Symbols for toric Eisenstein cocycles and arithmetic applications

Abstract

Using a complex parameterizing rational spherical chains, we construct explicit cocycles for $\mathrm{GL}_n(\Q)$ valued in the motivic cohomology of (open subsets of) the algebraic $n$-torus $\mathbb{G}_m^n$. The resulting cocycles directly generalize the work of Sharifi and Venkatesh from the case $n=2$ \cite{SV}. Even in this special case, our systematic use of pushforwards allows us to avoid the use of their ``connecting sequences,'' and allows us to refine the construction and Hecke properties of the Sharifi map $\varpi$ to the maximal expected statements, while inverting only the prime $2$. For general $n$, the $d\log$ regulator of our cocycle is related by convex conical duality to cocycles constructed from Shintani cones. This affords a systematic approach to $p$-adic $L$-functions for totally real fields without need for auxiliary data or logarithm sheaf coefficients, including a distribution-valued $\GL_n(\Z)$-cocycle specializing in a simple way to all such $p$-adic $L$-functions. It moreover provides a direct conceptual link between polylogarithmic constructions of Eisenstein classes (e.g., in \cite{BKL}), and those constructed using Shintani cones (e.g., in \cite{CDG}). We also show how our formalism gives an alternate proof of the exceptional divisibilities of the Deligne-Ribet $2$-adic $L$-function in almost all cases.