On Sato--Tate distributions, extremal traces, and real multiplication in genus 2

2020/12/19 by David Kohel, Kohel, David, Yih-Dar Shieh +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2012.10805

openalex publication_date 2020/12/19 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

The vertical Sato--Tate conjectures gives expected trace distributions for for families of curves. We develop exact expression for the distribution associated to degree-4 representations of USp(4), SU(2)\timesSU(2) and SU(2) in the neighborhood of the extremities of the Weil bound. As a consequence we derive qualitative distinctions between the extremal traces arising from generic genus-2 curves and genus-2 curves with real or quaternionic multiplication. In particular we show, in a specific sense, to what extent curves with real multiplication dominate the contribution to extremal traces.

Related

• The Sato-Tate Distribution in Thin Parametric Families of Elliptic Curves
• Sato-Tate groups of genus 2 curves
• Refinements on vertical Sato-Tate
• Sato-Tate Distributions of y²=x^p-1 and y²=x^2p-1
• The Sato-Tate conjecture for a Picard curve with Complex Multiplication