On the number of points on abelian and Jacobian varieties over finite fields

Yves Aubry (1, 2) , Safia Haloui (3) , Gilles Lachaud (2)

1 IMATH - Institut de Mathématiques de Toulon - EA 2134
2 IML - Institut de mathématiques de Luminy
3 Department of Mathematics (Kongens Lyngby, Denmark)

This article has roughly a threefold aim. The first is to provide a series of upper and lower bounds for the number of points on an abelian variety defined over a finite field. Our second aim is to obtain specific lower bounds in the special case where the abelian variety is the Jacobian of a smooth, projective, absolutely irreducible algebraic curve. The third aim is to give exact values for the maximum and the minimum number of rational points on Jacobian varieties of dimension 2.

Domaines: Géométrie algébrique [math.AG]

Yves Aubry, Safia Haloui, Gilles Lachaud. On the number of points on abelian and Jacobian varieties over finite fields. Acta Arithmetica, 2013, 160 (3), pp.201--241. ⟨10.4064/aa160-3-1⟩. ⟨hal-00978916⟩