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On the genus of a maximal curve. (English) Zbl 1018.11029

An \(\mathbb F_{q^2}\)-maximal curve \(\mathcal X\) of genus \(g\) is a projective, geometrically irreducible, non-singular algebraic curve defined over \(\mathbb F_{q^2}\) such that the number of its \(\mathbb F_{q^2}\)-rational points atttains the Hasse-Weil upper bound: \[ \#\mathcal X(\mathbb F_{q^2})=q^2+1+2qg. \] The genus \(g\) of a \(\mathbb F_{q^2}\)-maximal curve satisfies \(g\leq q(q-1)/2\) [Y. Ihara, J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 721-724 (1981; Zbl 0509.14019)]. Moreover R. Fuhrmann and F. Torres [Manuscr. Math. 89, 103-106 (1996; Zbl 0857.11032)] show that \[ \text{ either } \quad g \leq \lfloor(q-1)^2/4\rfloor \text{ or } \quad g=q(q-1)/2 . \]
The genus \(g\) attains the upper bound if and only if \(\mathcal X\) is \(\mathbb F_{q^2}\)-isomorphic to the Hermitian curve [H.-G. Rück and H. Stichtenoth, J. Reine Angew. Math. 457, 185-188 (1994; Zbl 0802.11053)].
For the second largest genus one has: In the case of \(q\) odd, \(g=(q-1)^2/4\) occurs if and only if \(\mathcal X\) is \(\mathbb F_{q^2}\)-isomorphic to the non-singular model of the plane curve of equation \(y^q+y=x^{(q+1)/2}\) [R. Fuhrmann, A. Garcia and F. Torres, J. Number Theory 67, 29-51 (1997; Zbl 0914.11036)]. For \(q\) even, a similar result is obtained in [M. Abdón and F. Torres, Manuscr. Math. 99, 39-53 (1999; Zbl 0931.11022)] under an extra-condition (which is shown here is always true) that \(\mathcal X\) has a particular Weierstrass point: \(g=\lfloor (q-1)^2/4\rfloor=q(q-2)/4\) if and only if \(\mathcal X\) is \(\mathbb F_{q^2}\)-isomorphic to the non-singular model of the plane curve of equation \(y^{q/2}+\ldots+y^2+y=x^{q+1}\).
In the range \(\lfloor (q-1)(q-3)/8\rfloor\leq g<\lfloor (q-1)^2/4\rfloor \) only twelve examples up to \(\mathbb F_{q^2} \)-isomorphisms are known to exist and the spectrum of their genera is listed below:
(I) \(g=\lfloor (q^2-q+4)/6\rfloor\) for \(q\equiv 0,1,2\pmod{3}\),
(II) \(g=(q^2-q-2)/6\) for \(q\equiv 2\pmod{3}\),
(III) \(g=\lfloor ((q-1)(q-2)/6\rfloor\) for \(q\equiv 0,2\pmod{3}\),
(IV) \(g=\lfloor (q^2-2q+5)/8\rfloor\) for \(q\equiv 0,1,3\pmod{4}\),
(V) \(g=\lfloor (q-1)(q-3)/8\rfloor\) for \(q\equiv 0,1,3\pmod{4}\).
These results make it plausible that only few \(\mathbb F_{q^2}\)-maximal curves can have genus close to the upper limit.
Curves with genera as in (III) and (V) are extremal in \(\mathbb P^4(\bar\mathbb F_{q^2})\) and in \(\mathbb P^5(\bar\mathbb F_{q^2})\) respectively, as such genera are Castelnuovo’s numbers \(c_0(q+1,r)\), \(r=4,5\). The genera in (I) and (IV) above coincide with Halphen’s number \(c_1(q+1,r)\), \(r=3,4\).
Having this evidence in mind, the authors conjecture that there is no \(\mathbb F_{q^2}\)-maximal curve of genus \(g\) such that \[ c_1(q+1,r)<g<c_0(q+1,r). \]
In the main Theorem, the authors show that: \[ \text{either}\quad g\leq \lfloor (q^2-q+4)/6\rfloor,\quad \text{or}\quad g=\lfloor(q-1)^2/4\rfloor,\quad\text{or}\quad g=q(q-1)/2 . \] They remark that this result is the best possible since the upper bound in the Theorem above cannot be improved as it is attained by the curves cited in (I) for every \(q\). In other words the third largest genus of an \(\mathbb F_{q^2}\)-maximal curve equals \(g=\lfloor (q^2-q+4)/6\rfloor\) independently of \(q\); by contrast, the fourth largest genus might heavily depend on \(q\): the gap between the third and fourth is only \(1\) for \(q\equiv 2\pmod{3}\).
The essential idea of the proof of this theorem is to show that every \(\mathbb F_{q^2}\)-maximal curve of genus \(g>\lfloor (q^2-q+4)/6\rfloor\) has a non-singular model \(\mathcal X\) over \(\mathbb F_{q^2}\) embedded in \(\mathbb P^3(\bar\mathbb F_{q^2})\) such that \(\mathcal X\) has degree \(q+1\) and lies on an \(\mathbb F_{q^2}\)-rational quadratic cone \(\mathcal Q\) whose vertex \(V\) belongs to \(\mathcal X\).
Finally, in the last section they investigate certain \(\mathbb F_{q^2}\)-maximal curves that are also extremal, that is the genus \(g\) is \(c_0(q+1,N)\). For the cases \(N=4\) or \(N=5\), they obtain a characterization for such curves.

MSC:

11G20 Curves over finite and local fields
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