4,169 research outputs found
Counting curves over finite fields
This is a survey on recent results on counting of curves over finite fields.
It reviews various results on the maximum number of points on a curve of genus
g over a finite field of cardinality q, but the main emphasis is on results on
the Euler characteristic of the cohomology of local systems on moduli spaces of
curves of low genus and its implications for modular forms.Comment: 25 pages, to appear in Finite Fields and their Application
On the cyclicity of the rational points group of abelian varieties over finite fields
We propose a simple criterion to know if an abelian variety defined over
a finite field is cyclic, i.e., it has a cyclic group of
rational points; this criterion is based on the endomorphism ring
End. We also provide a criterion to know if an isogeny
class is cyclic, i.e., all its varieties are cyclic; this criterion is based on
the characteristic polynomial of the isogeny class. We find some asymptotic
lower bounds on the fraction of cyclic -isogeny classes among
certain families of them, when tends to infinity. Some of these bounds
require an additional hypothesis. In the case of surfaces, we prove that this
hypothesis is achieved and, over all -isogeny classes with
endomorphism algebra being a field and where is an even power of a prime,
we prove that the one with maximal number of rational points is cyclic and
ordinary.Comment: 13 pages, this is a preliminary version, comments are welcom
Hypergeometric Functions over Finite Fields and their relations to Algebraic Curves
In this work we present an explicit relation between the number of points on
a family of algebraic curves over \F_{q} and sums of values of certain
hypergeometric functions over \F_{q}. Moreover, we show that these
hypergeometric functions can be explicitly related to the roots of the zeta
function of the curve over \F_{q} in some particular cases. A general
conjecture relating these last two is presented and advances toward its proof
are shown in the last section.Comment: 24 page
Counting points on curves over families in polynomial time
This note concerns the theoretical algorithmic problem of counting rational
points on curves over finite fields. It explicates how the algorithmic scheme
introduced by Schoof and generalized by the author yields an algorithm whose
running time is uniformly polynomial time for curves in families.Comment: 7 page
Maps between curves and arithmetic obstructions
Let X and Y be curves over a finite field. In this article we explore methods
to determine whether there is a rational map from Y to X by considering
L-functions of certain covers of X and Y and propose a specific family of
covers to address the special case of determining when X and Y are isomorphic.
We also discuss an application to factoring polynomials over finite fields.Comment: 8 page
Counting abelian varieties over finite fields via Frobenius densities
Let be a principally polarized abelian variety over a finite
field with commutative endomorphism ring; further suppose that either is
ordinary or the field is prime. Motivated by an equidistribution heuristic, we
introduce a factor for each place of , and
show that the product of these factors essentially computes the size of the
isogeny class of .
The derivation of this mass formula depends on a formula of Kottwitz and on
analysis of measures on the group of symplectic similitudes, and in particular
does not rely on a calculation of class numbers.Comment: Main text by Achter, Altug and Gordon; appendix by Li and Ru
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