724 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
Splitting full matrix algebras over algebraic number fields
Let K be an algebraic number field of degree d and discriminant D over Q. Let
A be an associative algebra over K given by structure constants such that A is
isomorphic to the algebra M_n(K) of n by n matrices over K for some positive
integer n. Suppose that d, n and D are bounded. Then an isomorphism of A with
M_n(K) can be constructed by a polynomial time ff-algorithm. (An ff-algorithm
is a deterministic procedure which is allowed to call oracles for factoring
integers and factoring univariate polynomials over finite fields.)
As a consequence, we obtain a polynomial time ff-algorithm to compute
isomorphisms of central simple algebras of bounded degree over K.Comment: 15 pages; Theorem 2 and Lemma 8 correcte
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
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
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