Sato-Tate distribution of p-adic hypergeometric functions

Recently Ono, Saad and the second author [21] 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 pth Hecke operators acting on the spaces of cusp forms of even weight \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 4$$\end{document}k≥4 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 [1, 2, 6].


Introduction
Let p be an odd prime and let F q denote the finite field with q = p r elements. In [11], Greene introduced Gaussian hypergeometric functions over finite fields using Jacobi sums. An important fact regarding these functions is that they satisfy many identities that are analogous to those satisfied by classical hypergeometric functions. Recently Ono, Saad and the second author [21] initiated a study of value distributions of certain families of these functions over random large finite fields F q . More precisely, they investigated the distributions of the normalized values of the following two Gaussian hypergeometric functions over F q that are defined by 2  A(x)B(1 − x) is the normalized Jacobi sum. Ono et al. [21] showed that for the 2 F 1 (λ) q functions the limiting distribution is semicircular whereas the distribution for the 3 F 2 (λ) q functions is the Batman distribution for the traces of the real orthogonal group O (3). In this paper our goal is to study similar questions for p-adic hypergeometric functions over random large finite fields.
McCarthy [17,18] defined p-adic hypergeometric functions using p-adic gamma functions extending Greene's hypergeometric functions [11] for wider classes of primes. These functions appear in the study of Frobenius trace of elliptic curves [17], Fourier coefficients of Hecke eigen forms [22], and proofs of supercongruence type identities [10].
Let p (·) be the Morita's p-adic gamma function (see [16]). Let ω be the Teichmüller character 2 of F p and ω denote its character inverse. For x ∈ Q let x denote the greatest integer less than or equal to x and x denote the fractional part of x, satisfying 0 ≤ x < 1. In these notation p-adic hypergeometric function is defined as follows.
In this paper we aim to investigate the value distributions of certain families of these functions over large finite fields F p . Namely, we study the value distributions of the following families of n G n -functions for n = 2, 6. Let ψ 6 = ω p−1 6 and ψ 3 = ω p−1 3 be characters of order 6 and 3, respectively. Then for λ ∈ F p we define 2 G 2 (λ) p := pψ 6 (2)φ(1 + λ)ψ 3 As our first theorem we obtain the moments of values of 2 G 2 (λ) p .

Theorem 1.2 Let m be a fixed positive integer and p ≡ 1 (mod 3) be a prime. Then as
We use these moments to conclude the limiting behaviour of 2 G 2 (λ) p as p → ∞. If we view the normalized values p −1/2 · 2 G 2 (λ) p ∈ [−2, 2] as random variables over F p then we obtain the limiting distribution of this function. More precisely we have the following distribution.
We also consider these problems for the 6 G 6 (λ) p functions.

Theorem 1.4 Let m be a fixed positive integer and p ≡ 2 (mod 3) be a prime. Then as
Similarly as in Corollary 1.3, we conclude the limiting distribution of 6 in the following Corollary.
Remark 1.6 It is important to note that these results can be extended to 2 G 2 (λ) q and 6 G 6 (λ) q over finite fields F q where q = p r using similar arguments. For simplicity we choose q = p.

Traces of Hecke operators and hypergeometric functions
It turns out that the hypergeometric functions 2 G 2 (λ) p and 6 G 6 (λ) p can be related to the traces of Hecke operators acting on the spaces of cusp forms. More precisely, for positive integers N and k, let S(N, k) be the space of cusp forms of weight k with respect to the congruence subgroup 0 (N ) and let Tr k ( 0 (N ), p) denote the trace of the pth Hecke operator acting on the space S(N, k). In a series of papers Ahlgren [1], Ahlgren-Ono [2] and Frechette-Ono-Papanikolas [6] studied the Eichler-Selberg type trace formulas for the pth Hecke operators acting on the spaces S(N, k) for N = 2, 4, 8 and established trace formulas using Gaussian hypergeometric functions over finite fields. Here we obtain the following results expressing the trace formulas Tr k ( 0 (4), p) and Tr k ( 0 (8), p) using p-adic hypergeometric functions 2 G 2 (λ) p and 6 G 6 (λ) p . Note that both these functions vanish at λ = −1, therefore, we define refinements of these functions. To this end we define the following two functions.
where A 4 is a character of order 4, . Moreover, let P k (s, p) be a polynomial defined as follows In this notation, we have the following results.
is an odd prime and k ≥ 4 is even then and

Theorem 2.2 If p ≡ 2 (mod 3) is an odd prime and k ≥ 4 is even then
and The paper is organized as follows. In Sect. 3 we provide some important results on p-adic gamma functions and Gauss sums. Namely, the Davenport-Hasse Relation and the Gross-Koblitz formula. Sect. 4 contains the proof of the main results.

Gauss sums and p-adic gamma function
In this section we discuss some important theorems, namely the Gross-Koblitz formula and Davenport-Hasse relation. We also recall some basic results regarding Gauss sums and p-adic gamma functions. We begin by recalling the orthogonality relations satisfied by multiplicative characters. Let F × p denote the cyclic group of multiplicative characters of F × p .

Lemma 3.1 ([13, Chapter 8]). If χ is a multiplicative character
x∈F p Let ζ p be a primitive pth root of unity and χ be a multiplicative character F × p . Then the Gauss sum is defined by Theorem 3.2 ([3, Davenport-Hasse Relation]). Let n be a positive integer and let p be a prime such that p ≡ 1 (mod n). For multiplicative characters χ, ψ ∈ F × p , we have Let Z p and Q p denote the ring of p-adic integers and the field of p-adic numbers, respectively. Let Q p be the algebraic closure of Q p and C p be the completion of Q p . For a positive integer n the p-adic gamma function p (n) is defined as The domain of definition of the p-adic gamma function can be extended to all x ∈ Z p by simply setting p (0) := 1 and for x = 0 where x n is any sequence of positive integers p-adically approaching to x. For more details on p-adic gamma functions we refer [16]. To this end we recall Gross-Koblitz formula which relates Gauss sums to p-adic gamma functions. Let π ∈ C p be the fixed root of the polynomial x p−1 + p, which satisfies the congruence condition π ≡ ζ p − 1 (mod (ζ p − 1) 2 ). Now we recall (see [16]) a p-adic analogue of the Davenport-Hasse relation and a product formula of p-adic gamma functions. If n ∈ Z + , p n and x = r p−1 with 0 ≤ r ≤ p − 1, then and we also note that

Proofs of theorems
be the Legendre normal form of elliptic curve over F p and be the trace of Frobenius of the elliptic curve E Leg λ . Our main idea is to express the functions 2 G 2 (λ) p and 6 G 6 (λ) p in terms of the traces of Frobenius a p (λ). Therefore, to obtain the asymptotic formulas of power moments of these functions it is sufficient if we have asymptotic formulas for the power moments of a p (λ). In [21] Ono et al. computed the asymptotic formulas for power moments of the functions 2 F 1 (λ) p . Here we reformulate their formulas in terms of power moments of a p (λ) in the following theorem.   [5,14] have refined Birch's result to arithmetic progressions.
Also it is easy to verify that Using these two facts in (4.2) we obtain Now, Gross-Koblitz formula allow us to write Replacing ω j by ω j ψ 3 we obtain Using the orthogonality of multiplicative characters and then replacing u by −(1 + λ)uy we deduce that .
If we use these three relations in the exponent of π present in H(λ) then the Gross-Koblitz formula allows to write By making use of Davenport-Hasse relation one can have g(ω 6j )g(φω 6j ) = g(ω 12j )ω 6j (2 −2 )g(φ). (4.11) Therefore, using (4.11) we have Since p ≡ 2 (mod 3) therefore transforming j → −j/3 we obtain Now observe that the above sum is equal to the sum involved in (4.4) up to a scalar multiple. Therefore following similar steps we obtain Therefore, if λ = 0, ±1 then 6 G 6 (λ) p = φ(−1) · a p (λ). (4.12) The rest of the proof relies on similar arguments as in the proof of Theorem 1.2.
If k ≥ 4 is even then Proposition 2.1 of [6] gives Therefore, we conclude the results by replacing a p (λ) by 2 G 2 (λ) p in the above two trace formulas.
Proof of Theorem 2.2 The proof follows similarly as in the proof of Theorem 2.1. The only requirement is to show for λ = 0, 1 6 G 6 (λ) p = a p (λ). This is straightforward by revisiting (4.12), (4.13) and the definition of 6 G 6 (−1) p .