Reciprocity laws and $K$-theory

TL;DR

This paper introduces a new framework linking $K$-theory and reciprocity laws on algebraic varieties, providing a unified approach to classical results like Weil and Parshin reciprocity through symbol maps associated with flags.

Contribution

It constructs symbol maps from $K$-theory spectra to formulate and prove a general reciprocity law for algebraic varieties of any dimension, unifying classical reciprocity laws.

Findings

Re-derivation of classical reciprocity laws from $K$-theory

Establishment of a general reciprocity law for flags in algebraic varieties

Extension of reciprocity concepts to higher-dimensional varieties

Abstract

We associate to a full flag $\mathcal{F}$ in an $n$-dimensional variety $X$ over a field $k$, a "symbol map" $\mu_{\mathcal{F}}:K(F_X) \to \Sigma^n K(k)$. Here, $F_X$ is the field of rational functions on $X$, and $K(\cdot)$ is the $K$-theory spectrum. We prove a "reciprocity law" for these symbols: Given a partial flag, the sum of all symbols of full flags refining it is $0$. Examining this result on the level of $K$-groups, we re-obtain various "reciprocity laws". Namely, when $X$ is a smooth complete curve, we obtain degree of a principal divisor is zero, Weil reciprocity, Residue theorem, Contou-Carr\`{e}re reciprocity. When $X$ is higher-dimensional, we obtain Parshin reciprocity.