On the number of rational points on some abelian varieties over finite fields

**Speaker**: Safia Haloui (Institut de Math. de Luminy, Marseille)

**Time**: 4:00PM**Date**: Mon 29th November 2010

**Location**: Mathematical Sciences Seminar Room

**Abstract**

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.)

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