[Paper Review] Greenberg's conjecture and units in multiple Z_p-extensions

William G. McCallum|ArXiv.org|Jul 13, 2000

Algebraic Geometry and Number Theory4 references5 citations

TL;DR

This paper proves Greenberg's conjecture for the cyclotomic field $\mathbb{Q}(\zeta_p)$ under conditions of index of irregularity 1, Vandiver's conjecture, and a unit group condition. It establishes that the Iwasawa module $A$ is pseudo-null when $p$ is irregular but satisfies these constraints, using a generalization of Iwasawa's unit theorem and class field theory to show the Galois group of the Kummer extension by $p$-power roots of $p$-units is torsion-free as a $\Lambda$-module.

ABSTRACT

In this paper we prove Greenberg's pseudo-null conjecture for the field of p-th roots of unity in the case that p exactly divides the class number and the index of the global units in the local units. We also generalize to the case of multiple Z_p-extensions a theorem of Iwasawa on the Kummer extension of the cyclotomic Z_p-extension generated by p-power roots of units .

Motivation & Objective

• To prove Greenberg's conjecture that the Iwasawa module $A$ is pseudo-null for $K = \mathbb{Q}(\zeta_p)$ under specific arithmetic conditions.
• To establish the torsion-freeness of the Galois group $Y' = \mathrm{Gal}(N_\infty/K_\infty)$, where $N_\infty$ is the Kummer extension by $p$-power roots of $p$-units.
• To show that Greenberg's conjecture implies that the maximal $p$-ramified pro-$p$ extension of $K$ has no free pro-$p$ quotient of rank $r$ unless $p$ is regular.
• To generalize Iwasawa's theorem on units in the context of multiple $\mathbb{Z}_p$-extensions.

Proposed method

• Use the Iwasawa algebra $\Lambda = \mathbb{Z}_p[[\Gamma]]$ with $\Gamma \cong \mathbb{Z}_p^r$ to study the structure of $A$ as a $\Lambda$-module.
• Apply the generalized Iwasawa unit theorem to show that the Galois group $Y' = \mathrm{Gal}(N_\infty/K_\infty)$ is torsion-free as a $\Lambda$-module under the conditions of one prime above $p$ and full $p$-power roots of unity in $K_\infty$.
• Leverage class field theory to identify $A \cong X = \mathrm{Gal}(L_\infty/K_\infty)$, so that Greenberg's conjecture becomes the statement that $X$ is pseudo-null.
• Use the equivalence between $X$ being pseudo-null and $Y = \mathrm{Gal}(M_\infty/K_\infty)$ being torsion-free, established via duality and cohomological techniques.
• Verify that the conditions of Theorem 1 (class group $A(K) \cong \mathbb{Z}/p\mathbb{Z}$, $(U/E)[p^\infty] \cong \mathbb{Z}/p\mathbb{Z}$) imply the hypotheses of Theorem 26, including $H^2(G, \mathbb{Z}/p\mathbb{Z}) \cong \mathbb{Z}/p\mathbb{Z}$ and $\mathrm{Kab} \subset N_\infty$.
• Use the reflection principle and known results on cyclotomic units to show that the Hilbert class field $H$ of $K$ is contained in $K_\infty$, which implies capitulation of the ideal class and allows $\alpha^{1/p} \in N_\infty$.

Experimental results

Research questions

• RQ1 Under what conditions on $K = \mathbb{Q}(\zeta_p)$ is Greenberg's conjecture that $A$ is pseudo-null true?
• RQ2 When is the Galois group of the Kummer extension by $p$-power roots of $p$-units torsion-free as a $\Lambda$-module?
• RQ3 How does Greenberg's conjecture constrain the existence of free pro-$p$ quotients of the maximal $p$-ramified pro-$p$ extension of $K$?
• RQ4 What is the relationship between the structure of $A$ and the unit group $U/E$ in the $p$-adic completion of $K$?
• RQ5 Can Iwasawa's unit theorem be generalized to multiple $\mathbb{Z}_p$-extensions?

Key findings

• Greenberg's conjecture holds for $K = \mathbb{Q}(\zeta_p)$ when $A(K) \cong \mathbb{Z}/p\mathbb{Z}$ and $(U/E)[p^\infty] \cong \mathbb{Z}/p\mathbb{Z}$, which is satisfied by primes such as 37, 59, 67, 101, 103, 131, and 149.
• The Galois group $Y' = \mathrm{Gal}(N_\infty/K_\infty)$ of the Kummer extension by $p$-power roots of $p$-units is torsion-free as a $\Lambda$-module when $K_\infty$ contains all $p^n$-th roots of unity and there is only one prime above $p$.
• The maximal abelian $p$-ramified pro-$p$ extension $M_\infty$ of $K_\infty$ has a Galois group $Y$ that is torsion-free as a $\Lambda$-module, which implies $A$ is pseudo-null.
• If $K = \mathbb{Q}(\zeta_p)$ satisfies Greenberg's conjecture, then the maximal $p$-ramified pro-$p$ extension $G = \mathrm{Gal}(\Omega/K)$ has a free pro-$p$ quotient of rank $g - s$ if and only if $p$ is regular.
• The Hilbert class field $H$ of $K$ is contained in $K_\infty$, so the generator of the class group capitulates, which implies that $\alpha^{1/p} \in N_\infty$ for a suitable $p$-unit $\alpha$.
• The conditions of Theorem 1 are equivalent to $p$ satisfying Vandiver's conjecture and dividing exactly one of $B_2, B_4, \dots, B_{p-1}$, with an additional congruence condition on the $p$-adic logarithm of the unit.