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On the number of curves of genus 2 over a finite field. (English) Zbl 1091.11023
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, E. Nart and 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 A. López, D. Maisner, E. Nart and 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 Y. Choie and 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 \(k\).

11G20 Curves over finite and local fields
14H25 Arithmetic ground fields for curves
14G15 Finite ground fields in algebraic geometry
Full Text: DOI
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