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

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 Sn-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 Sn-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'').