In this paper we establish certain identities connecting $p$-adic hypergeometric functions with 4... more In this paper we establish certain identities connecting $p$-adic hypergeometric functions with 4-th twisted Kloosterman sheaf sum. To prove these identities we express certain character sum over finite field in terms of special values of $p$-adic hypergeometric functions. One conjecture of Evans behaves as a bridge to connect $p$-adic hypergeometric functions with Kloosterman sheaf sum. We also connect $p$-adic hypergeometric functions with Fourier coefficients of certain modular forms.
In the 1980’s, Greene defined hypergeometric functions over finite fields using Jacobi sums. The ... more In the 1980’s, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Apéry-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the 2F1 functions, the limiting distribution is semicircular, whereas the distribution for the 3F2 functions is the more exotic Batman distribution.
Let p be an odd prime and Fp be the finite field with p elements. McCarthy [18] initiated a study... more Let p be an odd prime and Fp be the finite field with p elements. McCarthy [18] initiated a study of hypergeometric functions in the p-adic setting. This function can be understood as p-adic analogue of Gauss’ hypergeometric function, and also some kind of extension of Greene’s hypergeometric function over Fp. In this paper we investigate values of two generic families of McCarthy’s hypergeometric functions denoted by nGn(t), and nG̃n(t) for n ≥ 3, and t ∈ Fp. The values of the function nGn(t) certainly depend on whether t is n-th power residue modulo p or not. Similarly, the values of the function nG̃n(t) rely on the incongruent modulo p solutions of y − yn−1 + (n−1)n−1t nn ≡ 0 (mod p). These results generalize special cases of p-adic analogues of Whipple’s theorem and Dixon’s theorem of classical hypergeometric series. We examine zeros of the functions nGn(t), and nG̃n(t) over Fp. Moreover, we look into the values of t for which nGn(t) = 0 for infinitely many primes. For example, ...
In [14], McCarthy defined a function Gnn[...] using the Teichmuller character of finite fields an... more In [14], McCarthy defined a function Gnn[...] using the Teichmuller character of finite fields and quotients of the p-adic gamma function. He expressed the trace of Frobenius of elliptic curves in terms of special values of G22[...]. For d>=2, we establish four different expressions for the number of distinct zeros of the polynomials x^d+ax+b and x^d+ax^d^-^1+b over F"q in terms of special values of the function Gd-1d-1[...].
The classical AGM produces wonderful interdependent infinite sequences of arithmetic and geometri... more The classical AGM produces wonderful interdependent infinite sequences of arithmetic and geometric means with common limit. For finite fields Fq, with q ≡ 3 (mod 4), we introduce a finite field analogue AGMFq that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a jellyfish swarm, as the 3D renderings of the connected components resemble jellyfish (i.e. tentacles connected to a bell head). These swarms turn out to be more than the stuff of child’s play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of Fq-points, which can be used to prove that each swarm has at least (1/2 − ε) √ q many jellyfish. Additionally, this interpretation gives a description of the class numbers of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish. 1. ARITHMETIC AND GEOMETRIC MEANS. Beginning with positive real numbers a1 := a and b1 := b, ...
We classify all the zeros and non-zero values of three families of hypergeometric series in the p... more We classify all the zeros and non-zero values of three families of hypergeometric series in the p-adic setting. These values of hypergeometric series in the p-adic setting lead to transformations of hypergeometric series in the p-adic setting which can be described as p-adic analogues of Kummer's and Pfaff's linear transformations on classical hypergeometric series. We also evaluate certain summation identities for hypergeometric series in the p-adic setting as well as Gaussian hypergeometric series.
Mathematical Proceedings of the Cambridge Philosophical Society
We prove three more general supercongruences between truncated hypergeometric series and p-adic g... more We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.
We prove two transformations for the p-adic hypergeometric series which can be described as p-adi... more We prove two transformations for the p-adic hypergeometric series which can be described as p-adic analogues of a Kummer's linear transformation and a transformation of Clausen. We first evaluate two character sums, and then relate them to the p-adic hypergeometric series to deduce the transformations. We also find another transformation for the p-adic hypergeometric series from which many special values of the p-adic hypergeometric series as well as finite field hypergeometric functions are obtained.
We define four functions [Formula: see text] and [Formula: see text] as finite field analogues of... more We define four functions [Formula: see text] and [Formula: see text] as finite field analogues of Appell series [Formula: see text] and [Formula: see text], respectively using purely Gauss sums in the spirit of finite field hypergeometric series introduced by McCarthy. We establish relations among [Formula: see text] and [Formula: see text] analogous to those satisfied by the classical Appell series. Recently, several people have defined finite field analogues of Appell series using integral representations of Appell series. We show that our functions [Formula: see text] and [Formula: see text] are closely related to those functions.
We find summation identities and transformations for the Mc-Carthy's p-adic hypergeometric series... more We find summation identities and transformations for the Mc-Carthy's p-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family Z λ : x d 1 + x d 2 = dλx 1 x d−1 2 over a finite field Fp. A. Salerno expresses the number of points over a finite field Fp on the family Z λ in terms of quotients of p-adic gamma function under the condition that d|p − 1. In this paper, we first express the number of points over a finite field Fp on the family Z λ in terms of McCarthy's p-adic hypergeometric series for any odd prime p not dividing d(d − 1), and then deduce two summation identities for the p-adic hypergeometric series. We also find certain transformations and special values of the p-adic hypergeometric series. We finally find a summation identity for the Greene's finite field hypergeometric series.
We prove hypergeometric type summation identities for a function defined in terms of quotients of... more We prove hypergeometric type summation identities for a function defined in terms of quotients of the p-adic gamma function by counting points on certain families of hyperelliptic curves over Fq. We also find certain special values of that function.
Bulletin of the Australian Mathematical Society, 2016
We express the number of points on the Dwork hypersurface$X_{\unicode[STIX]{x1D706}}^{d}:x_{1}^{d... more We express the number of points on the Dwork hypersurface$X_{\unicode[STIX]{x1D706}}^{d}:x_{1}^{d}+x_{2}^{d}+\cdots +x_{d}^{d}=d\unicode[STIX]{x1D706}x_{1}x_{2}\cdots x_{d}$over a finite field of order$q\not \equiv 1\,(\text{mod}\,d)$in terms of McCarthy’s$p$-adic hypergeometric function for any odd prime$d$.
In [14], McCarthy defined a function Gnn[⋯] using the Teichmüller character of finite fields and ... more In [14], McCarthy defined a function Gnn[⋯] using the Teichmüller character of finite fields and quotients of the p -adic gamma function. He expressed the trace of Frobenius of elliptic curves in terms of special values of G22[⋯]. For d⩾2d⩾2, we establish four different expressions for the number of distinct zeros of the polynomials xd+ax+bxd+ax+b and xd+axd−1+bxd+axd−1+b over FqFq in terms of special values of the function Gd−1d−1[⋯].
ABSTRACT Let i and d be integers such that d≥2, 0id, and i|d. We explicitly find the number of so... more ABSTRACT Let i and d be integers such that d≥2, 0id, and i|d. We explicitly find the number of solutions of the polynomial equations x d +ax i +b=0 and x d +ax d−i +b=0 over \(\mathbb{F}_{q}\) in terms of special values of \(_{\frac{d-i}{i}}F_{\frac{d-2i}{i}}\) and \(_{\frac{d}{i}}F_{\frac{d-i}{i}}\) , Gaussian hypergeometric series with characters of orders \(\frac{d}{i}-1\) and \(\frac{d}{i}\) as parameters. This solves the problem posed by Ken Ono (Web of modularity: arithmetic of the coefficients of modular forms and q-series, vol. 102, p. 204, 2004) on special values of Gaussian hypergeometric series n+1F n for n>2.
ABSTRACT We express the trace of Frobenius of certain families of elliptic curves in terms of Gau... more ABSTRACT We express the trace of Frobenius of certain families of elliptic curves in terms of Gaussian hypergeometric functions. We also find some special values of F12 Gaussian hypergeometric functions containing characters of order 4 as parameters.
In [12], McCarthy defined a function n G n [• • •] using the Teichmüller character of finite fiel... more In [12], McCarthy defined a function n G n [• • •] using the Teichmüller character of finite fields and quotients of the p-adic gamma function, and expressed the trace of Frobenius of elliptic curves in terms of special values of 2 G 2 [• • •]. We establish two different expressions for the traces of Frobenius of elliptic curves in terms of the function 2 G 2 [• • •]. As a result, we obtain two transformation formulas of the function 2 G 2 [• • •] with different parameters.
In this paper we establish certain identities connecting $p$-adic hypergeometric functions with 4... more In this paper we establish certain identities connecting $p$-adic hypergeometric functions with 4-th twisted Kloosterman sheaf sum. To prove these identities we express certain character sum over finite field in terms of special values of $p$-adic hypergeometric functions. One conjecture of Evans behaves as a bridge to connect $p$-adic hypergeometric functions with Kloosterman sheaf sum. We also connect $p$-adic hypergeometric functions with Fourier coefficients of certain modular forms.
In the 1980’s, Greene defined hypergeometric functions over finite fields using Jacobi sums. The ... more In the 1980’s, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Apéry-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the 2F1 functions, the limiting distribution is semicircular, whereas the distribution for the 3F2 functions is the more exotic Batman distribution.
Let p be an odd prime and Fp be the finite field with p elements. McCarthy [18] initiated a study... more Let p be an odd prime and Fp be the finite field with p elements. McCarthy [18] initiated a study of hypergeometric functions in the p-adic setting. This function can be understood as p-adic analogue of Gauss’ hypergeometric function, and also some kind of extension of Greene’s hypergeometric function over Fp. In this paper we investigate values of two generic families of McCarthy’s hypergeometric functions denoted by nGn(t), and nG̃n(t) for n ≥ 3, and t ∈ Fp. The values of the function nGn(t) certainly depend on whether t is n-th power residue modulo p or not. Similarly, the values of the function nG̃n(t) rely on the incongruent modulo p solutions of y − yn−1 + (n−1)n−1t nn ≡ 0 (mod p). These results generalize special cases of p-adic analogues of Whipple’s theorem and Dixon’s theorem of classical hypergeometric series. We examine zeros of the functions nGn(t), and nG̃n(t) over Fp. Moreover, we look into the values of t for which nGn(t) = 0 for infinitely many primes. For example, ...
In [14], McCarthy defined a function Gnn[...] using the Teichmuller character of finite fields an... more In [14], McCarthy defined a function Gnn[...] using the Teichmuller character of finite fields and quotients of the p-adic gamma function. He expressed the trace of Frobenius of elliptic curves in terms of special values of G22[...]. For d>=2, we establish four different expressions for the number of distinct zeros of the polynomials x^d+ax+b and x^d+ax^d^-^1+b over F"q in terms of special values of the function Gd-1d-1[...].
The classical AGM produces wonderful interdependent infinite sequences of arithmetic and geometri... more The classical AGM produces wonderful interdependent infinite sequences of arithmetic and geometric means with common limit. For finite fields Fq, with q ≡ 3 (mod 4), we introduce a finite field analogue AGMFq that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a jellyfish swarm, as the 3D renderings of the connected components resemble jellyfish (i.e. tentacles connected to a bell head). These swarms turn out to be more than the stuff of child’s play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of Fq-points, which can be used to prove that each swarm has at least (1/2 − ε) √ q many jellyfish. Additionally, this interpretation gives a description of the class numbers of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish. 1. ARITHMETIC AND GEOMETRIC MEANS. Beginning with positive real numbers a1 := a and b1 := b, ...
We classify all the zeros and non-zero values of three families of hypergeometric series in the p... more We classify all the zeros and non-zero values of three families of hypergeometric series in the p-adic setting. These values of hypergeometric series in the p-adic setting lead to transformations of hypergeometric series in the p-adic setting which can be described as p-adic analogues of Kummer's and Pfaff's linear transformations on classical hypergeometric series. We also evaluate certain summation identities for hypergeometric series in the p-adic setting as well as Gaussian hypergeometric series.
Mathematical Proceedings of the Cambridge Philosophical Society
We prove three more general supercongruences between truncated hypergeometric series and p-adic g... more We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.
We prove two transformations for the p-adic hypergeometric series which can be described as p-adi... more We prove two transformations for the p-adic hypergeometric series which can be described as p-adic analogues of a Kummer's linear transformation and a transformation of Clausen. We first evaluate two character sums, and then relate them to the p-adic hypergeometric series to deduce the transformations. We also find another transformation for the p-adic hypergeometric series from which many special values of the p-adic hypergeometric series as well as finite field hypergeometric functions are obtained.
We define four functions [Formula: see text] and [Formula: see text] as finite field analogues of... more We define four functions [Formula: see text] and [Formula: see text] as finite field analogues of Appell series [Formula: see text] and [Formula: see text], respectively using purely Gauss sums in the spirit of finite field hypergeometric series introduced by McCarthy. We establish relations among [Formula: see text] and [Formula: see text] analogous to those satisfied by the classical Appell series. Recently, several people have defined finite field analogues of Appell series using integral representations of Appell series. We show that our functions [Formula: see text] and [Formula: see text] are closely related to those functions.
We find summation identities and transformations for the Mc-Carthy's p-adic hypergeometric series... more We find summation identities and transformations for the Mc-Carthy's p-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family Z λ : x d 1 + x d 2 = dλx 1 x d−1 2 over a finite field Fp. A. Salerno expresses the number of points over a finite field Fp on the family Z λ in terms of quotients of p-adic gamma function under the condition that d|p − 1. In this paper, we first express the number of points over a finite field Fp on the family Z λ in terms of McCarthy's p-adic hypergeometric series for any odd prime p not dividing d(d − 1), and then deduce two summation identities for the p-adic hypergeometric series. We also find certain transformations and special values of the p-adic hypergeometric series. We finally find a summation identity for the Greene's finite field hypergeometric series.
We prove hypergeometric type summation identities for a function defined in terms of quotients of... more We prove hypergeometric type summation identities for a function defined in terms of quotients of the p-adic gamma function by counting points on certain families of hyperelliptic curves over Fq. We also find certain special values of that function.
Bulletin of the Australian Mathematical Society, 2016
We express the number of points on the Dwork hypersurface$X_{\unicode[STIX]{x1D706}}^{d}:x_{1}^{d... more We express the number of points on the Dwork hypersurface$X_{\unicode[STIX]{x1D706}}^{d}:x_{1}^{d}+x_{2}^{d}+\cdots +x_{d}^{d}=d\unicode[STIX]{x1D706}x_{1}x_{2}\cdots x_{d}$over a finite field of order$q\not \equiv 1\,(\text{mod}\,d)$in terms of McCarthy’s$p$-adic hypergeometric function for any odd prime$d$.
In [14], McCarthy defined a function Gnn[⋯] using the Teichmüller character of finite fields and ... more In [14], McCarthy defined a function Gnn[⋯] using the Teichmüller character of finite fields and quotients of the p -adic gamma function. He expressed the trace of Frobenius of elliptic curves in terms of special values of G22[⋯]. For d⩾2d⩾2, we establish four different expressions for the number of distinct zeros of the polynomials xd+ax+bxd+ax+b and xd+axd−1+bxd+axd−1+b over FqFq in terms of special values of the function Gd−1d−1[⋯].
ABSTRACT Let i and d be integers such that d≥2, 0id, and i|d. We explicitly find the number of so... more ABSTRACT Let i and d be integers such that d≥2, 0id, and i|d. We explicitly find the number of solutions of the polynomial equations x d +ax i +b=0 and x d +ax d−i +b=0 over \(\mathbb{F}_{q}\) in terms of special values of \(_{\frac{d-i}{i}}F_{\frac{d-2i}{i}}\) and \(_{\frac{d}{i}}F_{\frac{d-i}{i}}\) , Gaussian hypergeometric series with characters of orders \(\frac{d}{i}-1\) and \(\frac{d}{i}\) as parameters. This solves the problem posed by Ken Ono (Web of modularity: arithmetic of the coefficients of modular forms and q-series, vol. 102, p. 204, 2004) on special values of Gaussian hypergeometric series n+1F n for n>2.
ABSTRACT We express the trace of Frobenius of certain families of elliptic curves in terms of Gau... more ABSTRACT We express the trace of Frobenius of certain families of elliptic curves in terms of Gaussian hypergeometric functions. We also find some special values of F12 Gaussian hypergeometric functions containing characters of order 4 as parameters.
In [12], McCarthy defined a function n G n [• • •] using the Teichmüller character of finite fiel... more In [12], McCarthy defined a function n G n [• • •] using the Teichmüller character of finite fields and quotients of the p-adic gamma function, and expressed the trace of Frobenius of elliptic curves in terms of special values of 2 G 2 [• • •]. We establish two different expressions for the traces of Frobenius of elliptic curves in terms of the function 2 G 2 [• • •]. As a result, we obtain two transformation formulas of the function 2 G 2 [• • •] with different parameters.
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