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

TL;DR

This paper develops exact trace distribution formulas for genus-2 curves related to Sato--Tate conjectures, highlighting how real multiplication influences extremal trace behavior and distinguishing it from generic cases.

Contribution

It provides explicit distribution expressions for degree-4 representations of symplectic and special unitary groups near extremities, revealing the impact of real multiplication on extremal traces in genus-2 curves.

Findings

Derived exact trace distributions for specific group representations.

Identified qualitative differences between generic and real multiplication curves.

Showed dominance of real multiplication curves in extremal trace contributions.

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 $\mathrm{USp}(4)$, $\mathrm{SU}(2)\times\mathrm{SU}(2)$ and $\mathrm{SU}(2)$ 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.