The étale cohomology ring of a punctured arithmetic curve
Eric Ahlqvist, Magnus Carlson
TL;DR
The paper computes the étale cohomology ring $H^*(U,\mathbb{Z}/n\mathbb{Z})$ for punctured arithmetic curves $U=\mathrm{Spec}\,\mathcal{O}_K\setminus S$, unifying cases with real places and without assuming $\mu_n\subset K$. It derives explicit cup-product formulas via compactly supported cohomology and Artin–Verdier duality, giving concrete identifications of $H^1$ and $H^2$ with idèle-class data and norms from extensions $L/K$. These tools enable efficient presentations of $Q_2(G_S)$ for varying $K$, and illuminate reciprocity laws such as Legendre symbol reciprocity through the graded-commutativity of cup products. The results broaden the computational and conceptual toolkit for Galois and class-field theoretic questions, with applications to class-field towers, unramified Fontaine–Mazur-type problems, and arithmetic Chern–Simons theory.
Abstract
We compute the cohomology ring $H^*(U,\mathbb{Z}/n\mathbb{Z})$ for $U=X\setminus S$ where $X$ is the spectrum of the ring of integers of a number field $K$ and $S$ is a finite set of finite primes. As a consequence, we obtain an efficient way to compute presentations of $Q_2(G_S)$, where $G_S$ is Galois group of the maximal extension of $K$ unramified outside of a finite set of primes $S$, for varying $K$. This includes the following cases (for $p$ any prime dividing $n$): $μ_p(\overline{K}) \not\subseteq K$; $S$ does not contain the primes above $p$; and $p=2$ with $K$ admitting real archimedean places. We also show how to recover the classical reciprocity law of the Legendre symbol from the graded commutativity of the cup product.