On the number of rational points on some abelian varieties over finite fieldsSpeaker: Safia Haloui (Institut de Math. de Luminy, Marseille)
Date: Mon 29th November 2010
Location: Mathematical Sciences Seminar Room
Let A be an abelian variety of dimension g defined over GF(q). By Weil conjectures, we have (q+1-2q^(1/2))^g <= #A(GF(q))<= (q+1+2q^(1/2))^g. It is actually possible (as for curves) to substitute 2q^(1/2) with its integer part in the previous inequality and the bounds obtained are often optimal. Lachaud, Martin-Deschamps and Perret gave better bounds when A is a Jacobian variety. We are interested in abelian varieties with small dimension, namely g=2 or g=3. We determine exactly the maximum and minimum number of rational points on Jacobians surfaces.
Next we describe the set of characteristic polynomials of abelian varieties of dimension 3 over finite fields.
(This talk is part of the Algebra/Claude Shannon Institute series.)