Consider a prime number \(p\) and a Galois extension \(L\) of a number field \(F\) with Galois group \(\mathfrak G = \text{Gal}(L/F)\). Let \(X_ L\) denote the Galois group of the maximal abelian pro-\(p\) extension of \(L\) and \(\Lambda (\mathfrak G)\) denote the Iwasawa algebra \({\mathbb Z}_ p[[\mathfrak G]]\) of \(\mathfrak G\). Then \(X_ L\) is a \(\Lambda (\mathfrak G)\)-module via conjugation. It is said that \(L/F\) is admissible if \(\mathfrak G\) has dimension larger than or equal to \(2\), \(L\) contains the cyclotomic \({\mathbb Z}_ p\)-extension \(K\) of \(F\), and \(L/F\) is unramified outside a finite set of prime divisors of \(F\). If in addition \(\mathfrak G\) is pro-\(p\) and contains no elements of order \(p\), \(L/F\) is called strongly admissible.
R. Greenberg [Adv. Stud. Pure Math. 30, 335–385 (2001; Zbl 0998.11054)] conjectured that when \(L\) is the compositum of all \({\mathbb Z} _ p\)-extensions of \(F\), the annihilator of \(X_ L\) has height at least \(2\) as \(\Lambda(\mathfrak G)\)-module. This is to say that \(X_ L\) is \(\Lambda(\mathfrak G)\)-pseudo-null.
A natural question is that if \(L/F\) is an admissible \(p\)-adic Lie extension, is \(X_ L\) pseudo–null as a \(\Lambda(\mathfrak G)\)-module?
If \(\Gamma = \text{Gal}(K/F)\), \(X_ K\) is a finitely generated torsion \(\Lambda= \Lambda(\Gamma)\)-module. It is a conjecture of Iwasawa that the \(\mu\)-invariant \(\mu(X_ K)\) is zero. This is true for \(F/{\mathbb Q}\) abelian. In case \(F={\mathbb Q}(\zeta _ p)\), \(\zeta _ p\) a primitive root of unity, the \(\lambda\)-invariant \(\lambda(X_ K)\) is positive if and only if \(p\) is an irregular prime. In particular, \(X_ K\) is often not pseudo-null as \(\Lambda\)-module, hence the reason of considering the dimension larger than or equal to \(2\).
In the paper under review the authors show that the answer to the question whether \(X_ L\) is a pseudo-null \(\Lambda(\mathfrak G)\)-module is “no” with counterexamples occurring frequently for CM-fields \(L\). In fact if \(p\) is odd, \(X_ L= X_ L^ + \oplus X_ L^ -\) as usual, and if \(G=\text{Gal}(L/K)\), it can be computed the \(\Lambda(G)\)-rank of \(X_ L^ -\) for any strongly admissible \(p\)-adic Lie extension \(L/F\) of CM-fields with \(\mu (X_ K^ -)=0\). This rank is zero if and only if \(X_ L^ -\) is \(\Lambda(G)\)-torsion, or equivalently, \(\Lambda(\mathfrak G)\) pseudo-null. In general the authors prove that \[ \text{rank} _ {\Lambda(G)}(X_ L^ -) = \lambda(X_ K^ -)-\delta + | Q_ {L/K}| \] where \(Q_ {L/K}\) is the set of prime divisors in the maximal real subfield \(K^ +\) of \(K\) that split in \(K\), ramify in \(L^ +\), and do not divide \(p\) and \(\delta\) is equal to \(1\) or \(0\) depending on whether \(F\) contains the \(p\)th roots of unity or not, respectively. In particular, \(X_ L^ -\) is not pseudo-null over \(\Lambda (\mathfrak G)\) if and only if \(\lambda( X_ K^ -)-\delta + | Q_{L/K}| \geq 1\).
The proof of this result uses Kida’s formula for \(\lambda\)-invariants in CM-extensions of cyclotomic \({\mathbb Z}_ p\)-extensions of number fields.
This result provides a negative answer to the question even in the case \(F={\mathbb Q}(\zeta _ p)\) and \(L\) is a \({\mathbb Z}_ p\)-extension of \(K\) with complex multiplication which is unramified outside \(p\). The smallest prime number \(p\) for which the \({\mathbb Z}_ p\)-rank of \(X_ K^ -\) is at least \(2\) is \(p=157\). Other examples occur for \(p= 353, 379, 467\) and \(491\).
For some cases where \(L\) is not a CM-field there exists some evidence that \(X_ L\) might be a pseudo-null \(\Lambda (\mathfrak G)\)-module. The paper contains an appendix by the second author on Iwasawa modules in procyclic extensions which is used for the new proof of Kida’s formula given in the paper which includes a weakening of the usual \(\mu=0\) assumption.