Further results on the number of rational points of hyperelliptic supersingular curves in characteristic 2
Given a prime power \(q=p^n\) and \(g>0\) an integer, for which integer \(N\) do there exist a (nonsingular, geometrically irreducible, projective, algebraic) curve \(\mathcal X\) of genus \(g\) over the finite field \(\mathbb F\) of order \(q\) such that \(N=\#\mathcal X(\mathbb F)\)? We have the constrain \(|N-(q+1)|\leq g\lfloor 2\sqrt{q}\rfloor\) by the Hasse-Weil-Serre bound; we always assume \(p=2\). In the paper under review, this problem is considered for some subclasses of curves. As a matter of fact let \(\mathcal X\) be hyperelliptic supersingular of genus \(g=1,2\) and \(n\) odd, then \(N-(q+1)\in \{0, \pm\sqrt{2q}\}\) (which are classical results due to Deuring-Waterhouse and Rück-Xing respectively); for \(g=4\), \(N-(q+1)\in \{0, \pm \sqrt{2q}, \pm 2\sqrt{2q}, \pm 4\sqrt{2q}\}\) [\textit{G. McGuire} and \textit{A. Zaytsev}, Finite Fields Appl. 18, No. 5, 886--893 (2012; Zbl 1256.14028)]. In this paper the cases \(g=5,6,8\) are worked out; we have \(N-(q+1)\in\{0, \pm\sqrt{2q}, \pm2\sqrt{2q}\}\) for \(g=5,5\) and \(N-(q+1)\in\{0, \pm\sqrt{2q}, \pm 2\sqrt{2q}, \pm 4\sqrt{2q}, \pm 8\sqrt{2q}\}\) for \(g=8\). In the last case \(N-(q+1)\in \{0, \pm \sqrt{q}, \pm 2\sqrt{q}, \pm 4\sqrt{q}, \pm 8\sqrt{q}, \pm 16\sqrt{q}\}\) whenever \(n\) is even. In general for a supersingular curve over \(\mathbb F\) with \(n\) odd, the number \(N-(q+1)\) must be of the form \(k\sqrt{2q}\) for some integer \(k\) [loc. cit.]. We notice that \(k=2^r\) is a power of 2 in all the results above being this fact an open problem for \(g>8\). We should mention that there are no hyperelliptic supersingular curves in characteristic 2 of genus 3 or 7, cf. [\textit{J. Scholten} and \textit{H. J. Zhu}, Int. Math. Res. Not. 2002, No. 17, 905--917 (2002; Zbl 1034.14013)]. Here also it is shown a family of hyperelliptic supersingular curves over \(\mathbb F\) of genus \(2^k\) such that \(N-(q+1)\in \{0, \pm\sqrt{2^{n+w}}\}\) with \(0\leq w\leq 2k\) and \(n+w\) even. Finally, as an interesting application (via characteristic polynomials of Frobenius morphisms on Jacobians) it is shown that there is no a hyperelliptic supersingular curve of genus 5 over \(\mathbb F\), \(n\) odd, whose Jacobian is \(\mathbb F\)-isogenous to an abelian variety of type \(E^5\) being \(E\) a supersingular elliptic curve.
scientific article; zbMATH DE number 6498069
scientific article