Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology

Counting points on Cab curves using Monsky-Washnitzer cohomology

We describe an algorithm to compute the zeta function of any Cab curve over any finite field Fpn . 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 generalizes Kedlaya’s algorithm for hyperelliptic curves. For fixed p the asymptotic running time for a Cab curve of genus g over Fpn i...

Computing Zeta Functions of Nondegenerate Curves

In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since in practice all known cases, e.g. hyperelliptic, superelliptic and Cab curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being no...

Integral Monsky-washnitzer Cohomology and the Overconvergent De Rham-witt Complex

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-defined integral cohomology groups, but this fact also does not appear to be explicitly written down anywhere. In this paper, we pro...

A Point Counting Algorithm for Cyclic Covers of the Projective Line

We present a Kedlaya-style point counting algorithm for cyclic covers y = f(x) over a finite field Fpn with p not dividing r, and r and deg f not necessarily coprime. This algorithm generalizes the Gaudry–Gürel algorithm for superelliptic curves to a more general class of curves, and has essentially the same complexity. Our practical improvements include a simplified algorithm exploiting the au...

An Algorithm for Computing p-Adic Heights Using Monsky-Washnitzer Cohomology