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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.

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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