symplectic group and the\n distribution of Frobenius
• Hyperelliptic curves, the scanning map, and moments of families of quadratic L-functions
• Cohomology of moduli spaces via a result of Chenevier and Lannes

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• The number of rational points of hyperelliptic curves over subsets of finite fields