Counting points on curves using a map to $\mathbf {P}^1$

Abstract

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.

Counting points on curves using a map to P1, II

“…This was then extended by others to characteristic 2 [9], superelliptic curves [13], C ab curves [8] and nondegenerate curves [6]. In [21] we proposed a much more general and practical extension of Kedlaya's algorithm. The goal of this paper is to further improve this algorithm.…”

“…The algorithm from [21] can be applied to generic, or in other words random, equations Q. However, there are equations to which it cannot be applied including some very interesting examples.…”

“…For example, when Q is (the reduction at some prime number p of) one of the defining equations computed for modular curves in [20,23], it turns out that the algorithm can almost never be applied. The reason is that in [21] we assume that Q, or rather its lift Q to characteristic zero, defines a smooth curve in the affine (x, y)-plane, i.e. that all the singularities of the plane curve defined by Q lie at infinity.…”

“…However, there are lots of small changes in many different places. To limit the amount of text overlap, we refer to [21] whenever a proof is the same or very similar. We have updated our implementation in Magma [4].…”

“…Comparing to [21], in the definition of the β i we have had to replace 1/r(x) by 1/∆(x) = g(x)/r(x) m . Note that the α i have not changed and still converge to F p (1/r(x)).…”