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 $\bm\xi_n$ with coefficients in a space of distributions that takes values in the Milnor $K_n$-group of the modular function field. The Siegel distribution $\bm\mu$ on $\mathbb Q^2$, with values in the modular function field, serves as the building block for $\bm\xi_n$; we define $\bm\xi_n$ essentially by taking the $n$-Steinberg product of $\bm\mu$. The most non-trivial part of this construction is the cocycle property of $\bm\xi_n$; we prove it by using an induction on $n$ based on the first two cases $\bm\xi_1$ and $\bm\xi_2$; 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.