#### Title of the thesis:

Computing abelian varieties over finite fields

#### Author of the thesis:

Stefano Marseglia

#### External reviewer:

Christophe Ritzenthaler (Université de Rennes)

#### Supervisor:

Jonas Bergström

#### Abstract

In this thesis we address the problem of developing effective algorithms to compute isomorphism classes of polarized abelian varieties over a finite field and of fractional ideals of an order in a finite product of number fields.

There are well-known methods to efficiently compute the classes of invertible ideals of an order in a number field, but not much has previously been known about non-invertible ideals. In Paper I we produce algorithms to compute representatives of all ideal classes of an order in a finite product of number fields. We also extend a theorem of Latimer and MacDuffee about conjugacy classes of integral matrices.

There are equivalences established by Deligne and Centeleghe-Stix between the category of abelian varieties over a finite field and the category of finitely generated free abelian groups with an endomorphism satisfying some easy-to-state axioms, which in certain cases can be described in terms of fractional ideals of orders in finite products of number fields.

In Paper II we use this method to produce an algorithm that computes the isomorphism classes of abelian varieties in an isogeny class determined by an ordinary square-free q-Weil polynomial or by a square-free p-Weil polynomial with no real roots (where p denotes a prime and q is a power of a prime). In the ordinary case we also produce an algorithm that computes the polarizations up to isomorphism and the automorphism groups of the polarized abelian varieties. If the polarization is principal, we can compute a period matrix of the canonical lift of the abelian variety.

In Paper III we extend the description of the second paper to the case when the Weil polynomial is a power of a square-free polynomial which fulfills the same requirements as in Paper II.

In Paper IV we use the results of the second and third papers to study questions related to base-field extension of the abelian varieties over finite fields.