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”).
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
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