arXiv subjects
Publications and source records attributed to Alan Adolphson.
We give a class of examples of $A$-hypergeometric systems that display integrality of mirror maps. Specifically, these systems have solutions $F(\lambda_1,\dots,\lambda_N) = 1$ and $\log\lambda^l + G(\lambda_1,\dots,\lambda_N)$ (for certain $l\in{\mathbb Z}^N$) such that $\exp G(\lambda)$ has integral coefficients. The proof requires only some elementary congruences.
By a generalized Delsarte polynomial we mean a Laurent polynomial whose exponent vectors are linearly independent. We consider certain monomial deformations of generalized Delsarte polynomials and study their associated differential modules. We determine the solutions at the origin, which are all classical ${}_kF_{k-1}$-hypergeometric functions with the variable raised to a power. Using standard changes of variable, we also obtain the solutions at infinity.
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.
By a codimension-one system we mean a system whose lattice of relations has rank one. We consider codimension-one $A$-hypergeometric systems and explicitly construct some of the logarithmic series solutions at the origin. When the parameter vector $\beta$ is nonresonant we obtain a full set of logarithmic series solutions at the origin by this procedure. We also determine when a codimension-one system with nonresonant parameter can have maximal unipotent monodromy at the origin.
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 $F(\Lambda^p)^{-1}F(\Lambda)$, where the entries in the matrix $F(\Lambda)$ are $A$-hypergeometric series with integral coefficients and $F(\Lambda)$ is independent of $p$.
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 return to some past studies of hyperkloosterman sums ([9,10]) via $p$-adic cohomology with an aim to improve earlier results. In particular, we work here with Dwork's $\theta_\infty$-splitting function and a better choice of basis for cohomology. To a large extent, we are guided to this choice of basis by our recent work on the $p$-integrality of coefficients of $A$-hypergeometric series[3]. In the earlier work, congruence estimates were limited to $p>n+2$. We are here able to remove all characteristic restrictions from earlier results.
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. From this we deduce further integrality results for hypergeometric series.
Let $X$ be the family of hypersurfaces in the odd-dimensional torus ${\mathbb T}^{2n+1}$ defined by a Laurent polynomial $f$ with fixed exponents and variable coefficients. We show that if $n\Delta$, the dilation of the Newton polytope $\Delta$ of $f$ by the factor $n$, contains no interior lattice points, then the Picard-Fuchs equation of $W_{2n}H^{2n}_{\rm DR}(X)$ has a full set of algebraic solutions (where $W_\bullet$ denotes the weight filtration on de Rham cohomology). We also describe a procedure for finding solutions of these Picard-Fuchs equations.
Landau has characterized the integrality of certain ratios of factorials. Delaygue has characterized the integrality of the Taylor coefficients of certain mirror maps constructed from series involving those ratios. Using the $A$-hypergeometric point of view, we express those characterizations in terms of the nonexistence of interior points in multiples of the associated lattice polytope.
The Hasse-Witt matrix of a hypersurface in ${\mathbb P}^n$ over a finite field of characteristic $p$ gives essentially complete mod $p$ information about the zeta function of the hypersurface. But if the degree $d$ of the hypersurface is $\leq n$, the zeta function is trivial mod $p$ and the Hasse-Witt matrix is zero-by-zero. We generalize a classical formula for the Hasse-Witt matrix to obtain a matrix that gives a nontrivial congruence for the zeta function for all $d$. We also describe the differential equations satisfied by this matrix and prove that it is generically invertible.
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 $F(\Lambda)/F(\Lambda^p)$, where $F(\Lambda)$ is a solution of the $A$-hypergeometric system of differential equations corresponding to the Picard-Fuchs equation of the family.
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.
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 $β= Av$. 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.
For an $A$-hypergeometric system with parameter $β$, a vector $v$ with minimal negative support satisfying $Av = β$ 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.
We associate to lattice points a_0,a_1,...,a_N in Z^n an A-hypergeometric series Φ(λ) 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 Φ(λ)/Φ(λ^p) has a p-adic analytic continuation to a closed unit polydisk minus a neighborhood of a hypersurface.
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.
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.