8 research outputs found
Counting points on curves using a map to P^1
We introduce a new algorithm to compute the zeta function of a curve over a
finite field. This method extends Kedlaya's algorithm to a very general class
of curves using a map to the projective line. We develop all the necessary
bounds, analyse the complexity of the algorithm and provide some examples
computed with our implementation
A Point Counting Algorithm for Cyclic Covers of the Projective Line
We present a Kedlaya-style point counting algorithm for cyclic covers over a finite field with not dividing , and
and not necessarily coprime. This algorithm generalizes the
Gaudry-G\"urel algorithm for superelliptic curves to a more general class of
curves, and has essentially the same complexity. Our practical improvements
include a simplified algorithm exploiting the automorphism of ,
refined bounds on the -adic precision, and an alternative pseudo-basis for
the Monsky-Washnitzer cohomology which leads to an integral matrix when . Each of these improvements can also be applied to the original
Gaudry-G\"urel algorithm. We include some experimental results, applying our
algorithm to compute Weil polynomials of some large genus cyclic covers
Computing zeta functions of arithmetic schemes
We present new algorithms for computing zeta functions of algebraic varieties
over finite fields. In particular, let X be an arithmetic scheme (scheme of
finite type over Z), and for a prime p let zeta_{X_p}(s) be the local factor of
its zeta function. We present an algorithm that computes zeta_{X_p}(s) for a
single prime p in time p^(1/2+o(1)), and another algorithm that computes
zeta_{X_p}(s) for all primes p < N in time N (log N)^(3+o(1)). These generalise
previous results of the author from hyperelliptic curves to completely
arbitrary varieties.Comment: 23 pages, to appear in the Proceedings of the London Mathematical
Societ
Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces
Let and be monomial deformations of two Delsarte
hypersurfaces in weighted projective spaces. In this paper we give a sufficient
condition so that their zeta functions have a common factor. This generalises
results by Doran, Kelly, Salerno, Sperber, Voight and Whitcher
[arXiv:1612.09249], where they showed this for a particular monomial
deformation of a Calabi-Yau invertible polynomial. It turns out that our factor
can be of higher degree than the factor found in [arXiv:1612.09249]
Computing Periods of Hypersurfaces
We give an algorithm to compute the periods of smooth projective
hypersurfaces of any dimension. This is an improvement over existing algorithms
which could only compute the periods of plane curves. Our algorithm reduces the
evaluation of period integrals to an initial value problem for ordinary
differential equations of Picard-Fuchs type. In this way, the periods can be
computed to extreme-precision in order to study their arithmetic properties.
The initial conditions are obtained by an exact determination of the cohomology
pairing on Fermat hypersurfaces with respect to a natural basis.Comment: 33 pages; Final version. Fixed typos, minor expository changes.
Changed code repository lin
Improvements to the Deformation Method for Counting Points on Smooth Projective Hypersurfaces
We present various improvements to the deformation method for computing the zeta function of smooth projective hypersurfaces over finite fields using p-adic cohomology. This includes new bounds for the p-adic and t-adic precisions required to obtain provably correct results and gains in the efficiency of the individual steps of the method. The algorithm that we thus obtain has lower time and space complexities than existing methods. Moreover, our implementation is more practical and can be applied more generally, which we illustrate with examples of quintic curves and quartic surfaces.status: publishe