13,988 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
Survey on counting special types of polynomials
Most integers are composite and most univariate polynomials over a finite
field are reducible. The Prime Number Theorem and a classical result of
Gau{\ss} count the remaining ones, approximately and exactly.
For polynomials in two or more variables, the situation changes dramatically.
Most multivariate polynomials are irreducible. This survey presents counting
results for some special classes of multivariate polynomials over a finite
field, namely the the reducible ones, the s-powerful ones (divisible by the
s-th power of a nonconstant polynomial), the relatively irreducible ones
(irreducible but reducible over an extension field), the decomposable ones, and
also for reducible space curves. These come as exact formulas and as
approximations with relative errors that essentially decrease exponentially in
the input size.
Furthermore, a univariate polynomial f is decomposable if f = g o h for some
nonlinear polynomials g and h. It is intuitively clear that the decomposable
polynomials form a small minority among all polynomials. The tame case, where
the characteristic p of Fq does not divide n = deg f, is fairly
well-understood, and we obtain closely matching upper and lower bounds on the
number of decomposable polynomials. In the wild case, where p does divide n,
the bounds are less satisfactory, in particular when p is the smallest prime
divisor of n and divides n exactly twice. The crux of the matter is to count
the number of collisions, where essentially different (g, h) yield the same f.
We present a classification of all collisions at degree n = p^2 which yields an
exact count of those decomposable polynomials.Comment: to appear in Jaime Gutierrez, Josef Schicho & Martin Weimann
(editors), Computer Algebra and Polynomials, Lecture Notes in Computer
Scienc
Counting hyperelliptic curves that admit a Koblitz model
Let k be a finite field of odd characteristic. We find a closed formula for
the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic
curves of genus g over k, admitting a Koblitz model. These numbers are
expressed as a polynomial in the cardinality q of k, with integer coefficients
(for pointed curves) and rational coefficients (for non-pointed curves). The
coefficients depend on g and the set of divisors of q-1 and q+1. These formulas
show that the number of hyperelliptic curves of genus g suitable (in principle)
of cryptographic applications is asymptotically (1-e^{-1})2q^{2g-1}, and not
2q^{2g-1} as it was believed. The curves of genus g=2 and g=3 are more
resistant to the attacks to the DLP; for these values of g the number of curves
is respectively (91/72)q^3+O(q^2) and (3641/2880)q^5+O(q^4)
Curves, dynamical systems and weighted point counting
Suppose X is a (smooth projective irreducible algebraic) curve over a finite
field k. Counting the number of points on X over all finite field extensions of
k will not determine the curve uniquely. Actually, a famous theorem of Tate
implies that two such curves over k have the same zeta function (i.e., the same
number of points over all extensions of k) if and only if their corresponding
Jacobians are isogenous. We remedy this situation by showing that if, instead
of just the zeta function, all Dirichlet L-series of the two curves are equal
via an isomorphism of their Dirichlet character groups, then the curves are
isomorphic up to "Frobenius twists", i.e., up to automorphisms of the ground
field. Since L-series count points on a curve in a "weighted" way, we see that
weighted point counting determines a curve. In a sense, the result solves the
analogue of the isospectrality problem for curves over finite fields (also know
as the "arithmetic equivalence problem"): it says that a curve is determined by
"spectral" data, namely, eigenvalues of the Frobenius operator of k acting on
the cohomology groups of all l-adic sheaves corresponding to Dirichlet
characters. The method of proof is to shown that this is equivalent to the
respective class field theories of the curves being isomorphic as dynamical
systems, in a sense that we make precise.Comment: 11 page
Two lectures on the arithmetic of K3 surfaces
In these lecture notes we review different aspects of the arithmetic of K3
surfaces. Topics include rational points, Picard number and Tate conjecture,
zeta functions and modularity.Comment: 26 pages; v4: typos corrected, references update
Point counting on curves using a gonality preserving lift
We study the problem of lifting curves from finite fields to number fields in
a genus and gonality preserving way. More precisely, we sketch how this can be
done efficiently for curves of gonality at most four, with an in-depth
treatment of curves of genus at most five over finite fields of odd
characteristic, including an implementation in Magma. We then use such a lift
as input to an algorithm due to the second author for computing zeta functions
of curves over finite fields using -adic cohomology
An upper bound on the number of rational points of arbitrary projective varieties over finite fields
We give an upper bound on the number of rational points of an arbitrary
Zariski closed subset of a projective space over a finite field. This bound
depends only on the dimensions and degrees of the irreducible components and
holds for very general varieties, even reducible and non equidimensional. As a
consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal
number of rational points of an equidimensional projective variety
- …