On rational connectedness and parametrization of finite Galois extensions

TL;DR

The article investigates $R$-equivalence on parametrizing spaces for finite $G$-Galois extensions over number fields, revealing how many rational families are needed to densely parametrize all extensions when a generic extension may not exist. By embedding into tori, employing flasque resolutions, and twisting cohomology, the authors derive concrete counts and dense parametrizations for basic groups, notably cyclic $2$-groups and dihedral groups, with the Brauer group playing a central role in the obstruction analysis. A key outcome is that for dihedral groups $D_{2^s}$ over number fields, $H^1(K,D_{2^s})$ is $R$-trivial, enabling dense parametrization even without a single universal extension, while explicit counterexamples (e.g., $C_8$ over certain quadratic fields) show that a single parametric family cannot always capture all cases. The work highlights deep connections between $R$-equivalence, Brauer-Manin obstructions, and weak approximation, illustrating both when parametrizations exist and when they necessarily require multiple, independently constructed families.

Abstract

Given two $G$-Galois extensions of $\mathbb Q$, is there an extension of $\mathbb Q(t)$ that specializes to both? The equivalence relation on $G$-Galois extension of $\mathbb Q$, induced by the above question, is called $R$-equivalence. The number of $R$-equivlance classes indicates how many rational spaces are required in order to parametrize all $G$-Galois extensions of $\mathbb Q$. We determine the $R$-equivalence classes for basic families of groups $G$, and consequently obtain parametrizations of the $G$-Galois extensions of $\mathbb Q$ in the absence of a generic extension for $G$.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.6
• Definition 2.1
• Example 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof