143 research outputs found

    AA-hypergeometric systems that come from geometry

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

    Hasse invariants and mod pp solutions of AA-hypergeometric systems

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    Igusa noted that the Hasse invariant of the Legendre family of elliptic curves over a finite field of odd characteristic is a solution mod pp 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 pp solutions of AA-hypergeometric systems.Comment: 22 page

    On the zeta function of a projective complete intersection

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    We compute a basis for the p-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give p-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.Comment: 24 pages, no figure

    Newton polytopes and algebraic hypergeometric series

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    Let XX be the family of hypersurfaces in the odd-dimensional torus T2n+1{\mathbb T}^{2n+1} defined by a Laurent polynomial ff with fixed exponents and variable coefficients. We show that if nΔn\Delta, the dilation of the Newton polytope Δ\Delta of ff by the factor nn, contains no interior lattice points, then the Picard-Fuchs equation of W2nHDR2n(X)W_{2n}H^{2n}_{\rm DR}(X) has a full set of algebraic solutions (where W∙W_\bullet denotes the weight filtration on de Rham cohomology). We also describe a procedure for finding solutions of these Picard-Fuchs equations.Comment: With an appendix by Nicholas Kat

    Dwork cohomology, de Rham cohomology, and hypergeometric functions

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    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

    On logarithmic solutions of A-hypergeometric systems

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    For an AA-hypergeometric system with parameter β\beta, a vector vv with minimal negative support satisfying Av=βAv = \beta gives rise to a logarithm-free series solution. We find conditions on vv 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 AA-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

    Exponential sums on A^n, III

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    We give two applications of our earlier work "Exponential sums on A^n, II" (math.AG/9909009). We compute the p-adic cohomology of certain exponential sums on A^n involving a polynomial whose homogeneous component of highest degree defines a projective hypersurface with at worst weighted homogeneous isolated singularities. This study was motivated by recent work of Garcia (Exponential sums and singular hypersurfaces, Manuscripta Math., v. 97 (1998), pp. 45-58). We also compute the p-adic cohomology of certain exponential sums on A^n whose degree is divisible by the characteristic.Comment: 15 pages, LaTeX2

    Distinguished-root formulas for generalized Calabi-Yau hypersurfaces

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    By a "generalized Calabi-Yau hypersurface" we mean a hypersurface in Pn{\mathbb P}^n of degree dd dividing n+1n+1. The zeta function of a generic such hypersurface has a reciprocal root distinguished by minimal pp-divisibility. We study the pp-adic variation of that distinguished root in a family and show that it equals the product of an appropriate power of pp times a product of special values of a certain pp-adic analytic function F{\mathcal F}. That function F{\mathcal F} is the pp-adic analytic continuation of the ratio F(Λ)/F(Λp)F(\Lambda)/F(\Lambda^p), where F(Λ)F(\Lambda) is a solution of the AA-hypergeometric system of differential equations corresponding to the Picard-Fuchs equation of the family.Comment: 33 page

    pp-adic estimates for multiplicative character sums

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    This article is an expanded version of the talk given by the first author at the conference "Exponential sums over finite fields and applications" (ETH, Z\"urich, November, 2010). We state some conjectures on archimedian and pp-adic estimates for multiplicative character sums over smooth projective varieties. We also review some of the results of J. Dollarhide, which formed the basis for these conjectures. Applying his results, we prove one of the conjectures when the smooth projective variety is Pn{\mathbb P}^n itself.Comment: 9 page

    On the pp-integrality of AA-hypergeometric series

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