Corrigendum to "Distribution of the traces of Frobenius on elliptic curves over function fields" (Acta Arith. 106 (2003), 255-263)
Corrigendum to: Improved upper bounds for the number of points on curves over finite fields
Counting points in medium characteristic using Kedlaya's algorithm.
Counting points on elliptic curves over finite fields
We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large ; it is based on Shanks's baby-step-giant-step strategy. The second algorithm is very efficient when the endomorphism ring of the curve is known. It exploits the natural lattice structure of this ring. The third algorithm is based on calculations with the torsion points of the elliptic curve [18]. This deterministic polynomial...
Coverings of Singular curves over Finite Fields.
Covers in -adic analytic geometry and log covers I: Cospecialization of the -tempered fundamental group for a family of curves
The tempered fundamental group of a -adic analytic space classifies covers that are dominated by a topological cover (for the Berkovich topology) of a finite étale cover of the space. Here we construct cospecialization homomorphisms between versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log reduction and...
Distribution of the traces of Frobenius on elliptic curves over function fields
Drinfeld modules of rank 1 and algebraic curves with many rational points. II
Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves.
Euclidean algorithm and Kummer covers with many points.
Explicit global function fields over the binary field with many rational places
Field of moduli versus field of definition for cyclic covers of the projective line
We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli that can not be defined over is also given.
Fields of moduli of three-point -covers with cyclic -Sylow, II
We continue the examination of the stable reduction and fields of moduli of -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic , where has a cyclic-Sylow subgroup of order . Suppose further that the normalizer of acts on via an involution. Under mild assumptions, if is a three-point -Galois cover defined over , then the th higher ramification groups above for the upper numbering of the (Galois closure of the) extension vanish,...
Finding the eigenvalue in Elkies' algorithm.
Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree
For each integer s ≥ 1, we present a family of curves that are -Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be -Frobenius nonclassical with respect to the linear system of conics. In the -Frobenius nonclassical cases, we determine the exact number of -rational points. In the remaining cases, an upper bound for the number of -rational points will follow from Stöhr-Voloch...
Galois towers over non-prime finite fields
We construct Galois towers with good asymptotic properties over any non-prime finite field ; that is, we construct sequences of function fields = (N₁ ⊂ N₂ ⊂ ⋯) over of increasing genus, such that all the extensions are Galois extensions and the number of rational places of these function fields grows linearly with the genus. The limits of the towers satisfy the same lower bounds as the best currently known lower bounds for the Ihara constant for non-prime finite fields. Towers with these properties...
Global function fields with many rational places over the quinary field. II
Ideal class group annihilators
Improved upper bounds for the number of points on curves over finite fields
We give new arguments that improve the known upper bounds on the maximal number of rational points of a curve of genus over a finite field , for a number of pairs . Given a pair and an integer , we determine the possible zeta functions of genus- curves over with points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus- curve over with points must have a low-degree map to another curve over , and often this is enough to...
Inegalite relative des genres.