Modular Symbols with Values in Beilinson-Kato Distributions

Abstract

For each integer $n\geq 1$, we construct a $\operatorname{GL}_n(\mathbb Q)$-invariant modular symbol $\bmξ_n$ with coefficients in a space of distributions that takes values in the Milnor $K_n$-group of the modular function field. The Siegel distribution $\bmμ$ on $\mathbb Q^2$, with values in the modular function field, serves as the building block for $\bmξ_n$; we define $\bmξ_n$ essentially by taking the $n$-Steinberg product of $\bmμ$. The most non-trivial part of this construction is the cocycle property of $\bmξ_n$; we prove it by using an induction on $n$ based on the first two cases $\bmξ_1$ and $\bmξ_2$; the first case is trivial, and the second case essentially follows from the fact that Beilinson-Kato elements in the Milnor $K_2$-group modulo torsion satisfy the Manin relations.