On maximal curves in characteristic two

Abstract:

The genus g of an q-maximal curve satisfies g=g ₁≔q(q−1)/2 or . Previously, q-maximal curves with g=g ₁ or g=g ₂, q odd, have been characterized up to q-isomorphism. Here it is shown that an q-maximal curve with genus g ₂, q even, is q-isomorphic to the non-singular model of the plane curve ∑ _{i =1}}^t y ^{q /2 i}=x ^{q +1}, q=2^t, provided that q/2 is a Weierstrass non-gap at some point of the curve.