Version 2.0 March 16, 2008These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form.

An abelian variety A defined over a finite field Fq admits sufficiently many complex multiplications, as Tate showed in [27]. For some details about complex multiplication, see §1.1. Is A the reduction of an abelian variety with sufficiently many complex multiplications in characteristic zero? We formulate several versions of this “CM-lifting problem ” in §1.2. Honda

Frey and Rück gave a method to map the discrete logarithm problem in the divisor class group of a curve over  ¢¡ into a finite field discrete logarithm problem in some extension. The discrete logarithm problem in the divisor class group can therefore be solved as long ¥ as is small. In the elliptic curve case it is known that for supersingular curves one ¥§¦© ¨ has. In this paper curves of higher genus are studied. Bounds on the possible values ¥ for in the case of supersingular curves are given. Ways to ensure that a curve is not supersingular are also given. 1.

. We describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm `a la Schoof for genus 2 using Cantor 's division polynomials. These are combined with a birthday paradox algorithm to calculate the cardinality. Our methods are practical and we give actual results computed using our current implementation. The Jacobian groups we handle are larger than those previously reported in the literature. Introduction In recent years there has been a surge of interest in algorithmic aspects of curves. When presented with any curve, a natural task is to compute the number of points on it with coordinates in some finite field. When the finite field is large this is generally difficult to do. Ren'e Schoof gave a polynomial time algorithm for counting points on elliptic curves i.e., those of genus 1, in his ground-...

Abstract. We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F q 2 whose number of F q 2-rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are F q 2-isomorphic to y q + y = x m for some m ∈ Z +. Goppa in [Go] showed how to construct linear codes from curves defined over finite fields. One of the main features of these codes is the fact that one can state a lower bound for the minimum distance of the codes. In fact, let CX(D, G) be a Goppa code defined over a curve X over the finite field Fq with q elements, where D = P1 +... + Pn,

Abstract. We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F q 2 whose number of F q 2-rational points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a rational point, we prove that maximal curves are F q 2-isomorphic to y q + y = x m, for some m ∈ Z +. As a consequence we show that a maximal curve of genus g = (q − 1) 2 /4 is F q 2-isomorphic to the curve y q + y = x (q+1)/2. The interest on curves over finite fields was renewed after Goppa [Go] showed their applications to Coding Theory. One of the main features of linear codes arising from curves is the fact that one can state a lower bound for their minimum distance. This lower bound is meaningful only if the curve has many rational points. The subject of this

Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes showed how to compute isogenies in small characteristic. The aim of this paper is to describe the first successful implementation of Couveignes’s algorithm. In particular, we describe the use of fast algorithms for performing incremental operations on series. We also insist on the particular case of the characteristic 2. 1.

It is well-known that an abelian variety is (absolutely) simple or is isogenous to a self-product of an (absolutely) simple abelian variety if and only if the center of its endomorphism algebra is a field. In this paper we prove that the center is a field if the field of definition of points of prime order ℓ is “big enough”. The paper is organized as follows. In §1 we discuss Galois properties of points of order ℓ on an abelian variety X that imply that its endomorphism algebra End 0 (X) is a central simple algebra over the field of rational numbers. In §2 we prove that similar Galois properties for two abelian varieties X and Y combined with the linear disjointness of the corresponding fields of definitions of points of order ℓ imply that X and Y are non-isogenous (and even Hom(X, Y) = 0). In §3 we give applications to endomorphism algebras of hyperelliptic jacobians. In §4 we prove that if X admits multiplications by a number field E and the dimension of the centralizer of E in End 0 (X) is “as large as possible ” then X is an abelian variety of CM-type isogenous to a self-product of an absolutely simple abelian variety. Throughout the paper we will freely use the following observation [21, p. 174]: if an abelian variety X is isogenous to a self-product Z d of an abelian variety Z then a choice of an isogeny between X and Z d defines an isomorphism between End 0 (X) and the algebra Md(End 0 (Z)) of d × d matrices over End 0 (Z). Since the center of End 0 (Z) coincides with the center of Md(End 0 (Z)), we get an isomorphism

We introduce the notion of a hypersymmetric abelian variety over a field of positive characteristic p. We show that every symmetric Newton polygon admits a hypersymmetric abelian variety having that Newton polygon; see 2.5 and 4.8. Isogeny classes of absolutely simple hypersymmetric abelian varieties are classified in terms of their endomorphism algebras and Newton polygons. We also discuss connections with abelian varieties of PEL-type, i.e. abelian varieties with extra symmetries, especially abelian varieties with real multiplications.

Abstract not found