Let p be a prime. A pro-p group G is said to be 1-smooth if it can be endowed with a homomorphism of pro-p groups G → 1+pZp satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro-p Galois groups of fields containing a root of 1 of order p, together with the cyclotomic character, are 1-smooth. We prove that a finitely generated p-adic analytic pro-p group is 1-smooth if, and only if, it occurs as the maximal pro-p Galois group of a field containing a root of 1 of order p. This gives a positive answer to De Clerq-Florence’s “Smoothness Conjecture” - which states that the surjectivity of the norm residue homomorphism (i.e., the “surjective half” of the Bloch-Kato Conjecture) follows from 1-smoothness - for the class of finitely generated p-adic analytic pro-p groups.

1-smooth pro-p groups and Bloch-Kato pro-p groups

Quadrelli, C
2022

Abstract

Let p be a prime. A pro-p group G is said to be 1-smooth if it can be endowed with a homomorphism of pro-p groups G → 1+pZp satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro-p Galois groups of fields containing a root of 1 of order p, together with the cyclotomic character, are 1-smooth. We prove that a finitely generated p-adic analytic pro-p group is 1-smooth if, and only if, it occurs as the maximal pro-p Galois group of a field containing a root of 1 of order p. This gives a positive answer to De Clerq-Florence’s “Smoothness Conjecture” - which states that the surjectivity of the norm residue homomorphism (i.e., the “surjective half” of the Bloch-Kato Conjecture) follows from 1-smoothness - for the class of finitely generated p-adic analytic pro-p groups.
Galois cohomology; maximal pro-p Galois groups; Bloch-Kato conjecture; cyclotomic character; p-adic analytic groups
Quadrelli, C
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2129358
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