%0 Journal Article
%T Optimal and maximal singular curves
%+ Institut de Mathématiques de Marseille (I2M)
%+ Institut de Mathématiques de Toulon - EA 2134 (IMATH)
%A Aubry, Yves
%A Iezzi, Annamaria
%< avec comité de lecture
%@ 0271-4132
%J Contemporary mathematics
%I American Mathematical Society
%S Arithmetic, Geometry and Coding Theory
%V 686
%P 31--43
%8 2017
%D 2017
%Z 1510.01853
%R 10.1090/conm/686/13776
%K finite field
%K rational point
%K maximal curve
%K Singular curve
%Z 14H20, 11G20, 14G15
%Z Mathematics [math]/Algebraic Geometry [math.AG]Journal articles
%X 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 $\pi$ for the non-existence of absolutely irreducible projective algebraic curves defined over $\mathbb F_q$ of geometric genus $g$, arithmetic genus $\pi$ and with $N_q(g)+\pi-g$ rational points.Moreover, for $q$ a square, we study the set of pairs $(g,\pi)$ for which there exists a maximal absolutely irreducible projective algebraic curve defined over $\mathbb F_q$ of geometric genus $g$ and arithmetic genus $\pi$, i.e. with $q+1+2g\sqrt{q}+\pi-g$ rational points.
%G English
%2 https://hal.science/hal-01212624/document
%2 https://hal.science/hal-01212624/file/Aubry_Iezzi.pdf
%L hal-01212624
%U https://hal.science/hal-01212624
%~ UNIV-TLN
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-
%~ IMATH
%~ AMIDEX
%~ TEST-HALCNRS
%~ ANR