11. Proof of Iwasawa theorem
This chapter proves Iwasawa's theorem, namely that the Coleman map induces an
isomorphism of \Lam(\GGam^+)-modules
\sU^+_{\infty,1}/\CC^+_{\infty,1} \;\xrightarrow{\sim}\; \Lam(\GGam^+)/I(\GGam^+)\zeta_p.
The strategy has four movements. First we put a \Lam(\GGam)-module structure on
the norm-coherent local units \sU_{\infty,1} and show the Coleman map is
equivariant for it. Next we compute the kernel and cokernel of the Coleman map by
analysing each constituent map; this is the fundamental exact sequence. Then we
exhibit explicit cyclic generators of the global and local cyclotomic units over
the relevant group rings. Finally we feed the generators through the Coleman map
to read off the image of the cyclotomic units, and conclude.
Throughout, p is an odd prime, F_\infty = \Q(\mu_{p^\infty}),
\GGam = \Gal(F_\infty/\Q) \cong \Zpx via the cyclotomic character \chi, and
\GGam^+ is the quotient by complex conjugation. We write
K_n = \Qp(\mu_{p^n}), \pi_n = \xi_{p^n}-1 for a uniformiser, \sU_n for
the local units of K_n, \sU_{n,1} for those congruent to 1 modulo
\pri_n, and \sU_\infty = \varprojlim_n \sU_n,
\sU_{\infty,1} = \varprojlim_n \sU_{n,1} for the norm-coherent towers. Recall
the Coleman map is the composition
\Col : \sU_\infty \xrightarrow{u\mapsto f_u} (\Zp[[T]]^\times)^{\cN=\mathrm{id}}
\xrightarrow{\dlog} \Zp[[T]] \xrightarrow{1-\varphi\circ\psi} \Zp[[T]]^{\psi=0}
\xrightarrow{\partial^{-1}} \Zp[[T]]^{\psi=0} \xrightarrow{\sA^{-1}} \Lam(\Zpx),
where \sA is the Mahler transform identifying measures with power series.