Raymond van Bommel

Abelian varieties of prescribed order over finite fields

The Hasse-Weil inequalities give a lower and upper bound for the number of rational points on an abelian variety over a finite field Fq in terms of its dimension and q. In this talk, I will discuss joint work with Edgar Costa, Wanlin Li, Bjorn Poonen, and Alexander Smith, in which we show the existence of sequences of simple abelian varieties whose point counts get close to these Hasse-Weil bounds. We also have more effective versions of the construction which gives an explicit upper bound in terms of q for the greatest integer which does not occur as the order of an abelian variety over Fq.

Our method uses Honda-Tate theory, which describes exactly which polynomials can occur as the Weil polynomial of an ordinary abelian variety over Fq. We construct such Weil polynomials by starting with certain real polynomials having good properties and approximating these real polynomials by integer polynomials. By imposing certain congruence conditions on these polynomials, we can force the abelian varieties that we construct to be ordinary, geometrically simple, and isogenous to a principally polarised abelian variety. If time allows, I will also discuss the situation where the dimension of the abelian variety is fixed and the prime power q goes to infinity.