Sato-Tate distributions of y2 = x − 1 and y2 = x2 − 1
Sato-Tate distributions of y2 = x − 1 and y2 = x2 − 1
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y2=x−1 和 y2=x2−1 的 Sato-Tate 分布
DOI：
10.1016/j.jalgebra.2022.01.002
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发表时间：
2022
影响因子：
0.9
通讯作者：
Heidi Goodson
中科院分区：
文献类型：
--
作者：
Melissa Emory;Heidi Goodson
本文确定了Sato-Tate群，并证明了y2 = xp − 1和y2 = x2 p− 1曲线的Jacobian的广义Sato-Tate猜想，其中p是奇素数.我们的结果依赖于这些曲线的雅可比矩阵是非退化的，这一事实，我们证明在文件中。此外，我们计算与Sato-Tate组相关的矩统计量。这些矩统计量可以用来验证等分布声明的广义Sato-Tate猜想，通过比较他们的矩统计量得到的迹线的归一化的L-多项式的曲线。
Abstract 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 and y 2= x 2 p− 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.