On Maximal Curves, Original Research Article

We study arithmetical and geometrical properties ofmaximal curves, that is, curves defined over the finite field imageq2whose number of imageq2-rational points reaches the Hasse–Weil upper bound. Under a hypothesis on non-gaps at a rational point, we prove that maximal curves are imageq2-isomorphic toyq+y=xm, for somemset membership, variantimage+. As a consequence we show that a maximal curve of genusg=(q−1)2/4 is imageq2-isomorphic to the curveyq+y=x(q+1)/2.