The first higher Chow groups of $\mathcal{M}_{1,n}$ for $n\leq 4$

TL;DR

This paper computes the indecomposable higher Chow groups of moduli spaces of genus one curves with up to four marked points, providing new proofs and formulas for related Chow rings and boundary classes.

Contribution

It explicitly calculates higher Chow groups for ,n, offering new proofs of Chow ring presentations and formulas for boundary classes.

Findings

Computed indecomposable higher Chow groups for ,n

Provided new proofs of Chow ring presentations for ,n

Derived formulas for boundary strata classes

Abstract

For $n\leq 4$, we compute the indecomposible higher Chow groups $\overline{\operatorname{CH}}(\mathcal{M}_{1,n},1)$ with integer coefficients. As an application, we give new proofs of presentations of the integral Chow rings $\operatorname{CH}(\overline{\mathcal{M}}_{1,n})$ for $n\leq 4$ and determine formulas for the classes of boundary strata in these rings.