271 research outputs found
On the zeta function of divisors for projective varieties with higher rank divisor class group
Given a projective variety X defined over a finite field, the zeta function
of divisors attempts to count all irreducible, codimension one subvarieties of
X, each measured by their projective degree. When the dimension of X is greater
than one, this is a purely p-adic function, convergent on the open unit disk.
Four conjectures are expected to hold, the first of which is p-adic meromorphic
continuation to all of C_p.
When the divisor class group (divisors modulo linear equivalence) of X has
rank one, then all four conjectures are known to be true. In this paper, we
discuss the higher rank case. In particular, we prove a p-adic meromorphic
continuation theorem which applies to a large class of varieties. Examples of
such varieties are projective nonsingular surfaces defined over a finite field
(whose effective monoid is finitely generated) and all projective toric
varieties (smooth or singular).Comment: Significant Improvements. 13 page
Meromorphy of the rank one unit root L-function revisited
We demonstrate that Wan's alternate description of Dwork's unit root
L-function in the rank one case may be modified to give a proof of meromorphy
that is classical, eliminating the need to study sequences of uniform
meromorphic functions.Comment: 9 page
Families of generalized Kloosterman sums
We construct p-adic relative cohomology for a family of toric exponential
sums which generalize the classical Kloosterman sums. Under natural hypotheses
such as quasi-homogeneity and nondegeneracy, this cohomology is acyclic except
in the top dimension. Our construction enables sufficiently sharp estimates for
the action of Frobenius on cohomology so that our earlier work may be applied
to the L-functions coming from linear algebra operations on these families to
deduce a number of basic properties.Comment: 36 pages, 4 figure
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