[Paper Review] Iwasawa Invariants of Some Non-Cyclotomic $\mathbb Z_p$-extensions

David Hubbard, Lawrence C. Washington|arXiv (Cornell University)|Mar 20, 2017

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper explicitly computes Iwasawa invariants (λ, μ, ν) for non-cyclotomic ℤₚ-extensions of imaginary quadratic fields when p = 2 and p = 3, using Iwasawa’s construction involving anticyclotomic ℤₚ-extensions and cyclic extensions of degree p. The key contribution is proving that μ = s − 1 for s primes inert in the base field and ramified in the cyclic extension, with explicit computations confirming this in multiple cases.

ABSTRACT

Iwasawa showed that there are non-cyclotomic $\mathbb Z_p$-extensions with positive $μ$-invariant. We show that these $μ$-invariants can be evaluated explicitly in many situations when $p=2$ and $p=3$.

Motivation & Objective

• To determine explicit values of Iwasawa invariants μ, λ, ν for non-cyclotomic ℤₚ-extensions when p = 2 and p = 3.
• To investigate whether Iwasawa’s lower bound μ ≥ s − 1 is sharp in non-cyclotomic settings.
• To compute class numbers of specific number fields Kₙ = kₙ(√[p]{q}) to evaluate the growth of p-primary class groups.
• To analyze the role of unit indices and fixed ideal classes under Galois action in determining μ-invariants.
• To explore conditions under which μ > s − 1, based on local and global norm conditions.

Proposed method

• Uses Iwasawa’s construction: start with an imaginary quadratic field k₀, take its anticyclotomic ℤₚ-extension k∞/k₀, and base-change via a cyclic degree-p extension K₀/k₀ to form K∞/K₀.
• Applies Chevalley’s formula for the size of the G-fixed subgroup of the class group in cyclic extensions to estimate the p-part of the class number of Kₙ.
• Employs the inverse limit X = lim← Aₙ of p-Sylow subgroups of class groups, viewed as a Λ = ℤₚ[[T]]-module, to analyze μ-invariants via module structure.
• Leverages the fact that primes inert in k₀/ℚ split completely in k∞/k₀, so ramification in K∞/K₀ arises only from primes above q in K₀/k₀.
• Performs explicit class number computations for small n (especially n = 1) using algebraic number theory tools, including unit group indices and local norms.
• Uses inequalities involving h₂′ (the class number of a related field) and bounds on eₙ to constrain possible values of μ, especially when μ = 0 or μ = 1.

Experimental results

Research questions

• RQ1 Is Iwasawa’s lower bound μ ≥ s − 1 sharp for non-cyclotomic ℤₚ-extensions when p = 3 or p = 2?
• RQ2 Can the exact value of the μ-invariant be computed explicitly in non-cyclotomic settings, and if so, under what conditions?
• RQ3 What determines whether μ = s − 1 or μ > s − 1 in Iwasawa’s construction?
• RQ4 How do local norm conditions and unit group indices influence the μ-invariant in ℤₚ-extensions?
• RQ5 Can explicit class number computations for Kₙ = kₙ(√[p]{q}) be used to deduce μ, λ, ν invariants?

Key findings

• For p = 3 and q ≡ 2 or 5 mod 9, the class number of ℚ(ζ₃, √[3]{q}) is not divisible by 3, so μ = λ = ν = 0 for the ℤ₃-extension.
• When q ≡ 8 mod 9, the class number of K₀ = ℚ(ζ₃, √[3]{q}) is not divisible by 3, so A₀^τ = 1 and μ = λ = ν = 0.
• For q ≡ 8 mod 9 and A₁ = 9 × 3, the unit index is 3 and e₁ − e₀ = 3, so 0 ≤ μ ≤ 1, but μ = 0 is implied under plausible assumptions on h₂.
• For q = 89, 449, 431, 647, 719, 683, 773, with A₁ = 3², the unit index is 3² and 0 ≤ μ ≤ 1; μ = 0 is likely under assumptions on h₂.
• For q = 269, 521, 809 with A₁ = 9² × 3², the unit index is 3² and 0 ≤ μ ≤ 3; μ ≤ 1 is likely for q = 269, 809 and μ ≤ 2 for q = 521.
• For q = 827 with A₁ = 27 × 9 × 3², the unit index is 3 and 0 ≤ μ ≤ 3; under assumption h₂ ∣ (h₂′)², μ = 0 is deduced.