Frobenius trace distributions for K3 surfaces

2021/02/21 by Andreas-Stephan Elsenhans, Elsenhans, Andreas-Stephan, Jörg Jahnel +1
Mathematics · #11T06 secondary #14F20 #14J10 #14J20 #14J28 primary #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11T06 #msc:14F20 #msc:14J10 #msc:14J20 #msc:14J28

paper · pdf · doi:10.48550/arxiv.2102.10620

Revised and extended version

openalex publication_date 2021/02/21 · arxiv created 2022/11/14 · arxiv updated 2022/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the distribution of the Frobenius traces on K3 surfaces. We compare experimental data with the predictions made by the Sato--Tate conjecture, i.e. with the theoretical distributions derived from the theory of Lie groups assuming equidistribution. Our sample consists of generic K3 surfaces, as well as of such having real and complex multiplication. Each time, the theoretical density and the histogram obtained by counting points match in the range of visible accuracy. Thus, we report evidence for the Sato-Tate conjecture for the surfaces considered.

Citations

• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques
• Modular Elliptic Curves and Fermat's Last Theorem

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