3.7. Locally analytic functions and distributions
Finally we sketch locally analytic distributions, the dual of locally analytic
functions, which extend the Mahler correspondence from \OL[[T]] to all
everywhere-convergent power series on the open unit disc. This section is used only
peripherally (to study values of \zeta_p near s = 1).
The p-adic open unit ball is B(0,1) = \set{z \in \Cp : \abs{z} < 1}. An
L-valued function on B(0,1) is rigid analytic if it is given by a power
series \sum_n a_n T^n \in L[[T]] everywhere convergent on B(0,1)
(i.e. \abs{a_n} r^n \to 0 for all r < 1); write \rp \subset L[[T]] for
these. A rigid analytic function is bounded if the \abs{a_n} are bounded.
The bounded rigid analytic functions form \OL[[T]] \otimes_{\OL} L, which by
Theorem 3.4.4 is \sM(\Zp, L): measures on \Zp are precisely
the bounded rigid analytic functions on B(0,1). It is natural to drop
boundedness, extending the correspondence to all of \rp.
A function f : \Zp \to L is locally analytic at z if there exist n_z \geq 0
and a_k(z) \in L with \sum_{k} a_k(z)(x-z)^k = f(x) for all
x \in z + p^{n_z}\Zp; it is locally analytic if so at every z \in \Zp. Write
\cCla(\Zp, L) for the L-vector space of such functions. Setting
\cC^{n\text{-an}}(\Zp, L) to be those with a uniform radius n_z = n — a Banach
space under \norm{f}_n = \sup_{z} \sup_k \abs{a_k(z)} p^{-nk} — one has
\cCla(\Zp, L) = \varinjlim_n \cC^{n\text{-an}}(\Zp, L) with the direct limit
topology.
Locally analytic functions are continuous, so \cCla(\Zp, L) \subset \cC(\Zp, L)
densely (locally constant functions are locally analytic), though the locally
analytic topology is finer than the induced one.
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The space \sDla(\Zp, L) of locally analytic distributions on \Zp is the
continuous dual \mathrm{Hom}_{\mathrm{cts}}(\cCla(\Zp, L), L), using
Definition 3.7.2. We write \int_{\Zp} \phi \cdot \mu := \mu(\phi)
and, since the binomial polynomials Definition 3.4.1 are locally
analytic, extend the Mahler transform by
\Am_\mu(T) = \int_{\Zp}(1+T)^x \cdot \mu = \sum_n (\int_{\Zp}\binomc{x}{n}\cdot\mu) T^n \in L[[T]].
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The Mahler transform induces a bijection
\sDla(\Zp, L) \xrightarrow{\ \sim\ } \rp \subset L[[T]],
an isomorphism of Fréchet spaces. This extends Theorem 3.4.4 and
rests on Definition 3.7.3 and Definition 3.7.1.
This is a theorem of Amice–Colmez. Both sides are inverse limits of Banach spaces.
On the function side, \cCla = \varinjlim_n \cC^{n\text{-an}}, so
\sDla = \varprojlim_n \sD^{n\text{-an}} with each \sD^{n\text{-an}} a Banach
space under the strong dual topology. On the power-series side, B(0,1) is the
increasing union of the closed discs B(0,r), r < 1, so
\rp = \varprojlim_{r < 1} \cO(B(0,r)) is an inverse limit of Banach spaces of
analytic functions; both carry Fréchet topologies. The growth condition
\abs{a_n} r^n \to 0 defining \rp is exactly dual to the radius-p^{-n}
analyticity norms on \cC^{n\text{-an}}, so the term-by-term Mahler map is a
topological isomorphism at each level and hence in the limit.
Restricting a measure \mu \in \sM(\Zp, L) to \cCla(\Zp, L) gives a locally
analytic distribution \widetilde\mu; by density of \cCla this is injective, so
\sM(\Zp, L) \subset \sDla(\Zp, L). Comparing Theorem 3.4.4 with
Theorem 3.7.4, this matches the inclusion of bounded power series
\OL[[T]] \otimes_{\OL} L \subset \rp. Every operation of the toolbox carries over
verbatim to distributions.
There is a multiplicative analogue for \Zpx. The weight space is
\cW(\Cp) = \mathrm{Hom}_{\mathrm{cts}}(\Zpx, \Cp^\times); using
\Zpx \cong \mu_{p-1} \times (1 + p\Zp), evaluation at a topological generator of
1 + p\Zp identifies it with p - 1 copies of B(0,1),
\cW(\Cp) = \bigsqcup_{\nu \in (\mu_{p-1})^\vee} U_\nu. A measure
\mu on \Zpx is the bounded rigid analytic function
F_\mu(\chi) = \int_{\Zpx} \chi \cdot \mu on \cW, multiplicative convolution
becoming pointwise multiplication; a pseudo-measure Definition 3.6.1 is then
a rigid function on \cW with at worst a simple pole at the trivial character.
The decomposition \Zpx \cong \mu_{p-1} \times (1+p\Zp) splits a continuous
character into its restriction to the torsion part \mu_{p-1} (one of p-1
characters \nu) and to the pro-p part 1+p\Zp; evaluating the latter at a
fixed topological generator lands in B(0,1), giving the disjoint-union
description. The pairing \chi \mapsto \int_{\Zpx} \chi \cdot \mu is, by the
multiplicative analogue of Theorem 3.4.4 (Amice), a bounded rigid
function on \cW, and convolution dualises to multiplication. For a pseudo-measure
\lambda = \mu/([a]-[1]) with a a topological generator,
\int_{\Zpx} \chi \cdot ([a]-[1]) = \chi(a) - 1 vanishes only when \chi(a) = 1,
i.e. for the trivial character; hence \lambda is analytic away from the trivial
character, where it may have a simple pole, by Proposition 3.6.6.