Isogeny Clases of Abelian Varieties over Finite Fields

Abstract

The aim of this thesis is to determine the isogeny classification of abelian varieties over a finite field F_q with q = p^n elements, particularly for dimensions 3, 4 and 5. For a given dimension g, each isogeny class has a distinguished characteristic polynomial, which is a q-Weil polynomial of degree 2g satisfying certain conditions regarding its factorisation over the p-adic integers Z_p. A q-Weil polynomial is a polynomial with integer coefficients such that all of its roots have absolute value sqrt(q). Enumerating all isogeny classes is a two-steps procedure. The first step is determining all q-Weil polynomials of degree 2g and the second step is determining the conditions for which a given q-Weil polynomial is the characteristic polynomial of some abelian variety over F_q. This process has been carried out for a fixed dimension up to g = 5 in recent articles by various authors. However, a few of these results contained some mistakes, particularly in the first step for dimensions 3, 4 and 5. This thesis contains a correction of these specific results and also explains the second step for these dimensions.