Related papers: Counting Points on Dwork Hypersurfaces and $p$-adi…

We provide a formula for the number of $\mathbb{F}_{p}$-points on the Dwork hypersurface $$x_1^n + x_2^n \dots + x_n^n - n \lambda \, x_1 x_2 \dots x_n=0$$ in terms of a $p$-adic hypergeometric function previously defined by the author.…

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 $$X_{\lambda}^d: \hspace{.1in} x_1^d+x_2^d+\ldots+x_d^d=d\lambda x_1x_2\cdots x_d$$…

We give an expression for number of points for the family of Dwork K3 surfaces $$X_{\lambda}^4: \hspace{.1in} x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4$$ over finite fields of order $q\equiv 1\pmod 4$ in terms of Greene's finite field…

Let $D_\lambda^d$ denote the family of monomial deformations of diagonal hypersurface over a finite field $\mathbb{F}_q$ given by \begin{align*} D_\lambda^d: X_1^d+X_2^d+\cdots+X_n^d=\lambda d X_1^{h_1}X_2^{h_2}\cdots X_n^{h_n},…

We consider the family of diagonal hypersurfaces with monomial deformation $$D_{d, \lambda, h}: x_1^d + x_2^d \dots + x_n^d - d \lambda \, x_1^{h_1} x_2^{h_2} \dots x_n^{h_n}=0$$ where $d = h_1+h_2 +\dots + h_n$ with $\gcd(h_1, h_2, \dots…

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…

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…

Let $D_\lambda^{d,k}$ denote the family of diagonal hypersurface over a finite field $\mathbb{F}_q$ given by \begin{align*} D_\lambda^{d,k}:X_1^d+X_2^d=\lambda dX_1^kx_2^{d-k}, \end{align*} where $d\geq2$, $1\leq k\leq d-1$, and…

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…

This article reports on an approach to point counting on algebraic varieties over finite fields that is based on a detailed investigation of the $2$-adic orthogonal group. Combining the new approach with a $p$-adic method, we count the…

We present various improvements to the deformation method for computing the zeta function of smooth projective hypersurfaces over finite fields using $p$-adic cohomology. This includes new bounds for the $p$-adic and $t$-adic precisions…

We compute the Artin $L$-function of a diagonal hypersurface D_{\lambda} over a finite field associated to a character of a finite group acting on D_{\lambda} , and under some condition, express it in terms of hypergeometric functions and…

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 $$Z_{\lambda}: x_1^d+x_2^d=d\lambda x_1x_2^{d-1}$$ over a…

We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to…

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…

Given a smooth cubic hypersurface $X$ over a finite field of characteristic greater than 3 and two generic points on $X$, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number…

The number of points on a certain one parameter family of algebraic surface over a finite field $\F_p$ can be expressed as $p^2+A_p(\lambda),$ where $A_p(\lambda)$ is a character sum and $\lambda$ is an element of the finite field $\F_p.$…

We use bounds of mixed character sums modulo a prime $p$ to estimate the density of integer points on the hypersurface $$ f_1(x_1) + \ldots + f_n(x_n) =a x_1^{k_1} \ldots x_n^{k_n} $$ for some polynomials $f_i \in {\mathbb Z}[X]$, nonzero…

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…

Let X be a non-singular projective hypersurface of degree 4, which is defined over the rational numbers. Assume that X has dimension 39 or more, and that X contains a real point and p-adic points for every prime p. Then X is shown to…