Rational cohomology of the moduli spaces of pointed genus 1 curves

TL;DR

This paper provides a combinatorial description of the rational cohomology of moduli spaces of pointed genus 1 curves with level N structures, detailing the Leray spectral sequence and mixed Hodge structures involved.

Contribution

It explicitly describes the E2 term of the Leray spectral sequence for these moduli spaces and establishes an isomorphism with their rational cohomology as mixed Hodge structures.

Findings

Explicit description of the E2 term of the Leray spectral sequence.

Isomorphism between the spectral sequence result and the rational cohomology.

Inclusion of symmetric group actions in the cohomology structure.

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.