Exponential sums and finite field AA-hypergeometric functions

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.Comment: 4 page

Hasse invariants and mod pp solutions of AA-hypergeometric systems

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.Comment: 22 page

AA-hypergeometric series associated to a lattice polytope with a unique interior lattice point

We associate to lattice points a_0,a_1,...,a_N in Z^n an A-hypergeometric series \Phi(\lambda) with integer coefficients. If a_0 is the unique interior lattice point of the convex hull of a_1,...,a_N, then for every prime p\neq 2 the ratio \Phi(\lambda)/\Phi(\lambda^p) has a p-adic analytic continuation to a closed unit polydisk minus a neighborhood of a hypersurface.Comment: 12 page

On the pp-integrality of AA-hypergeometric series

Let $A$ be a set of $N$ vectors in ${\mathbb Z}^n$ and let $v$ be a vector in ${\mathbb C}^N$ that has minimal negative support for $A$. Such a vector $v$ gives rise to a formal series solution of the $A$-hypergeometric system with parameter $\beta = Av$. If $v$ lies in ${\mathbb Q}^n$, then this series has rational coefficients. Let $p$ be a prime number. We characterize those $v$ whose coordinates are rational, $p$-integral, and lie in the closed interval $[-1,0]$ for which the corresponding normalized series solution has $p$-integral coefficients.Comment: Expanded introduction, Sections 2 and 5 rewritten, Section 7 added, small changes elsewher

A-hypergeometric series and the Hasse-Witt matrix of a hypersurface

We give a short combinatorial proof of the generic invertibility of the Hasse-Witt matrix of a projective hypersurface. We also examine the relationship between the Hasse-Witt matrix and certain $A$-hypergeometric series, which is what motivated the proof.Comment: 7 page

On logarithmic solutions of A-hypergeometric systems

For an $A$-hypergeometric system with parameter $\beta$, a vector $v$ with minimal negative support satisfying $Av = \beta$ gives rise to a logarithm-free series solution. We find conditions on $v$ analogous to `minimal negative support' that guarantee the existence of logarithmic solutions of the system and we give explicit formulas for those solutions. Although we do not study in general the question of when these logarithmic solutions lie in a Nilsson ring, we do examine the $A$-hypergeometric systems corresponding to the Picard-Fuchs equations of certain families of complete intersections and we state a conjecture regarding the integrality of the associated mirror maps.Comment: 23 page

Dwork cohomology, de Rham cohomology, and hypergeometric functions

In the 1960s, Dwork developed a p-adic cohomology theory of de Rham type for varieties over finite fields, based on a trace formula for the action of a Frobenius operator on certain spaces of p-adic analytic functions. One can consider a purely algebraic analogue of Dwork's theory for varieties over a field of characteristic zero and ask what is the connection between this theory and ordinary de Rham cohomology. N. Katz showed that Dwork cohomology coincides with the primitive part of de Rham cohomology for smooth projective hypersurfaces, but the exact relationship for varieties of higher codimension has been an open question. In this article, we settle the case of smooth affine complete intersections.Comment: 20 page

A cohomological property of Lagrange multipliers

The method of Lagrange multipliers relates the critical points of a given function f to the critical points of an auxiliary function F. We establish a cohomological relationship between f and F and use it, in conjunction with the Eagon-Northcott complex, to compute the sum of the Milnor numbers of the critical points in certain situations.Comment: 15 page

AA-hypergeometric systems that come from geometry

We establish some connections between nonresonant $A$-hypergeometric systems and de Rham-type complexes. This allows us to determine which of these $A$-hypergeometric systems "come from geometry."Comment: 10 page

On the Jacobian ring of a complete intersection

Let f_1,...,f_r be homogeneous polynomials in K[x_1,...,x_n], K a field. Put F=y_1f_1+...+y_rf_r in K[x,y] and let I be the ideal of K[x,y] generated by the partials of F relative to the x_i and y_j. The Jacobian ring of F is the quotient J:=K[x,y]/I. We describe J by computing the cohomology of a certain complex whose top cohomology group is J.Comment: 28 pages, no figure