Cohomology of moduli spaces of curves of genus three via point counts

Jonas Bergström 1
  • 1 Department of Mathematics, KTH, 100 44 Stockholm, Sweden. e-mail:


In this article we consider the moduli space of smooth n-pointed non-hyperelliptic curves of genus 3. In the pursuit of cohomological information about this space, we make -equivariant counts of its numbers of points defined over finite fields for n ≦ 7. Combining this with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2, and the moduli space of smooth hyperelliptic curves of genus 3, we can determine the -equivariant Galois and Hodge structure of the (ℓ-adic respectively Betti) cohomology of the moduli space of stable curves of genus 3 for n ≦ 5 (to obtain n ≦ 7 we would need counts of “8-pointed curves of genus 2”).

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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.