Pro-$\ell$-by-cyclotomic and tamely ramified variants of the Neukirch-Uchida Theorem

TL;DR

This work strengthens the rigidity of Neukirch-Uchida-type results by proving that the maximal pro-$\ell$-by-cyclotomic quotient of the absolute Galois group and the maximal tamely ramified quotient suffice to recover a number field, contrasting with negative results for pronilpotent quotients. The authors develop an $\ell$-sealed cohomology framework, establish a Brauer-type exact sequence, and use abundance-based prime-characteristic detection to connect local data to global field isomorphisms. They prove a sequence of results culminating in a full Neukirch-Uchida theorem for $\ell$-sealed and abundant extensions, including weak and full lifting of group isomorphisms to field isomorphisms, and they derive corollaries for pro-$\ell$-by-cyclotomic and tamely ramified variants. The methods illuminate how restricted ramification and $\ell$-adic structure control arithmetic equivalence and field determination, suggesting new directions toward prosupersolvable variants. $K$ is thereby determined from quotients like $G_{\Omega/K}$ when $\Omega$ is the maximal pro-$\ell$-by-cyclotomic or tamely ramified extension, with implications for arithmetic geometry and Galois-reconstruction problems.

Abstract

We prove a generalization of the Neukirch-Uchida Theorem. In particular, we show that the isomorphism type of a number field $K$ can be recovered from the maximal pro-$\ell$-by-cyclotomic quotient of its absolute Galois group $G_{\overline{K}/K}$. This should be contrasted with the previous result that the isomorphism type cannot, in general, be recovered from the maximal pronilpotent quotient. We also show that the isomorphism type can be recovered from the maximal tamely ramified quotient.

Theorems & Definitions (69)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Example 1
• proof