论文标题
$ y^2 = x^p-1 $和$ y^2 = x^{2p} -1 $的sato-tate分布
Sato-Tate Distributions of $y^2=x^p-1$ and $y^2=x^{2p}-1$
论文作者
论文摘要
我们确定了Sato-tate组，并证明了表格$$ y^2 = x^p-1 \ text {and} y^2 = x^{2p} -1，$ p $其中$ p $的曲线的广义sato-tate猜想。我们的结果依赖于这些曲线的雅各布人的事实是非排定的，我们在论文中证明了这一事实。此外，我们计算与Sato-Tate组相关的力矩统计数据。这些时刻的统计信息可用于验证广义sato-tate猜想的等分分配声明，通过将它们与曲线归一化$ L $ l $ - 多种物质的痕迹获得的矩统计进行比较。
We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form $$ y^2=x^p-1 \text{ and } y^2=x^{2p}-1,$$ where $p$ is an odd prime. Our results rely on the fact the Jacobians of these curves are nondegenerate, a fact that we prove in the paper. Furthermore, we compute moment statistics associated to the Sato-Tate groups. These moment statistics can be used to verify the equidistribution statement of the generalized Sato-Tate conjecture by comparing them to moment statistics obtained for the traces in the normalized $L$-polynomials of the curves.
