$S_n$-extensions with prescribed norms

TL;DR

The paper analyzes the distribution of $S_n$-n-ics over a number field $k$ with prescribed norms from a finitely generated subgroup $\mathcal{A} \subseteq k^\times$. It shows that, for $n\in\{3,4,5\}$ (and conjecturally for $n\ge 6$), the leading density equals a product over primes of local masses $m_{\mathcal{A},\mathfrak{p}}$, yielding explicit Euler products in the odd-prime and $4$-quartic cases. The authors develop a comprehensive local-to-global framework based on étale algebras, splitting symbols, and local conditions, and they provide detailed formulas and algorithms to compute these local masses, including tame and wild ramification regimes and $2$-adic subtleties. These results enable practical computation of densities and offer precise asymptotics for the proportion of $S_n$-n-ics with $\mathcal{A}$ inside their norm groups, with substantial algorithmic contributions for quartic extensions. The work advances arithmetic statistics in a non-abelian setting by making the Malle–Bhargava heuristics concrete for norm-restricted $S_n$-extensions and by delivering concrete, implementable mass computations.

Abstract

Given a number field $k$, a finitely generated subgroup $\mathcal{A}\subseteq k^\times$, and an integer $n\geq 3$, we study the distribution of $S_n$-extensions of $k$ such that the elements of $\mathcal{A}$ are norms. For $n\leq 5$, and conjecturally for $n \geq 6$, we show that the density of such extensions is the product of so-called ``local masses'' at the places of $k$. When $n$ is an odd prime, we give formulas for these local masses, allowing us to express the aforementioned density as an explicit Euler product. For $n=4$, we determine almost all of these masses exactly and give an efficient algorithm for computing the rest, again yielding an explicit Euler product.

Theorems & Definitions (218)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• proof : Proof of Theorem \ref{['thm-mass-infinite-prime']}