zbMATH — the first resource for mathematics

On maximal curves in characteristic two. (English) Zbl 0931.11022
The authors study $${\mathbb F}_{q^2}$$-maximal curves when $$q$$ is even, that is non-singular geometrically irreducible projective algebraic curves $$\mathcal X$$ over $${\mathbb F}_{q^2},$$ whose number of $${\mathbb F}_{q^2}$$-rational points attains the Hasse-Weil bound $$q^2+1+2qg$$, where $$g$$ is the genus of $$\mathcal X.$$
For general $$q$$ it is known that either $$g=q(q-1)/2,$$ in which case up isomorphism $$\mathcal X$$ is the Hermitian curve, or $$g\leq g_2:=\lfloor (q-1)^2/4\rfloor.$$ For $$q$$ odd it was recently shown in joint work by the second author, that up to isomorphism there is a unique $${\mathbb F}_{q^2}$$-maximal curve of genus $$g,$$ with $$(q-1)(q-2)/4<g\leq g_2.$$ This curve is the non-singular model determined by $$y^q+y=x^{(q+1)/2}$$ and has $$g=g_2.$$
In this paper the authors extend this result to even characteristic, provided there exists a point $$P\in {\mathcal X}$$ such that $$q/2$$ is a Weierstrass non-gap at $$P.$$ Making this assumption they show that for $$q=2^t,$$ if $$(q-1)(q-2)/4<g\leq g_2=q(q-2)/4,$$ then $$\mathcal X$$ is $${\mathbb F}_{q^2}$$-isomorphic to the non-singular model determined by $$\sum_{i=1}^t y^{q/2^i} = x^{q+1},$$ and here again $$g=g_2.$$
The paper is well written, using special properties of maximal curves and their linear systems, in particular properties of Frobenius orders and the theory of Weierstrass points.

MSC:
 11G20 Curves over finite and local fields 14H55 Riemann surfaces; Weierstrass points; gap sequences 14H25 Arithmetic ground fields for curves 11R58 Arithmetic theory of algebraic function fields
Full Text: