Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a smooth projective surface over a finite field, by computing the Frobenius action on p-adic cohomology to a small degree of p-adic accuracy. We have implemented this procedure in Magma; using this implementation, we exhibit several examples, such as smooth quartics over F2 and F3 with arithmetic Picard number 1, and a smooth quintic over F2 with geometric Picard number 1. We also produce some examples of smooth quartics with geometric Picard number 2, which by a construction of van Luijk also have trivial geometric automorphism group.

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. We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We prove that every curve of genus g ≤ 4 over an algebraically closed field is nondegenerate in the above sense. More generally, let be the locus of nondegenerate curves inside the moduli space of curves of genus g ≥ 2. Then we show that dim Mnd g = min(2g + 1,3g − 3), except for g = 7 where dim Mnd 7 = 16; thus, a generic curve of genus g is nondegenerate M nd g if and only if g ≤ 4.

Abstract. We apply deformation theory to compute zeta functions in a family of Ca,b curves over a finite field of small characteristic. The method combines Denef and Vercauteren’s extension of Kedlaya’s algorithm to Ca,b curves with Hubrechts ’ recent work on point counting on hyperelliptic curves using deformation. As a result, it is now possible to generate Ca,b curves suitable for use in cryptography in a matter of minutes. 1

Abstract. We develop explicitly computable bounds for the order of the Jacobian of a cubic function field. We use approximations via truncated Euler products and thus derive effective methods of computing the order of the Jacobian of a cubic function field. Also, a detailed discussion of the zeta function of a cubic function field extension is included. 1.

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.

Abstract. Coleman’s theory of p-adic integration figures prominently in several number-theoretic applications, such as finding torsion and rational points on curves, and computing p-adic regulators in K-theory (including p-adic heights on elliptic curves). We describe an algorithm for computing Coleman integrals on hyperelliptic curves, and its implementation in Sage. 1

Let p be prime and Zpn a degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast deterministic algorithms for common operations in Zpn modulo pN. Combining existing methods with recent work of Kedlaya and Umans about modular composition of polynomials, we achieve quasi-linear time algorithms in the parameters n and N, and quasi-linear or quasi-quadratic time in log p, for most basic operations on these fields, including Galois conjugation, Teichmüller lifting and computing minimal polynomials.

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

Abstract. Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly well-suited to work with polynomials in small characteristic that have few monomials (relative to their dimension). Our method covers toric, affine, and projective hypersurfaces and also can be used to compute the L-function of an exponential sum. Let p be prime and let Fq be a finite field with q = p a elements. Let V be a variety defined over Fq, described by the vanishing of a finite set of polynomial equations with coefficients in Fq. We encode the number of points #V (Fqr) on V over the extensions Fqr of Fq in an exponential generating series, called the zeta function of V: