143 research outputs found
-hypergeometric systems that come from geometry
We establish some connections between nonresonant -hypergeometric systems
and de Rham-type complexes. This allows us to determine which of these
-hypergeometric systems "come from geometry."Comment: 10 page
Hasse invariants and mod solutions of -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 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
solutions of -hypergeometric systems.Comment: 22 page
On the zeta function of a projective complete intersection
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
Let be the family of hypersurfaces in the odd-dimensional torus defined by a Laurent polynomial with fixed exponents and
variable coefficients. We show that if , the dilation of the Newton
polytope of by the factor , contains no interior lattice
points, then the Picard-Fuchs equation of has a full
set of algebraic solutions (where 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
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
For an -hypergeometric system with parameter , a vector with
minimal negative support satisfying gives rise to a logarithm-free
series solution. We find conditions on 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 -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
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
By a "generalized Calabi-Yau hypersurface" we mean a hypersurface in
of degree dividing . The zeta function of a generic
such hypersurface has a reciprocal root distinguished by minimal
-divisibility. We study the -adic variation of that distinguished root in
a family and show that it equals the product of an appropriate power of
times a product of special values of a certain -adic analytic function
. That function is the -adic analytic
continuation of the ratio , where is a
solution of the -hypergeometric system of differential equations
corresponding to the Picard-Fuchs equation of the family.Comment: 33 page
-adic estimates for multiplicative character sums
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
-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 itself.Comment: 9 page
On the -integrality of -hypergeometric series
Let be a set of vectors in and let be a vector in
that has minimal negative support for . Such a vector
gives rise to a formal series solution of the -hypergeometric system with
parameter . If lies in , then this series has
rational coefficients. Let be a prime number. We characterize those
whose coordinates are rational, -integral, and lie in the closed interval
for which the corresponding normalized series solution has
-integral coefficients.Comment: Expanded introduction, Sections 2 and 5 rewritten, Section 7 added,
small changes elsewher
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