Geodesic
Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves
Documenta mathematica, Tome 14 (2009), pp. 259-296
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We consider the moduli space Hg,n​ of n-pointed smooth hyperelliptic curves of genus g. In order to get cohomological information we wish to make Sn​-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfil. Thus, if we can make Sn​-equivariant counts of Hg,n​ for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for Hg,n​ for all g and for n≤7. These results are applied to the moduli spaces of stable curves of genus 2 with up to 7 points, and this gives us the Sn​-equivariant Galois (resp. Hodge) structure of their l-adic (resp. Betti) cohomology.
DOI : 10.4171/dm/273
Classification : 11G20, 14H10
Mots-clés : cohomology of moduli spaces of curves, curves over finite fields 
@article{10_4171_dm_273,
     author = {Jonas Bergstr\"om},
     title = {Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves},
     journal = {Documenta mathematica},
     pages = {259--296},
     year = {2009},
     volume = {14},
     doi = {10.4171/dm/273},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/273/}
}
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Jonas Bergström. Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves. Documenta mathematica, Tome 14 (2009), pp. 259-296. doi: 10.4171/dm/273

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