Sato-Tate distribution of \(p\)-adic hypergeometric functions. (English) Zbl 1518.11051
Let \(p\) be an odd prime, and let \(\mathbb{F}_q\) be the finite field with \(q=p^r\) elements. For positive integer \(n, k (1\leq k\leq n)\) let \(a_k, b_k\in\mathbb{Q}\cap\mathbb{Z}_p\), D. McCarthy [Pac. J. Math. 261, No. 1, 219–236 (2013; Zbl 1296.11079); Int. J. Number Theory 8, No. 7, 1581–1612 (2012; Zbl 1253.33024)] defined, using \(p\)-adic Gamma function, \(p\)-adic hypergeometric functions \(_{n}G_{n}\left[\begin{matrix} a_1& a_2\cdots &a_n \\
b_1& b_2& \cdots& b_n\end{matrix} |\, t \right]_p\).
The main results of this article are about \(_nG_n\) for \(n=2, 6\) and are formulated in the following theorem.
Theorem: Let \(m\) be a fixed positive integer and \(p\equiv 1\pmod 3\) for \(n=2\) (resp. \(p\equiv 2\pmod 3\) for \(n=6\)) be a prime. Then as \(p\to\infty\), \(\sum_{\lambda\in\mathbb{F}_p}nG_{n}(\lambda)_p^m\) is equal to \(a_m(p^{\frac{m}{2}+1})\) if \(m\) is odd, and to \(\frac{(2n)!}{n!(n+1)!}p^{n+1}+a_m(p^{n+1})\) if \(m=2n\) is even.
Corollary: Let \(-2\leq 2<b\leq 2\) and \(p\equiv 1\pmod 3\) for \(n=2\) (resp. \(p\equiv 2\pmod 3\) for \(n=6\)). Then \[ \lim_{p\to\infty}\frac{|\{\lambda\in\mathbb{F}_p\,|\, p^{-\frac{1}{2}}_{n}G_{n}(\lambda)_p\in[a,b]\,\}|}{p} =\displaystyle\frac{1}{2\pi} \int_a^b\sqrt{4-t^2}dt. \] As applications, the trace formulas for the \(p\)th Hecke operators acting on the space of cusp forms of weight \(4\) (resp. \(8\)) in terms of \(_2G_2\) (resp. \(_6G_6\)) are derived.
The main results of this article are about \(_nG_n\) for \(n=2, 6\) and are formulated in the following theorem.
Theorem: Let \(m\) be a fixed positive integer and \(p\equiv 1\pmod 3\) for \(n=2\) (resp. \(p\equiv 2\pmod 3\) for \(n=6\)) be a prime. Then as \(p\to\infty\), \(\sum_{\lambda\in\mathbb{F}_p}nG_{n}(\lambda)_p^m\) is equal to \(a_m(p^{\frac{m}{2}+1})\) if \(m\) is odd, and to \(\frac{(2n)!}{n!(n+1)!}p^{n+1}+a_m(p^{n+1})\) if \(m=2n\) is even.
Corollary: Let \(-2\leq 2<b\leq 2\) and \(p\equiv 1\pmod 3\) for \(n=2\) (resp. \(p\equiv 2\pmod 3\) for \(n=6\)). Then \[ \lim_{p\to\infty}\frac{|\{\lambda\in\mathbb{F}_p\,|\, p^{-\frac{1}{2}}_{n}G_{n}(\lambda)_p\in[a,b]\,\}|}{p} =\displaystyle\frac{1}{2\pi} \int_a^b\sqrt{4-t^2}dt. \] As applications, the trace formulas for the \(p\)th Hecke operators acting on the space of cusp forms of weight \(4\) (resp. \(8\)) in terms of \(_2G_2\) (resp. \(_6G_6\)) are derived.
Reviewer: Noriko Yui (Kingston)
MSC:
| 11G20 | Curves over finite and local fields |
| 11T24 | Other character sums and Gauss sums |
| 33E50 | Special functions in characteristic \(p\) (gamma functions, etc.) |
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