Related papers: Dwork hypersurfaces of degree six and Greene's hyp…
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 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 compute the number of real forms of Fermat hypersurfaces for degree $d \ge 2$ except the degree 4 surface case, and give explicit descriptions of them.
Classical hypergeometric functions are well-known to play an important role in arithmetic algebraic geometry. These functions offer solutions to ordinary differential equations, and special cases of such solutions are periods of…
The Green's functions for the Laplace equation respectively satisfying the Dirichlet and Neumann boundary conditions on the upper side of an infinite plane with a circular hole are introduced and constructed. These functions enables…
In this paper, we build on the work of [T. Hughes, G. Sangalli, VARIATIONAL MULTISCALE ANALYSIS: THE FINE-SCALE GREENS' FUNCTION, PROJECTION, OPTIMIZATION, LOCALIZATION, AND STABILIZED METHODS, SIAM Journal of Numerical Analysis, 45(2),…
In this note we give exact formulas (and asymptotics) for the number of rational points of bounded height on weighted projective stacks over global function fields.
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…
Given a hypersurface, $X$, prime $p$, the zeta function is a generating function for the number of $\mathbb{F}_{p}$ rational points of $X$. Until now, there is no algorithm for computing hypersurfaces with ADE singularities. Scott Stetson…
We give an upper bound for the number of points of a hypersurface over a finite field that has no lines on, in terms of the dimension, the degree, and the number of the elements of the finite field.
We calculate the two-point Green's functions of operators dual to fermions of maximal gauged supergravity in four and five dimensions, in finite temperature backgrounds with finite charge density. The numerical method used in these…
Motivated by a recent question of Peyre, we apply the Hardy-Littlewood circle method to count "sufficiently free" rational points of bounded height on arbitrary smooth projective hypersurfaces of low degree that are defined over the…
We sharpen to nearly optimal the known asymptotic and explicit bounds for the number of $\mathbb{F}_q$-rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic…
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…
In this paper, we consider a problem of counting multiplicities. We fix a counting function of multiplicity of rational points in a hypersurface of a projective space over a finite field, and we give an upper bound for the sum with respect…
We study the set of $D$ such that a given irreducible hypersurface $C$ of degree $d$ has infinitely many points of degree $D$ over $\mathbb{Q}$. We give a new explicit proof that this set contains all (positive) multiples of the index of…
We consider an example of tubes of hypersurfaces in Euclidean space and generalise the tube formula to supercase. By this we assign to a point of the hypersurface in superspace a rational characteristic function. Does this rational function…
In this paper we describe an algorithm for implicitizing rational hypersurfaces in case there exists at most a finite number of base points. It is based on a technique exposed in math.AG/0210096, where implicit equations are obtained as…
The paper reviews Dwork's p-adic analytic methods used in the Weil Conjectures. The first two chapters review a version of his proof of the rationality conjecture. The rest of the paper is devoted to Dwork's original cohomological methods,…
For any finite field k of characteristic exceeding 3, the Hasse principle and weak approximation is established for non-singular cubic hypersurfaces X over the function field k(t), provided that X has dimension at least 6.