The \'etale cohomology ring of a punctured arithmetic curve

TL;DR

This paper computes the étale cohomology ring of punctured arithmetic curves, providing new methods for understanding Galois groups and classical reciprocity laws in number theory.

Contribution

It introduces an explicit computation of the cohomology ring for punctured arithmetic curves and offers an efficient way to determine presentations of related Galois groups.

Findings

Computed the cohomology ring $H^*(U,\mathbb{Z}/n\mathbb{Z})$ for punctured arithmetic curves.

Provided a method to compute presentations of $Q_2(G_S)$ for various number fields.

Connected the cup product structure to classical reciprocity laws.

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 $n$): $\mu_p(\overline{K}) \not\subseteq K$; $S$ 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.