[Paper Review] On the Fontaine-Mazur Conjecture for CM-Fields

Kay Wingberg|ArXiv.org|Jul 19, 2000

Mathematical Dynamics and Fractals4 citations

TL;DR

This paper investigates the Fontaine-Mazur conjecture for CM-fields by analyzing unramified p-extensions and their Galois groups. Using duality theorems and Iwasawa theory, it proves that if the Galois group of the maximal unramified p-extension of a CM-field is p-powerful, then it must be finite unless specific conditions on the p-rank of the ideal class group or the Iwasawa μ-invariant are violated. The key result establishes a conditional finiteness criterion for p-adic analytic Galois groups in CM-extensions.

ABSTRACT

Fontaine and Mazur conjecture that a number field k has no infinite unramified Galois extension such that its Galois group is a p-adic analytic pro-p-group. We consider this conjecture for the maximal unramified p-extension of a CM-field k.

Motivation & Objective

• To investigate the Fontaine-Mazur conjecture in the context of CM-fields, particularly regarding the existence of infinite unramified p-extensions with p-adic analytic Galois groups.
• To determine conditions under which the Galois group of the maximal unramified p-extension of a CM-field is finite, especially when it is p-powerful or uniform.
• To extend Boston's result on finite unramified p-extensions to CM-fields with additional arithmetic constraints on the ideal class group and Iwasawa invariants.
• To establish a duality-theoretic framework linking unit groups, class groups, and Galois cohomology in infinite Galois extensions.
• To prove that if the Galois group of the maximal unramified p-extension of a CM-field is p-adic analytic, then the corresponding group over its maximal totally real subfield must be finite.

Proposed method

• Applies Tate duality and Poitou-Tate duality to the Galois cohomology of S-units and S-ideal class groups in infinite Galois extensions.
• Uses the exact sequence relating S-units, S-ideles, and S-class groups to derive cohomological isomorphisms for the Galois group of the maximal unramified S-extension.
• Applies the duality between cohomology groups of the Galois group of the maximal unramified p-extension and the p-adic integers, via Pontryagin duality.
• Employs Iwasawa theory and the structure of Z_p-extensions to analyze the behavior of p-ranks and μ-invariants in towers of number fields.
• Applies the theory of p-powerful and uniform pro-p groups, particularly using the dimension formula and the fact that such groups with small rank cannot surject onto Z_p unless finite.
• Uses the fact that if a pro-p group surjects onto Z_p, then the class number finiteness implies the group must be finite, to derive contradiction in the infinite case.

Experimental results

Research questions

• RQ1 Under what conditions is the Galois group of the maximal unramified p-extension of a CM-field finite when it is p-powerful?
• RQ2 How does the p-rank of the ideal class group of the maximal totally real subfield affect the structure of unramified p-extensions?
• RQ3 What role does the Iwasawa μ-invariant play in determining the finiteness of unramified p-extensions in CM-extensions?
• RQ4 Can the Fontaine-Mazur conjecture be verified for CM-fields using duality and cohomological techniques?
• RQ5 Is it possible for the Galois group of the maximal unramified p-extension of a CM-field to be infinite and p-adic analytic?

Key findings

• If the p-rank of the ideal class group of the maximal totally real subfield of a CM-field is not equal to 1, then the Galois group of the maximal unramified p-extension is finite whenever it is p-powerful.
• When the p-rank of the ideal class group of the maximal totally real subfield is 1, the Galois group is still finite if the first step of the cyclotomic Z_p-extension is ramified and the group is uniform.
• For a CM-field k with μ-invariant zero in its cyclotomic Z_p-extension, there exists n₀ such that for all n ≥ n₀, the Galois group of the maximal unramified p-extension of the n-th layer of the cyclotomic Z_p-extension is finite if it is p-powerful.
• The result extends to S-extensions where S contains the archimedean primes and no prime of S splits in k/k⁺, with the ideal class group replaced by the S-ideal class group.
• If the Galois group of the maximal unramified p-extension of a CM-field is p-adic analytic, then the Galois group over its maximal totally real subfield must be finite.
• The proof relies on showing that an infinite p-powerful pro-p group with small rank (≤2) would surject onto Z_p, contradicting the finiteness of the class number unless the group is finite.