论文标题
在Sato-Tate分布，极端轨迹和属2中的真实乘法下
On Sato--Tate distributions, extremal traces, and real multiplication in genus 2
论文作者
论文摘要
垂直的Sato-Tate猜想为曲线家族提供了预期的痕量分布。我们开发了与$ \ Mathrm {USP}（4）$，$ \ MATHRM {SU}（2）（2）\ Times \ Mathrm {Su}（Su}（2）$和$ \ Mathrm {Surm {Surm {Surm {Surm {Surm {Surm {Surm {Surm {Surm {su}（2）$，我们在$ \ mathrm {usp}（4）$，$ \ MATHRM {USP}（4）$中开发精确表达式。结果，我们得出了由通用属 - $ 2 $曲线和属的$ 2 $曲线产生的极端痕迹之间的定性区别，具有真实或Quaternionic乘法。特别是，我们从特定意义上说明了实际乘法的曲线在多大程度上占主导地位的极端痕迹。
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.
