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

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.