205 research outputs found
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 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
Introduction to Arithmetic Mirror Symmetry
We describe how to find period integrals and Picard-Fuchs differential
equations for certain one-parameter families of Calabi-Yau manifolds. These
families can be seen as varieties over a finite field, in which case we show in
an explicit example that the number of points of a generic element can be given
in terms of p-adic period integrals. We also discuss several approaches to
finding zeta functions of mirror manifolds and their factorizations. These
notes are based on lectures given at the Fields Institute during the thematic
program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics
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