We present a probabilistic Las Vegas algorithm for computing the local zeta
function of a hyperelliptic curve of genus g defined over Fq. It
is based on the approaches by Schoof and Pila combined with a modeling of the
ℓ-torsion by structured polynomial systems. Our main result improves on
previously known complexity bounds by showing that there exists a constant
c>0 such that, for any fixed g, this algorithm has expected time and space
complexity O((logq)cg) as q grows and the characteristic is large
enough.Comment: To appear in Foundations of Computational Mathematic