Enric Nart

Abelian Surfaces over Finite Fields as Jacobians

For any finite field k = F q , we explicitly describe the k-isogeny classes of abelian surfaces defined over k and their behavior under finite field extension. In particular, we determine the absolutely simple abelian surfaces. Then, we analyze numerically what surfaces are k-isogenous to the Jacobian of a smooth projective curve of genus 2 defined over k. We prove some partial results suggested by these numerical data. For instance, we show that every absolutely simple abelian surface is k-isogenous to a Jacobian. Other facts suggested by these numerical computations are that the polynomials t 4 + (1 – 2q)t 2 + q 2 (for all q) and t 4 + (2 – 2q)t 2 + q 2 (for q odd) are never the characteristic polynomial of Frobenius of a Jacobian. These statements have been proved by E. Howe. The proof for the first polynomial is attached in an appendix.

The Bloch Complex in Codimension One and Arithmetic Duality

The importance of this point of view is apparent, since it suggests one possible way of extending (abelian) class field theory to more general schemes. For instance, Artin-Verdier duality holds also for X a smooth proper curve over a finite field [S], again with G, as the dualizing sheaf. However, G, is not able to play this role in more general situations, when we allow singularities or higher dimension. In fact, Deninger [6] recently has extended the l-dimensional duality to the singular case, being then forced to take as a dualizing object a complex G of sheaves

A New Computational Approach to Ideal Theory in Number Fields

Let K be the number field determined by a monic irreducible polynomial f(x) with integer coefficients. In previous papers we parameterized the prime ideals of K in terms of certain invariants attached to Newton polygons of higher order of f(x). In this paper we show how to carry out the basic operations on fractional ideals of K in terms of these constructive representations of the prime ideals. From a computational perspective, these results facilitate the manipulation of fractional ideals of K avoiding two heavy tasks: the construction of the maximal order of K and the factorization of the discriminant of f(x). The main computational ingredient is the Montes algorithm, which is an extremely fast procedure to construct the prime ideals.

On a theorem of Ore

Abstract O. Ore (Math. Ann. 99, 1928, 84–117) developed a method for obtaining the absolute discriminant and the prime-ideal decomposition of the rational primes in a number field K. The method, based on Newtons polygon techniques, worked only when certain polynomials ƒ S (Y) , attached to any side S of the polygon, had no multiple factors. These results are generalized in this paper finding a much weaker condition, effectively computable, under which it is still possible to give a complete answer to the above questions. The multiplicities of the irreducible factors of the polynomials ƒ S (Y) play then an essential role.

Complexity of OM factorizations of polynomials over local fields

Let

Hyperelliptic curves of genus three over finite fields of even characteristic

k

Zeta function and cryptographic exponent of supersingular curves of genus 2

be a locally compact complete field with respect to a discrete valuation

Zeta functions of supersingular curves of genus 2

v

Local computation of differents and discriminants

. Let

On the index of a number field

\oo