13.3. Consequences
We have already seen one application (in stating the Main Conjecture): if one class
number in a \Zp-extension is prime to p, so are all the others. For a finite
abelian group A, the p-rank is \rk_p(A) = \dim_{\Fp}(A/pA) = \dim_{\Fp}(A[p]),
the number of cyclic summands of p-power order.
Let F_\infty/F be a \Zp-extension. Then \mu = 0 if and only if
\rk_p(\Cl(F_n)) is bounded independently of n. This uses
Proposition 13.2.2.2.
By Lemma 13.2.2.1 and Proposition 13.2.1.2, \Cl(F_n)\otimes\Zp =
\sY_n sits in an exact sequence 0 \to C_n \to \sY_n \to \sA_n \to B_n \to 0 with
\sA_n = \sA/\varphi^n(T) and \abs{B_n}, \abs{C_n} bounded; so it suffices to
bound \dim_{\Fp}(\sA_n/p\sA_n). Now
\sA/(p,\varphi^n(T)) = \Big(\bigoplus_{i=1}^s \Lam/(p,\varphi^n(T))\Big) \oplus \Big(\bigoplus_{j=1}^t \Lam/(p,g_j,\varphi^n(T))\Big).
For n large that p^n \ge \deg g_j, both g_j and \varphi^n(T) are
distinguished, so \Lam/(p,\varphi^n(T)) = \Lam/(p,T^{p^n}) and
\Lam/(p,g_j,\varphi^n(T)) = \Lam/(p,T^{\deg g_j}). Hence the total is
(\Z/p\Z)^{s p^n + t g} with g = \sum_j \deg g_j, whose dimension is bounded in
n iff s = 0, i.e. iff \mu = \sum_i m_i = 0.
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(Ferrero–Washington.) If F is an abelian number field and F_\infty/F is
its cyclotomic \Zp-extension, then \mu = 0.
The proof is genuinely analytic and orthogonal to the algebraic theory above. For
F abelian over \Q, the Iwasawa Main Conjecture identifies the characteristic
power series of the relevant component of \sY_\infty with a branch of the
Kubota–Leopoldt p-adic L-function Definition 4.3.2; concretely,
each Dirichlet character \chi of F contributes a power series g_\chi(T) \in
\Zp[[T]] interpolating the values L_p(\chi\omega^j, s), and the \mu-invariant
of \sY_\infty is the minimum over \chi of the largest power of p dividing
g_\chi. Ferrero and Washington show this minimum is 0, i.e. that g_\chi is
not divisible by p, by exhibiting one coefficient that is a p-adic unit. The
coefficients are explicit sums of fractional parts \{a/p^n\} weighted by \chi,
and the key input is a normality statement: the base-p digits of these
generalised Bernoulli/Stickelberger expressions are equidistributed, so they cannot
all be divisible by p simultaneously. Hence \mu = 0 for the analytic side, and
the Main Conjecture transports the vanishing to the algebraic module \sY_\infty.
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(Greenberg's conjecture.) For any totally real field F and any
\Zp-extension F_\infty/F, one expects \mu = \lambda = 0; equivalently, the
class numbers \#\Cl(F_n) are bounded as n \to \infty. This remains open.