On the number of curves of genus 2 over a finite field.
The author computes the number of nonsingular curves of genus 2 defined over a finite field \(k\) of odd characteristic up to isomorphisms defined over \(k\), and gives an explicit representative for each isomorphism class corresponding to such a curve with a nontrivial reduced automorphism group. The case of characteristic 2 is treated in a paper by the author, \textit{E.~Nart} and \textit{J.~Pujolàs} [Math. Z. 250, No. 1, 177--201 (2005; Zbl 1097.11033)]. Previously, the number of such curves up to isomorphisms defined over \(k\) and quadratic twists was computed by \textit{A.~López, D.~Maisner, E.~Nart} and \textit{X.~Xarles} [Finite Fields Appl. 8, No. 2, 193--206 (2002; Zbl 1036.14011)], and the number of such curves, up to isomorphisms defined over \(k\), that have a \(k\)-rational Weierstrass point was computed by \textit{Y.~Choie} and \textit{D.~Yun} [Lect. Notes Comput. Sci. 2384, 190--202 (2002; Zbl 1022.11029)]. Most of the current article deals with computing the cohomology set \(H^1(\text{Gal}(\bar k/k),\text{Aut}(C))\) for the various possibilities for the automorphism group of a nonsingular curve \(C\) of genus 2 defined over \(k\). This cohomology set is in one-to-one correspondence with the set of twists of \(C\) over \(k\).
scientific article; zbMATH DE number 2005897