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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 \(k\).

MSC:
11G20 Curves over finite and local fields
14H25 Arithmetic ground fields for curves
14G15 Finite ground fields in algebraic geometry
Software:
GAP
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References:
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