We describe an algorithm for counting points on an arbitrary hyperelliptic curve over a finite
field of odd characteristic, using Monsky-Washnitzer cohomology to compute a p-adic
approximation to the characteristic polynomial of Frobenius. For fixed p, the asymptotic …

We describe an algorithm to compute the zeta function of any C ab curve over any finite field
F p n. The algorithm computes a p-adic approximation of the characteristic polynomial of
Frobenius by computing in the Monsky–Washnitzer cohomology of the curve and thus …

… Those cohomology groups, based on work of B. Dwork, are called the Monsky-Washnitzer
cohomology. The first four sections of this paper give a survey of the papers of Monsky and
Washnitzer … 2. Definition of the Monsky-Washnitzer cohomology …

… (see for instance [52]). 2.2 Monsky-Washnitzer Cohomology … i=1 ∑ j1,...,jn≥0 jiej1,...,jn (t j1 1 ···
tji−1 i ··· tjn n )dti. Then the Monsky-Washnitzer cohomology (or MW-cohomology) Hi MW(X) of
X is the cohomology of the “de Rham complex” ··· d → Ωi A† ⊗W W[ 1 p ] d → …

0. Introduction. Let k be a field of characteristic 0 and A be a finitely generated smooth k
algebra. Let O? A/k be the complex of algebraic differential forms on A. QA/k admits a
degree 1 exterior differentiation, d. The homology groups of the complex (QA/A, d) will be …

Satoh p≥ 5 O (n3+ε) O (n3) Skjernaa p= 2 O (n3+ε) O (n3) Fouquet-Gaudry-Harley p= 2, 3 O
(n3+ε) O (n3) Vercauteren all p O (n3+ε) O (n2) Mestre-Harley (AGM) p= 2 O (n3+ε) O (n2)
Satoh-Skjernaa-Taguchi all p O (n2+ 1/2+ ε) O (n2) Gaudry p= 2 O (n2+ 1/2+ ε) O (n2) Carls …

In their paper which introduced Monsky-Washnitzer cohomology, Monsky and Washnitzer
described conditions under which the definition can be adapted to give integral cohomology
groups. It seems to be well-known among experts that their construction always gives well …

Introduction Throughout this paper, k is a perfect field of characteristic p > 0, R is a complete
discrete valuation ring with residue field k and quotient field of characteristic zero, and Z is a
connected smooth prescheme of finite type over k. In their papers [6,8] Monsky and Washnitzer …

… apply similar ideas to certain smooth afline curves. For in the case of affine curves,
one can work in the setting of Monsky-Washnitzer cohomology, an older special
case of rigid cohomology. This cohomology theory is a p-adic …