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

David Kohel (1) , Yih-Dar Shieh

1 I2M - Institut de Mathématiques de Marseille

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) × SU(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.

Domaines: Théorie des nombres [math.NT]

David Kohel, Yih-Dar Shieh. On Sato-Tate distributions, extremal traces, and real multiplication in genus 2. Arithmetic, Geometry, Cryptography and Coding Theory, AMS Contemporary Mathematics, 2021, 2021. ⟨hal-03089551⟩