Abelian surfaces over finite fields with prescribed groups

Abelian Surfaces over Finite Fields as Jacobians

Jacobians in isogeny classes of abelian surfaces over finite fields

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.

Principally Polarizable Isogeny Classes of Abelian Surfaces over Finite Fields

Let A be an isogeny class of abelian surfaces over Fq with Weil polynomial x4+ax3+bx2+aqx+q2. We show that A does not contain a surface that has a principal polarization if and only if a2 − b = q and b < 0 and all prime divisors of b are congruent to 1 modulo 3.

Groups of Points on Abelian Varieties over Finite Fields

Fix an isogeny class of abelian varieties with commutative endomorphism algebra over a finite field. This isogeny class is determined by a Weil polynomial fA without multiple roots. We give a classification of groups of k-rational points on varieties from this class in terms of Newton polygons of fA(1− t).

Groups of Rational Points on Abelian Varieties over Finite Fields

Fix an isogeny class of abelian varieties with commutative endomorphism algebra over a finite field. This isogeny class is determined by a Weil polynomial fA without multiple roots. We give a classification of groups of rational points on varieties from this class in terms of Newton polygons of fA(1− t).