We define the notion of {\it Dwork family of logarithmic $F$-crystals}, a typical example of which is the family of Gauss hypergeometricdifferential systems, viewed as parametrized by their exponents of algebraic monodromy. The $p$-adic analytic dependence of the Frobenius operation upon those exponents, is Dwork's "Boyarsky Principle". We discuss, in favorable cases, the $p$-adic analytic continuation of the unit root $F$-subcrystal in the open tube of a singularity, uniformly w.r.t. the exponents. We obtain a conceptual proof of the Koblitz-Diamond formula $p$-adically analog to Gauss' evaluation of $F(a,b,c;1)$.
We study the $p$-adic (generalized) hypergeometric equations by using the theory of multiplicative convolution of arithmetic $\mathscr{D}$-modules. As a result, we prove that the hypergeometric isocrystals with suitable rational parameters have a structure of overconvergent $F$-isocrystals.
To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $p$-adic counterpart of the GKZ hypergeometric system. The $p$-adic GKZ hypergeometric complex is a twisted relative de Rham complex of over-convergent differential forms with logarithmic poles. It is an over-holonomic object in the derived category of arithmetic $\mathcal D$-modules with Frobenius structures. Traces of Frobenius on fibers at Techmüller points …
We identify the $p$-adic unit roots of the zeta function of a projective hypersurface over a finite field of characteristic $p$ as the eigenvalues of a product of special values of a certain matrix of $p$-adic series. That matrix is a product $F(Λ^p)^{-1}F(Λ)$, where the entries in the matrix $F(Λ)$ are $A$-hypergeometric series with integral coefficients and $F(Λ)$ is independent of $p$.
We classify all the zeros and non-zero values of a family 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.
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 two transformations for the $p$-adic hypergeometric functions which can be described as $p$-adic analogues of a Euler's transformation and a transformation of Clausen. We first evaluate certain character sums, and then relate them to the $p$-adic hypergeometric functions to deduce the transformations. We use a character sum identity proved by Ahlgren, Ono, and Penniston to deduce the $p$-adic Clausen's transformation. We also deduce special values of certain $p$-adic hypergeometric functions.
We prove hypergeometric type identities for a function defined in terms of quotients of the $p$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $_4F_3$ hypergeometric series and the Fourier coefficients of certain weight four modular form.
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.
We study classical hypergeometric series as a p-adic function of its parameters inspired by a problem in the American Mathematical Monthly solved by D. Zagier. This is an extended abstract of a talk given at the workshop "Hypergeometric motives and Calabi-Yau differential equations" at the Mathematical Research Institute (MATRIX) of The University of Melbourne in Creswick, Australia in January of 2017.
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 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 $\mathbb{F}_{q}$. We also find certain special values of that function.
We define a function which extends Gaussian hypergeometric series to the $p$-adic setting. This new function allows results involving Gaussian hypergeometric series to be extended to a wider class of primes. We demonstrate this by providing various congruences between the function and truncated classical hypergeometric series. These congruences provide a framework for proving the supercongruence conjectures of Rodriguez-Villegas.
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.
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…
Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$ elements. This paper focuses on the study of values of a generic family of hypergeometric functions in the $p$-adic setting which we denote by ${_{3n-1}G_{3n-1}}(p, t),$ where $n\geq1$ and $t\in\mathbb{F}_p$. These values are expressed in terms of numbers of zeros of certain polynomials over $\mathbb{F}_p$. These results lead to certain $p$-adic analogues of classical hypergeometric identities. Namely, we obtain $p$-adic analogues of particular cases of a Gauss' theorem and a Kummer's theorem. Moreover, we examine the zero…
Using an identity due to Gessel and Stanton and some properties of the $p$-adic Gamma function, we establish a $p$-adic supercongruence for truncated hypergeometric series ${}_7F_6$. From it we deduce some related supercongruences, which extend certain recent results and confirm a supercongruence conjecture.
Dwork's $p$-adic hypergeometric function is defined to be a ratio ${}_sF_{s-1}(t)/{}_sF_{s-1}(t^p)$ of hypergeometric power series. Dwork showed that it is a uniform limit of rational functions, and hence one can define special values on $|t|_p=1$. However to compute the value modulo $p^n$ in the naive method, the bit complexity increases by exponential when $n\to\infty$. In this paper we present a certain algorithm whose complexity increases at most $O(n^4(\log n)^3)$.
Using Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number p and not only for almost all primes. Along the way, using Christol's functions, we provide an explicit formula for the "Eisenstein constant" of any globally bounded hypergeometric series with rational parameters. As an application of these results, we obtain an arithmetic statement of a new type concerning the integrality of Tay…
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…
Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$ elements. McCarthy \cite{mccarthy-pacific} 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 $\mathbb{F}_p$. In this paper we investigate values of two generic families of McCarthy's hypergeometric functions denoted by ${_nG_n}(t)$, and ${_n\widetilde{G}_n}(t)$ for $n\geq3$, and $t\in\mathbb{F}_p$. The values of the function ${_nG_n}(t)$ certai…
We give an exposition of Dwork's construction of Frobenius structures associated to generalized hypergeometric equations via the interpretation of the latter due to Gelfand-Kapranov-Zelevinsky in the language of A-hypergeometric systems. As a consequence, we extract some explicit formulas for the degeneration at 0 in terms of the Morita p-adic gamma function.
This is a sequel of the paper "A generalization of the Ross symbols in higher K-groups and hypergeometric functions I" where we introduced higher Ross symbols in higher $K$-groups of the hypergeometric schemes, and discussed the Beilinson regulators. In this paper we give its p-adic counterpart and an application to the $p$-adic Beilinson conjecture for K3 surfaces of Picard number 20.
In this paper, we confirm several conjectures posed by Sun recently; for example, we prove that for any odd prime $p$ we have $$ \sum_{k=0}^{p-1}A_k\equiv\begin{cases}4x^2-2p\pmod{p^2}\quad&\text{if $p=x^2+2y^2\ (x,y\in\mathbb{Z})$},\\ 0\pmod{p^2}\quad&\text{if $p\equiv5,7\pmod{8}$},\end{cases} $$ where $A_n:=\sum_{k=0}^n\binom{n+k}{k}^2\binom{n}{k}^2$ are the Apéry numbers.
We study a family of hypergeometric motives $H(α,β|t)$ attached to a pair of tuples $α=(1/4,1/2,3/4)$, $β=(0,0,0)$. To each such motive we can attach a system of $\ell$--adic realisations with the trace of geometric Frobenius given by the evaluation of the finite field analogue of complex hypergeometric function. Geometry of elliptic fibrations makes it possible to to realise the motive $H(α,β|t)$ as a pure Chow motive attached to a suitable K3 surface $V_{t}$.
We establish some supercongruences for the truncated ${}_2F_1$ and ${}_3F_2$ hypergeometric series involving the $p$-adic Gamma functions. Some of these results extend the four Rodriguez-Villegas supercongruences on the truncated ${}_3F_2$ hypergeometric series. The corresponding conjectural supercongruences modulo $p^3$ are also proposed for further research.
For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of $p$-adic Gamma functions that are stronger than those one can expect to prove using commutative formal group laws. We prove a number of such supercongruences by using classical hypergeometric transformation formulae. These formulae (see the appendix), most of which are decades or centuries old, allow us to write the terminating series as the ratio of products of of $Γ$-values. At this point sums have become quotients. Writing these $Γ$-quotients as $Γ_p$-quotients, we are …
We define a function in terms of quotients of the $p$-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the $p$-adic setting. We prove, for primes $p > 3$, that the trace of Frobenius of any elliptic curve over $\mathbb{F}_p$, whose $j$-invariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of Fuselier and Lennon which evaluate the trace of Frobenius in terms of hypergeometric functions over $\mathbb{F}_p$ when $p \equiv 1 \pmod {12}$.
Let $D_λ^d$ denote the family of monomial deformations of diagonal hypersurface over a finite field $\mathbb{F}_q$ given by \begin{align*} D_λ^d: X_1^d+X_2^d+\cdots+X_n^d=λd X_1^{h_1}X_2^{h_2}\cdots X_n^{h_n}, \end{align*} where $d,n\geq2$, $h_i\geq1$, $\sum_{i=1}^n h_i=d$, and $\gcd(d,h_1,h_2,\ldots, h_n)=1$. The Dwork hypersurface is the case when $d=n$, that is, $h_1=h_2=\cdots =h_n=1$. Formulas for the number of $\mathbb{F}_q$-points on the Dwork hypersurfaces in terms of McCarthy's $p$-adic hypergeometric functions are known. In this article we provide a formula for the number of $\mathbb…
We express the number of points on the Dwork hypersurface $$X_λ^d: x_1^d+x_2^d+\cdots +x_d^d=dλx_1x_2\cdots x_d$$ over a finite field of order $q \not \equiv 1 \pmod{d}$ in terms of McCarthy's $p$-adic hypergeometric function for any odd prime $d$.
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.
During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured $p$-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures.
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.
We provide a formula for the number of $\mathbb{F}_{p}$-points on the Dwork hypersurface $$x_1^n + x_2^n \dots + x_n^n - n λ\, x_1 x_2 \dots x_n=0$$ in terms of a $p$-adic hypergeometric function previously defined by the author. This formula holds in the general case, i.e for any $n, λ\in \mathbb{F}_p^{*}$ and for all odd primes $p$, thus extending results of Goodson and Barman et al which hold in certain special cases.
We give a formula for the cyclotomic valuation of $q$-Pochhammer symbols in terms of (generalized) Dwork maps. We also obtain a criterion for the $q$-integrality of basic hypergeometric series in terms of certain step functions, which generalize Christol step functions. This provides suitable $q$-analogs of two results proved by Christol: a formula for the $p$-adic valuation of Pochhammer symbols and a criterion for the $N$-integrality of hypergeometric series.
By a "generalized Calabi-Yau hypersurface" we mean a hypersurface in ${\mathbb P}^n$ of degree $d$ dividing $n+1$. The zeta function of a generic such hypersurface has a reciprocal root distinguished by minimal $p$-divisibility. We study the $p$-adic variation of that distinguished root in a family and show that it equals the product of an appropriate power of $p$ times a product of special values of a certain $p$-adic analytic function ${\mathcal F}$. That function ${\mathcal F}$ is the $p$-adic analytic continuation of the ratio $F(Λ)/F(Λ^p)$, where $F(Λ)$ is a solution of the $A$-hypergeome…
Dwork's conjecture, now proven by Wan, states that unit root L-functions "coming from geometry" are p-adic meromorphic. In this paper we study the p-adic variation of a family of unit root L-functions coming from a suitable family of toric exponential sums. In this setting, we find that the unit root L-functions each have a unique p-adic unit root. We then study the variation of this unit root over the family of unit root L-functions. Surprisingly, we find that this unit root behaves similarly to the classical case of families of exponential sums. That is, the unit root is essentially a ratio …
In the late '60s, B. Dwork studied a Frobenius structure compatible with the classical hypergeometric differential equation with parameters $\left(\frac{1}{2},\frac{1}{2} ; 1 \right)$ by analyzing behavior of solutions of the differential equation under Frobenius transformation. Recently, P. Scholze conjectured the existence of $q$-de Rham cohomology groups for any $\mathbb{Z}$-scheme. In this paper, we give a Frobenius structure compatible with the $q$-hypergeometric differential equation with parameters $(q^{\frac12},q^{\frac12};q)$ by showing a $q$-analogue of some results of Dwork. This co…
A well-known formula of Whipple relates certain hypergeometric values $_7F_6(1)$ and $_4F_3(1)$. In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data $HD$, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data $HD$ are primitive and self-dual. If the data are also defined over $\mathbb Q$, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of $\ell$-adic representations of the absolute Galois group of …
Starting from a classical generating series for Bessel functions due to Schlomilch, we use Dwork's relative dual theory to broadly generalize unit-root results of Dwork on Kloosterman sums and Sperber on hyperkloosterman sums. In particular, we express the (unique) p-adic unit root of an arbitrary exponential sum on the torus in terms of special values of the p-adic analytic continuation of a ratio of A-hypergeometric functions. In contrast with the earlier works, we use noncohomological methods and obtain results that are valid for arbitrary exponential sums without any hypothesis of nondegen…
We give a generalization of $p$-adic congruences for truncated period functions, that were originally discovered for a class of hypergeometric functions by Bernard Dwork.
Several new $q$-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Wadim Zudilin. More concretely, the results in this paper include $q$-analogues of supercongruences (referring to $p$-adic identities remaining valid for some higher power of $p$) established by Long, by Long and Ramakrishna, and several other $q$-supercongruences. The six basic hypergeometric transformation formulas which ar…
