论文标题
$ p $ - adiC超几何功能
Sato-Tate Distribution of $p$-adic hypergeometric functions
论文作者
论文摘要
最近，Ono，Saad和第二作者\ Cite {Khn}开始了一项研究，研究了高斯高几何函数的某些家族在大的有限域中的价值分布。他们调查了两个高斯高几何功能的家族，并表明它们满足了半圆形和蝙蝠侠的分布。受其结果的激励，我们旨在研究在大型有限领域的$ p $ - 亚种设置中某些高几何功能的分布。特别是，我们考虑了$ p $ - adiC设置中的两个和六个参数族，并获得其限制分布在大的有限磁场上是半圆形的。在这样做的过程中，我们还表达了$ p $ th hecke操作员的痕迹，这些操作员在cusp形式的偶数$ k \ geq4 $和4和8级别的the of the of $ p $ - ad-adic高几何功能方面的痕迹。这些结果可以看作是$ p $ - adic类似于\ cite {ah，ah-ono，fop}的痕迹公式。
Recently Ono, Saad and the second author \cite{KHN} initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric functions and showed that they satisfy semicircular and Batman distributions. Motivated by their results we aim to study distributions of certain families of hypergeometric functions in the $p$-adic setting over large finite fields. In particular, we consider two and six parameters families of hypergeometric functions in the $p$-adic setting and obtain that their limiting distributions are semicircular over large finite fields. In the process of doing this we also express the traces of $p$th Hecke operators acting on the spaces of cusp forms of even weight $k\geq4$ and levels 4 and 8 in terms of $p$-adic hypergeometric function which is of independent interest. These results can be viewed as $p$-adic analogous of some trace formulas of \cite{ah, ah-ono, fop}.
