10. Iwasawa theorem on the zeros of the p-adic zeta function
The Coleman map of the previous chapter lets us construct the Kubota–Leopoldt
p-adic L-function \zeta_p from a tower of cyclotomic units. We now state a
theorem of Iwasawa that puts this on a deeper footing: it describes the zeros of
\zeta_p — packaged as a canonical ideal in the Iwasawa algebra — in terms of the
module of cyclotomic units sitting inside the local units. To move all of the
analytic information to the Galois side, we first reinterpret \zeta_p as a
pseudo-measure on the Galois group, then introduce the global and local modules of
cyclotomic units and their classical link to class numbers, and finally state
Iwasawa's theorem.
Throughout, p is an odd prime, F_\infty = \bigcup_{n\ge 1}\Q(\mu_{p^n}) is the
cyclotomic \Z_p-extension's ambient field, and \GG = \Gal(F_\infty/\Q). The
cyclotomic character is an isomorphism \chi : \GG \xrightarrow{\sim} \Zpx.