2015/10/07 by Aubry, Yves, Iezzi, Annamaria · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1510.01853
Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field \mathbb F_q.This bound enables us to provide explicit conditions on q, g and π for the non-existence of absolutely irreducible projective algebraic curves defined over \mathbb F_q of geometric genus g, arithmetic genus π and with N_q(g)+π-g rational points.Moreover, for q a square, we study the set of pairs (g,π) for which there exists a maximal absolutely irreducible projective algebraic curve defined over \mathbb F_q of geometric genus g and arithmetic genus π, i.e. with q+1+2g√(q)+π-g rational points.