Abstract. We introduce a short signature scheme based on the Computational Diffie-Hellman assumption on certain elliptic and hyper-elliptic curves. The signature length is half the size of a DSA signature for a similar level of security. Our short signature scheme is designed for systems where signatures are typed in by a human or signatures are sent over a low-bandwidth channel. 1

Version 19980315 Abstract. In this paper we prove several theorems about abelian varieties over finite fields by studying the set of monic real polynomials of degree 2n all of whose roots lie on the unit circle. In particular, we consider a set Vn of vectors in R n that give the coefficients of such polynomials. We calculate the volume of Vn and we find a large easily-described subset of Vn. Using these results, we find an asymptotic formula — with explicit error terms — for the number of isogeny classes of n-dimensional abelian varieties over Fq. We also show that if n> 1, the set of group orders of n-dimensional abelian varieties over Fq contains every integer in an interval of length roughly q n − 1 2 centered at q n + 1. Our calculation of the volume of Vn involves the evaluation of the integral over the simplex { (x1,..., xn) ∣ 0 ≤ x1 ≤ · · · ≤ xn ≤ 1} of the determinant of the n × n matrix [ x e] i−1 j, where the ei are positive real numbers. 1.

Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g dened over a nite eld Fq come either from Serre's renement of the Weil bound if the genus is small compared to q, or from Oesterle's optimization of the explicit formulae method if the genus is large. This paper presents three methods for improving these bounds. The arguments used are the indecomposability of the theta divisor of a curve, Galois descent, and Honda-Tate theory. Examples of improvements on the bounds include lowering them for a wide range of small genus when q = 2 3 ; 2 5 ; 2 13 ; 3 3 ; 3 5 ; 5 3 ; 5 7 , and when q = 2 2s , s > 1. For large genera, isolated improvements are obtained for q = 3; 8; 9. 1

We show that for all finite fields Fq, there exists a curve C over Fq of genus 3 such that the number of rational points on C is within 3 of the Serre-Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over Fq.

Abstract. In this expository paper we show how one can, in a uniform way, calculate the weight distributions of some well-known binary cyclic codes. The codes are related to certain families of curves, and the weight distributions are related to the distribution of the number of rational points on the curves. 1.

Abstract. We present an algorithm for computing the cardinality of the Jacobian of a random Picard curve over a finite field. If the underlying field is a prime field Fp, the algorithm has complexity O ( √ p). 1.

Abstract. We study the behaviour near s = 1 of zeta functions of varieties over finite fields Fq 2 with q a square. The main result is an Euler-characteristic formula for the square of the special value at s = 1. The Euler-characteristic is constructed from the Weil-étale cohomology of a certain 2 supersingular elliptic curve.

Abstract. We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-2 curves over finite fields. 1.

Abstract. We give a function F(d, n, p) such that if K/Qp is a degree n field extension and A/K is a d-dimensional abelian variety with potentially good reduction, then #A(K)[tors] ≤ F(d, n, p). Separate attention is given to the prime-to-p torsion and to the case of purely additive reduction. These latter bounds are applied to classify rational torsion on CM elliptic curves over number fields of degree at most 3, on elliptic curves over Q with integral j (recovering a theorem of Frey), and on abelian surfaces over Q with integral moduli. In the last case, our efforts leave us with 11 numbers which may, or may not, arise as the order of the full torsion subgroup. The largest such number is 72. 1.

Abstract. We study the question of whether algebraic curves of a given genus g defined over a field K must have points rational over the maximal abelian extension K ab of K. We give: (i) an explicit family of diagonal plane cubic curves without Q ab-points, (ii) for every number field K, a genus one curve C /Q with no K ab-points, and (iii) for every g ≥ 4 an algebraic curve C /Q of genus g with no Q ab-points. In an appendix, we discuss varieties over Q((t)), obtaining in particular a curve of genus 3 without (Q((t)) ab-points. Convention: All varieties over a field K are assumed to be nonsingular, projective and (as is especially important for what follows) geometrically irreducible. 1.