Rational cohomology of the moduli spaces of pointed genus 1 curves

Abstract

We give a combinatorial description of the rational cohomology of the moduli spaces of pointed genus 1 curves with $n$ marked points and level $N$ structures. More precisely, we explicitly describe the $E_2$ term of the Leray spectral sequence of the forgetful mapping $\EuScript{M}_{1,n}(N)\to\EuScript{M}_{1,1}(N)$ and show that the result is isomorphic to the rational cohomology of $\EuScript{M}_{1,n}(N)$ as a rational mixed Hodge structure equipped with an action of the symmetric group ${\mathfrak{S}}_n$. The classical moduli space $\EuScript{M}_{1,n}$ is the particular case N=1.

27 pages. Remarks & comments very much appreciated

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