1,732 research outputs found
A p-adic quasi-quadratic point counting algorithm
In this article we give an algorithm for the computation of the number of
rational points on the Jacobian variety of a generic ordinary hyperelliptic
curve defined over a finite field of cardinality with time complexity
and space complexity , where . In the latter
complexity estimate the genus and the characteristic are assumed as fixed. Our
algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and
the canonical lifting method of T. Satoh. We canonically lift a certain
arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of
theta constants. The theta null values are computed with respect to a
semi-canonical theta structure of level where is an integer
and p=\mathrm{char}(\F_q)>2. The results of this paper suggest a global
positive answer to the question whether there exists a quasi-quadratic time
algorithm for the computation of the number of rational points on a generic
ordinary abelian variety defined over a finite field.Comment: 32 page
Computing fundamental domains for the Bruhat-Tits tree for GL2(Qp), p-adic automorphic forms, and the canonical embedding of Shimura curves
We describe an algorithm for computing certain quaternionic quotients of the
Bruhat-Tits tree for GL2(Qp). As an application, we describe an algorithm to
obtain (conjectural) equations for the canonical embedding of Shimura curves.Comment: Accepted for publication in LMS Journal of Computation and
Mathematic
Quasi-quadratic elliptic curve point counting using rigid cohomology
We present a deterministic algorithm that computes the zeta function of a
nonsupersingular elliptic curve E over a finite field with p^n elements in time
quasi-quadratic in n. An older algorithm having the same time complexity uses
the canonical lift of E, whereas our algorithm uses rigid cohomology combined
with a deformation approach. An implementation in small odd characteristic
turns out to give very good results.Comment: 14 page
An extension of Kedlaya's algorithm for hyperelliptic curves
In this paper we describe a generalisation and adaptation of Kedlaya's
algorithm for computing the zeta function of a hyperelliptic curve over a
finite field of odd characteristic that the author used for the implementation
of the algorithm in the Magma library. We generalise the algorithm to the case
of an even degree model. We also analyse the adaptation of working with the
rather than the differential basis. This basis has the
computational advantage of always leading to an integral transformation matrix
whereas the latter fails to in small genus cases. There are some theoretical
subtleties that arise in the even degree case where the two differential bases
actually lead to different redundant eigenvalues that must be discarded.Comment: v3: some minor changes and addition of a reference to a paper by Theo
van den Bogaar
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
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