Abstract. We provide a subexponential algorithm for solving the discrete logarithm problem in Jacobians of high-genus hyperelliptic curves over finite fields. Its expected running time for instances with genus g and underlying finite field Fq satisfying g ≥ ϑ log q for a positive constant ϑ is given by

Abstract. A curve X over Q is modular if it is dominated by X1(N) for some N; if in addition the image of its jacobian in J1(N) is contained in the new subvariety of J1(N), then X is called a new modular curve. We prove that for each g ≥ 2, the set of new modular curves over Q of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de Franchis-Severi Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of J0(N) new with N divisible by a prescribed prime, and for modular curves (new or not) with levels in a restricted set. We study new modular hyperelliptic curves in detail. In particular, we find all new modular curves of genus 2 explicitly, and construct what might be the complete list of all new modular hyperelliptic curves of all genera. Finally we prove that for each field k of characteristic zero and g ≥ 2, the set of genus-g curves over k dominated by a Fermat curve is finite and computable. 1. Introduction. Let X1(N) be the usual modular curve over Q; see Section 3.1 for a definition. (All curves and varieties in this paper are smooth, projective, and geometrically integral, unless otherwise specified. When we write an affine equation for a curve, its smooth projective model is implied.) A curve X

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 odd characteristic. For a genus g hyperelliptic curve defined over F2 n , the average-case time complexity is O(g ) and the average-case space complexity is O(g ), whereas the worst-case time and space complexities are O(g ) and ) respectively.

Abstract. In this paper, we provide tight estimates for the divisor class number of hyperelliptic function fields. We extend the existing methods to any hyperelliptic function field and improve the previous bounds by a factor proportional to g with the help of new results. We thus obtain a faster method of computing regulators and class numbers. Furthermore, we provide experimental data and heuristics on the distribution of the class number within the bounds on the class number. These heuristics are based on recent results by Katz and Sarnak. Our numerical results and the heuristics imply that our approximation is in general far better than the bounds suggest. 1.

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

Abstract. In this report, we discuss the problem of computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo p m of the zeta function of a hypersurface, where p is the characteristic of the finite field. 1991 Mathematics Subject Classification: 11Y16, 11T99, 14Q15. 1.

We give an introductory account of the general algorithmic theory of the zeta function of an algebraic set defined over a finite field.

Abstract. We describe the Klein quartic � and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing the order of the automorphism group of �; an explicit identification of � with the modular curve X(7); and applications to the class number 1 problem and the case n = 7 of Fermat.

. This is a introduction to some aspects of the arithmetic of elliptic curves, intended for readers with little or no background in number theory and algebraic geometry. In keeping with the rest of this volume, the presentation has an algorithmic slant. We also touch lightly on curves of higher genus. Readers desiring a more systematic development should consult one of the references for further reading suggested at the end. 1. Plane curves Let k be a field. For instance, k could be the field Q of rational numbers, the field R of real numbers, the field C of complex numbers, the field Q p of p-adic numbers (see [Kob] for an introduction), or the finite field F q of q elements (see Chapter I of [Ser1]). Let k be an algebraic closure of k. A plane curve 1 X over k is defined by an equation f(x, y) = 0 where f(x, y) = # a ij x i y j # k[x, y] is irreducible over k. One defines the degree of X and of f by deg X = deg f = max{i + j : a ij #= 0}. A k-rational point (or simply k-point) on X is a point (a, b) with coordinates in k such that f(a, b) = 0. The set of all k-rational points on X is denoted X(k). Example: The equation x 2 y - 6y 2 - 11 = 0 defines a plane curve X over Q of degree 3, and (5, 1/2) # X(Q). Already at this point we can state an open problem, one which over the centuries has served as motivation for the development of a huge amount of mathematics. Question. Is there an algorithm, that given a plane curve X over Q, determines X(Q), or at least decides whether X(Q) is nonempty? Although X(Q) need not be finite, we will see later that it always admits a finite description, so this problem of determining X(Q) can be formulated precisely using the notion of Turing machine: see [HU] for a definition. For the relationship of this questi...

We prove two theorems and raise a few questions concerning discrete logarithms and algebraic groups.