Abelian surfaces over finite fields with prescribed groups

Abstract

Let A be an abelian surface over F q , the field of q elements. The rational points on A / F q form an abelian group A ( F q ) ≃ Z / n 1 Z × Z / n 1 n 2 Z × Z / n 1 n 2 n 3 Z × Z / n 1 n 2 n 3 n 4 Z . We are interested in knowing which groups of this shape actually arise as the group of points on some abelian surface over some finite field. For a fixed prime power q, a characterization of the abelian groups that occur was recently found by Rybakov. One can use this characterization to obtain a set of congruences on certain combinations of coefficients of the corresponding Weil polynomials. We use Rybakov's criterion to show that groups Z / n 1 Z × Z / n 1 n 2 Z × Z / n 1 n 2 n 3 Z × Z / n 1 n 2 n 3 n 4 Z do not occur if n 1 is very large with respect to n 2 , n 3 , n 4 (Theorem 1.1), and occur with density zero in a wider range of the variables (Theorem 1.2).