We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental properties and prove summation formulas, transformation formulas and product formulas. An application to zeta functions of K3-surfaces is given. In the appendix, we give an elementary proof of the Davenport-Hasse multiplication formula for Gauss sums.
Many product formulas are known classically for generalized hypergeometric functions over the complex numbers. In this paper, we establish some analogous formulas for generalized hypergeometric functions over finite fields.
We define finite field $A$-hypergeometric functions and show that they are Fourier expansions of families of exponential sums on the torus. For an appropriate choice of $A$, our finite field $A$-hypergeometric function can be specialized to the finite field ${}_kF_{k-1}$-hypergeometric function defined by McCarthy.
Finite hypergeometric functions are functions of a finite field ${\bf F}_q$ to ${\bf C}$. They arise as Fourier expansions of certain twisted exponential sums and were introduced independently by John Greene and Nick Katz in the 1980's. They have many properties in common with their analytic counterparts, the hypergeometric functions. One restriction in the definition of finite hypergeometric functions is that the hypergeometric parameters must be rational numbers whose denominators divide $q-1$. In this note we use the symmetry in the hypergeometric parameters and an extension of the exponent…
Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions over finite fields in a manner that is parallel to that of the classical hypergeometric functions. Using a comparison between the classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of hypergeometric transformation and evaluation formulas over finite fields and interpre…
We give an expression for the trace of Frobenius for the family of curves \[ y^3 = x (x-1)(x-λ)(x-μ) \] over finite fields in terms of finite field hypergeometric functions.
We compute the Artin $L$-function of a diagonal hypersurface D_λ over a finite field associated to a character of a finite group acting on D_λ , and under some condition, express it in terms of hypergeometric functions and Jacobi sums over the finite field. As an application, we consider the Dwork hypersurfaces and obtain relations among certain hypergeometric functions over different finite fields.
It is a well known result that the number of points over a finite field on the Legendre family of elliptic curves can be written in terms of a hypergeometric function modulo $p$. In this paper, we extend this result, due to Igusa, to a family of monomial deformations of a diagonal hypersurface. We find explicit relationships between the number of points and generalized hypergeometric functions as well as their finite field analogues.
We prove finite field analogues of integral representations of Appell- Lauricella hypergeometric functions in many variables. We consider certain hypersurfaces having a group action and compute the numbers of rational points associated with characters of the group, which will be expressed in terms of Appell-Lauricella functions over finite fields.
We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the well-poised classical series. We also discuss this function's relationship to other finite field analogues of the classical series, most notably those defined by Greene and Katz.
In 1984, the second author conjectured a quadratic transformation formula which relates two hypergeometric 2F1 functions over a finite field F_q. We prove this conjecture and give an application. The proof depends on a new linear transformation formula for pseudo hypergeometric functions over F_q.
We consider natural polynomial truncations of hypergeometric power series defined over finite fields. For these truncations, we establish asymptotic upper bounds of order $O(p^{11/12})$ on the number of roots in the prime field $\mathbb{F}_p$. We discuss the correspondence to families of elliptic curves and K3 surfaces of certain such hypergeometric polynomials, for which sharp bounds are obtained in some cases. We include some computations to illustrate and supplement our results.
Although links between values of finite field hypergeometric functions and eigenvalues of elliptic modular forms are well known, we establish in this paper that there are also connections to eigenvalues of Siegel modular forms of higher degree. Specifically, we relate the eigenvalue of the Hecke operator of index $p$ of a Siegel eigenform of degree 2 and level 8 to a special value of a ${}_4F_3$-hypergeometric function.
Finite hypergeometric functions are complex valued functions on finite fields which are the analogue of the classical analytic hypergeometric functions. From the work of N.M.Katz it follows that their values are traces of Frobenius on certain l-adic sheafs. More concretely, in many instances their values can be used to give formulas for pointcounts of F_q-rational points on certain varieties. In this paper we work out the case of one-variable functions whose monodromy in the analytic case can be defined over the rational integers.
In \cite{mccarthy2}, McCarthy defined a function $_{n}G_{n}[\cdots]$ using the Teichmüller character of finite fields and quotients of the $p$-adic gamma function. This function extends hypergeometric functions over finite fields to the $p$-adic setting. In this paper, we give certain transformation formulas for the function $_{n}G_{n}[\cdots]$ which are not implied from the analogous hypergeometric functions over finite fields.
For non-negative integers $l_{1}, l_{2},\ldots, l_{n}$, we define character sums $φ_{(l_{1}, l_{2},\ldots, l_{n})}$ and $ψ_{(l_{1}, l_{2},\ldots, l_{n})}$ over a finite field which are generalizations of Jacobsthal and modified Jacobsthal sums, respectively. We express these character sums in terms of Greene's finite field hypergeometric series. We then express the number of points on the hyperelliptic curves $y^2=(x^m+a)(x^m+b)(x^m+c)$ and $y^2=x(x^m+a)(x^m+b)(x^m+c)$ over a finite field in terms of the character sums $φ_{(l_{1}, l_{2}, l_{3})}$ and $ψ_{(l_{1}, l_{2}, l_{3})}$, and finally ob…
We define a finite-field version of Appell-Lauricella hypergeometric functions built from period functions in several variables, paralleling the development by Fuselier, et. al in the single variable case. We develop geometric connections between these functions and the family of generalized Picard curves. In our main result, we use finite-field Appell-Lauricella functions to establish a finite-field analogue of Koike and Shiga's cubic transformation for the Appell hypergeometric function $F_1$, proving a conjecture of Ling Long. We use our multivariable period functions to construct formulas …
We give an expression for number of points for the family of Dwork K3 surfaces $$X_λ^4: \hspace{.1in} x_1^4+x_2^4+x_3^4+x_4^4=4λx_1x_2x_3x_4$$ over finite fields of order $q\equiv 1\pmod 4$ in terms of Greene's finite field hypergeometric functions. We also develop hypergeometric point count formulas for all odd primes using McCarthy's $p$-adic hypergeometric function. Furthermore, we investigate the relationship between certain period integrals of these surfaces and the trace of Frobenius over finite fields. We extend this work to higher dimensional Dwork hypersurfaces.
We prove explicit formulas for certain first and second moment sums of families of Gaussian hypergeometric functions $_{n+1}F_n$, $n\ge1$, over finite fields with $q$ elements where $q$ is an odd prime. This enables us to find an estimate for the value $_6F_5(1)$. In addition, we evaluate certain second moments of traces of the family of Clausen elliptic curves in terms of the value $_3F_2(-1)$. These formulas also allow us to express the product of certain $_2F_1$ and $_{n+1}F_n$ functions in terms of finite field Appell series which generalizes current formulas for products of $_2F_1$ functi…
We find summation identities and transformations for the McCarthy's $p$-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family $$Z_λ: x_1^d+x_2^d=dλx_1x_2^{d-1}$$ over a finite field $\mathbb{F}_p$. A. Salerno expresses the number of points over a finite field $\mathbb{F}_p$ 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 $\mathbb{F}_p$ on the family $Z_λ$ in terms of McCarthy's $p$-adic hypergeometric series for a…
Igusa noted that the Hasse invariant of the Legendre family of elliptic curves over a finite field of odd characteristic is a solution mod $p$ of a Gaussian hypergeometric equation. We show that any family of exponential sums over a finite field has a Hasse invariant which is a sum of products of mod $p$ solutions of $A$-hypergeometric systems.
In this article we find finite field analogues of certain product formulas satisfied by the classical hypergeometric series. We express product of two ${_2}F_1$-Gaussian hypergeometric series as ${_4}F_3$- and ${_3}F_2$-Gaussian hypergeometric series. We use properties of Gauss and Jacobi sums and our earlier works on finite field Appell series to deduce these product formulas satisfied by the Gaussian hypergeometric series. We then use these transformations to evaluate explicitly some special values of ${_4}F_3$- and ${_3}F_2$-Gaussian hypergeometric series. By counting points on CM elliptic …
For an odd prime $p$, let $ϕ$ denote the quadratic character of the multiplicative group ${\mathbb F}_p^\times$, where ${\mathbb F}_p$ is the finite field of $p$ elements. In this paper, we will obtain evaluations of the hypergeometric functions $ {}_2F_1\left(\begin{matrix} ϕψ& ψ\\ & ϕ\end{matrix};x\right)$, $x\in {\mathbb F}_p$, $x\neq 0, 1$, over ${\mathbb F}_p$ in terms of Hecke character attached to CM elliptic curves for characters $ψ$ of ${\mathbb F}_p^\times$ of order $3$, $4$, $6$, $8$, and $12$.
In this paper, we develop a finite field analogue for one of the Lauricella series, $F^{(n)}_A$. Extending results of Greene, a finite field analog for the multinomial coefficient is developed in order to express the Lauricella series in terms of binomial coefficients. We have further deduced certain transformation and reduction formulas for the Lauricella series $F^{(n)}_A$. Finally, we have obtained a number of generating functions for the Lauricella series $F^{(n)}_A$.
In this work we present an explicit relation between the number of points on a family of algebraic curves over $\F_{q}$ and sums of values of certain hypergeometric functions over $\F_{q}$. Moreover, we show that these hypergeometric functions can be explicitly related to the roots of the zeta function of the curve over $\F_{q}$ in some particular cases. A general conjecture relating these last two is presented and advances toward its proof are shown in the last section.
In this paper, we give a formula for the number of rational points on the Dwork hypersurfaces of degree six over finite fields by using Greene's finite-field hypergeometric function, which is a generalization of Goodson's formula for the Dwork hypersurfaces of degree four. Our formula is also a higher-dimensional and a finite field analogue of Matsumoto-Terasoma-Yamazaki's formula. Furthermore, we also explain the relation between our formula and Miyatani's formula.
We extend our previous work on hypergeometric point count formulas by proving that we can express the number of points on families of Dwork hypersurfaces $$X_λ^d: \hspace{.1in} x_1^d+x_2^d+\ldots+x_d^d=dλx_1x_2\cdots x_d$$ over finite fields of order $q\equiv 1\pmod d$ in terms of Greene's finite field hypergeometric functions. We prove that when $d$ is odd, the number of points can be expressed as a sum of hypergeometric functions plus $(q^{d-1}-1)/(q-1)$ and conjecture that this is also true when $d$ is even. The proof rests on a result that equates certain Gauss sum expressions with finite …
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 l…
We consider algebraic transformations of hypergeometric functions from a geometric point of view. Hypergeometric functions are shown to arise from the deRham realization of a hypergeometric motive. The $\ell$-adic realization of the motive gives rise to hypergeometric characters sums over finite fields. This helps to unify and explain some recent results about transformations of hypergeometric character sums.
Let $q$ be a prime power and $\mathbb{F}_q$ be a finite field with $q$ elements. Let $e$ and $d$ be positive integers. In this paper, for $d\geq2$ and $q\equiv1(\mathrm{mod}~ed(d-1))$, we calculate the number of points on an algebraic curve $E_{e,d}:y^e=x^d+ax+b$ over a finite field $\mathbb{F}_q$ in terms of $_dF_{d-1}$ Gaussian hypergeometric series with multiplicative characters of orders $d$ and $e(d-1)$, and in terms of $_{d-1}F_{d-2}$ Gaussian hypergeometric series with multiplicative characters of orders $ed(d-1)$ and $e(d-1)$. This helps us to express the trace of Frobenius endomorphis…
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.
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.
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 ${_7}F_6$ truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.
For a prime $p\equiv 3\pmod{4}$ and a positive integer $t$, let $q=p^{2t}$. Let $g$ be a primitive element of the finite field $\mathbb{F}_q$. The Peisert graph $P^\ast(q)$ is defined as the graph with vertex set $\mathbb{F}_q$ where $ab$ is an edge if and only if $a-b\in\langle g^4\rangle \cup g\langle g^4\rangle$. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in $P^\ast(q)$. We also give a new proof for the number of complete subgraphs of order three contained in $P^\ast(q)$ by evaluating certain characte…
In the 1980's, Greene defined {\it 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 $_2F_1$ functions, t…
The discrete tori are graph analogues of the real tori, which are defined by the Cayley graphs of a finite product of finite cyclic groups. In this paper, using the theory of the heat kernel on the discrete tori established by Chinta, Jorgenson and Karlsson, we derive an explicit prime geodesic theorem for the discrete tori, which is not an asymptotic formula. To describe the formula, we need generalizations of the classical Jacobi polynomials, which are defined by the Lauricella multivariable hypergeometric function of type C.
The purpose of this article is to give an explicit description, in terms of hypergeometric functions over finite fields, of zeta function of a certain type of smooth hypersurfaces that generalizes Dwork family. The point here is that we count the number of rational points employing both character sums and the theory of weights, which enables us to enlighten the calculation of the zeta function.
We provide formulas for traces of p-th Hecke operators in level 1 in terms of values of finite field 2F1-hypergeometric functions, extending previous work of the author to all odd primes p, instead of only those p=1 (mod 12). We first give a general level 1 trace formula in terms of the trace of Frobenius on a family of elliptic curves, and then we draw on recent work of Lennon to produce level 1 trace formulas in terms of hypergeometric functions for all primes p >3.
The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have…
We establish the supercongruences for the fourteen rigid hypergeometric Calabi--Yau threefolds over $\mathbb Q$ conjectured by Rodriguez-Villegas in 2003. Our first method is based on Dwork's theory of $p$-adic unit roots and it allows us to establish the supercongruences between the truncated hypergeometric series and the corresponding unit roots for ordinary primes. The other method makes use of the theory of hypergeometric motives, in particular, adapts the techniques from the recent work of Beukers, Cohen and Mellit on finite hypergeometric sums over $\mathbb Q$. Essential ingredients in e…
Let $λ\in \mathbb{Q}\setminus \{0, 1\}$ and $l \geq 2$, and denote by $C_{l,λ}$ the nonsingular projective algebraic curve over $\mathbb{Q}$ with affine equation given by $$y^l=x(x-1)(x-λ).$$ In this paper we define $Ω(C_{l, λ})$ analogous to the real periods of elliptic curves and find a relation with ordinary hypergeometric series. We also give a relation between the number of points on $C_{l, λ}$ over a finite field and Gaussian hypergeometric series. Finally we give an alternate proof of a result of \cite{rouse}.
We present here explicit relations between the traces of Frobenius endomorphisms of certain families of elliptic curves and special values of ${_{2}}F_1$-hypergeometric functions over $\mathbb{F}_q$ for $q \equiv 1 (\text{mod} 6)$ and $q \equiv 1 (\text{mod} 4)$.
