Let $\mathbb{F}_q$ be a finite field with $q$ elements, $ψ$ a non-zero element of $\mathbb{F}_q$, and $n$ an integer $\geq 3$ prime to $q$. The aim of this article is to show that the zeta function of the projective variety over $\mathbb{F}_q$ defined by $X_ψ\colon x_1^n+...+x_n^n - n ψx_1... x_n=0$ has, when $n$ is prime and $X_ψ$ is non singular (i.e. when $ψ^n \neq 1$), an explicit decomposition in factors coming from affine varieties of odd dimension $\leq n-4$ which are of hypergeometric type. The method we use consists in counting separately the number of points of $X_ψ$ and of some varieties of the preceding type and then compare them. This article answers, at least when $n$ is prime, a question asked by D. Wan in his article "Mirror Symmetry for Zeta Functions".
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.
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 $O(n^4(\log n)^3)$.
We describe the action of the Dwork-Frobenius operator on certain $A$-hypergeometric series. As a consequence, we obtain an integrality result for the coefficients of those series. This implies an integrality result for classical hypergeometric series.
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.
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 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.
We use the Dwork-Frobenius operator to prove an integrality result for $A$-hypergeometric series whose coefficients are factorial ratios. As a special case, we generalize one direction of a classical result of Landau on the integrality of factorial ratios.
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.
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…
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 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…
Using an intrinsic $q$-hypergeometric strategy, we generalise Dwork-type congruences $H(p^{s+1})/H(p^s)\equiv H(p^s)/H(p^{s-1})\pmod{p^3}$ for $s=1,2,\dots$ and $p$ a prime, when $H(N)$ are truncated hypergeometric sums corresponding to the periods of rigid Calabi--Yau threefolds.
We prove that if two Calabi-Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard--Fuchs differential equation. The factor in the zeta function is defined over the rationals and has degree at least the order of the Picard--Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.
The $j$th divisor function $d_j$, which counts the ordered factorisations of a positive integer into $j$ positive integer factors, is a very well-known arithmetic function; in particular, $d_2(n)$ gives the number of divisors of $n$. However, the $j$th non-trivial divisor function $c_j$, which counts the ordered proper factorisations of a positive integer into $j$ factors, each of which is greater than or equal to 2, is rather less well-studied. We also consider associated divisor functions $c_j^{(r)}$, whose definition is motivated by the sum-over divisors recurrence for $d_j$. After reviewin…
