9. The Coleman map
Throughout this chapter p is an odd prime and we work with coefficient field
\Qp. We bring the arithmetic of cyclotomic fields into the picture through the
theory of local units. The main result is a theorem of Coleman identifying
norm-coherent systems of local units with a distinguished space of power series
over \Zp; under the Mahler transform these become p-adic measures. Applied to
cyclotomic units, this machinery reconstructs the Kubota–Leopoldt p-adic
L-function \zeta_p on the algebraic side, packaged as the Coleman map.
This is the key bridge between the analytic object \zeta_p and arithmetic, and is
the first step towards the Iwasawa Main Conjecture. See Coleman (1979)
for Coleman's original work and Jacinto and Williams (2023) for the treatment we
follow.