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, for example, hyperelliptic, superelliptic, and Cab curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over Fpn, the expected running time is � O(n3g6 + n2g6.5), whereas the space complexity amounts to �O(n 3g4), assuming p is fixed. 1

We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for small odd characteristic. For a genus g hyperelliptic curve over n , the asymptotic running time of the algorithm is O(g ) and the space complexity is O(g ).

Let ĒΓ be a family of hyperelliptic curves over F2 alg cl with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm for computing the zeta function of Ē¯γ, with ¯γ in a degree n extension field of F, which has as time complexity Õ(n3) bit operations and memory requirements O(n2) bits. With a slightly different algorithm we can get time O(n2.667) and memory O(n2.5), and the computation for n curves of the family can be done in time Õ(n3.376). All of these algorithms are polynomial-time in the genus.

Abstract. In this paper we present an extension of Kedlaya’s algorithm for computingthe zeta function of an Artin-Schreier curve over a finite field Fq of characteristic 2. The algorithm has running time O(g 5+ε log 3+ε q) and needs O(g 3 log 3 q) storage space for a genus g curve. Our first implementation in MAGMA shows that one can now generate hyperelliptic curves suitable for cryptography in reasonable time. We also compare our algorithm with an algorithm by Lauder and Wan which has the same time and space complexity. Furthermore, the method introduced in this paper can be used for any hyperelliptic curve over a finite field of characteristic 2. Keywords: Hyperelliptic curves, Monsky-Washnitzer cohomology, Kedlaya’s algorithm, Lauder & Wan algorithm, cryptography

We describe an algorithm to compute the zeta function of a Cab curve over a 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 is O(g5+"n3+") and the space complexity is O(g³n³).

In this note we correct a gap in the proof of the complexity estimates appearing in our papers [1],[2]. The complexity estimates are correct, but the proof was incomplete at a certain point. The same gap appears in our paper [3], but there the estimate for the space complexity has to be multiplied with a power of log(g), where g is the genus of the curve. First we fill in the gap in [2]. In this paper we gave an algorithm to compute the zeta function of an hyperelliptic curve of genus g over a field with q = 2 n elements. We proved that the worst-case time complexity is O(g 5+ɛ n 3+ɛ), and that the worst-case space complexity is O(g 4 n 3). For the average-case time and space complexity we obtained O(g 4+ɛ n 3+ɛ) and O(g 3 n 3), respectively. These estimates are correct, but there is a gap in the proof, which we will now correct. The gap in [1] can be treated in the same way. The gap has to do with the precision estimates. The matrix M of the small Frobenius (induced by squaring) on the given basis of the Monsky-Washnitzer cohomology H 1 does not have integral entries. We proved in [2] that the denominators have valuation O(log(g)). Hence the denominators in the norm MF of M have valuation at most O(n log(g)). This

Abstract. We provide a number of results that can be used to derive approximations for the Euler product representation of the zeta function of an arbitrary algebraic function field. Three such approximations are given here. Our results have two main applications. They lead to a computationally suitable algorithm for computing the class number of an arbitrary function field. The ideas underlying the class number algorithms in turn can be used to analyze the distribution of the zeros of its zeta function. 1.

These are the lecture notes from a series of six lectures given at a summer school on p-adic cohomology held in Mainz in the fall of 2008. (They may be viewed as a sequel to the author’s notes from the Arizona Winter School in 2007 [49].) The goal of the notes is to describe how to use p-adic cohomology to make effective, provably correct numerical computations of zeta functions. More specifically, we discuss three techniques in detail: • use of the Hodge filtration to infer the zeta function from point counts; • the “direct cohomological method ” of computing the Frobenius action on the p-adic cohomology of a single variety; • the “deformation method ” of computing the Frobenius structure on the p-adic cohomologies of a one-parameter family of varieties, using the associated Picard-Fuchs differential equation. We make these methods explicit for cyclic cubic threefolds (cubic threefolds in P 4 admitting an automorphism of order 3), and demonstrate with a numerical example over the field F7. To demonstrate these calculations, we use the open-source computer algebra system Sage [77]; however, some of the calculations depend on the nonfree system Magma [63], which may

One of the more difficult and central problems in computational algebraic number theory is the computation of certain invariants of a field and its maximal order. In this thesis, we consider this problem where the field in question is a purely cubic function field, K/Fq(x), with char(K) ≥ 5. In addition, we will give a divisor-theoretic treatment of the infrastructures of K, including a description of its arithmetic, and develop arithmetic on the ideals of the maximal order, O, of K. Historically, the infrastructure, RC, of an ideal class, C ∈ Cl(O) has been defined as a set of reduced ideals in C. However, we extend work of Paulus and Rück [PR99] and Jacobson, Scheidler, and Stein [JSS07b] to define RC as a certain subset of the divisor class group, JK, of a cubic function field, K, specifically, the subset of distinguished divisors whose classes map to C via JK → Cl(O). Our definition of distinguished generalizes the same notion by Bauer for purely cubic function fields of unit rank 0 [Bau04] to those of unit rank 1 and 2 as well. Further, we prove a bijection between RC, as a set of distinguished divisors, and the infrastructure of C defined by “reduced” ideals, as in [Sch00, SS00, Sch01, LSY03, Sch04]. We describe the arithmetic on RC, providing new results on the baby step and giant step operations and generalizing notions of the inverse of a divisor in R [O] from quadratic infrastructures in [JSS07b] to cubic infrastructures. We also give algorithms to

Memory efficient hyperelliptic curve point counting