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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
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