Joint IAS/Princeton University Number Theory Seminar

We explain the magic of the entire function $\Psi_p \in \mathbb{Z}[x] \cap \mathbb{Q}_p\{x\}$ defined by the functional equation \[x = \sum_{j=0}^{\infty} p^{-j}\Psi_p (p^{j} x)^{p^{j}}\] which satisfies $\Psi_p (\mathbb{Q}_p) \subset \mathbb{Z}_p$ and admits an integral addition law. We then use the Artin-Hasse exponential \[F(T) = \exp \sum_{i=0}^\infty T^{p^i}/p^i \in \mathbb{Z}_{(p)}[[T]]\] to deduce from $\Psi_p$ a topological basis $\{ G_q \}_{q \in S}$, where $S = \mathbb{Z}[1/p] \cap \mathbb{R}_{\geq 0}$, of the Fréchet space of continuous functions $\mathbb{Q}_p \to \mathbb{Q}_p$, which consists of entire functions $G_q: \mathbb{C}_p \to \mathbb{C}_p$ defined over $\mathbb{Z}_{(p)}$ taking $p$-adic integral values all over $\mathbb{Q}_p$. They satisfy $G_0(x) = 1$, \[G_q(x+y) = \sum_{q_1 +q_2 = q}G_{q_1}(x)G_{q_2}(y)\;\;\mbox{and}\;\; G_{pq}(px)= G_q(x)\;,\;\;\forall q \in S \;.\] The convergence of the previous sum is uniform on compact subsets of $\mathbb{Q}_p$ along the filter of cofinite subsets of $S$. We identify the $(p,T)$-adic completion $\mathscr{D}$ of the ring $\mathbb{Z}_{(p)}[T^{1/p^\infty}]$ with a topological Hopf algebra of $\mathbb{Z}_p$-valued measures on the uniformly open subsets of $\mathbb{Q}_p$, equipped with the topology of uniform convergence on the families of balls of equal radius, in such a way that, for any $i\in \mathbb{Z}$, \[\lim_{n \to +\infty} F(T^{p^{i-n}})^{p^n} = \Delta_{p^i}\] is the Dirac mass at $p^i$. So, for the inverse series $E$ of $F$, let \[\mu_{\mathrm{can}} := \lim_{n \to +\infty} E(\Delta_{p^{-n}} - \Delta_0)^{p^n}\] be the $\mathbb{Z}_p$-valued measure on $\mathbb{Q}_p$ corresponding to $T$. Then, for any $q \in S$, the measure $\mu_{\mathrm{can}}^q$ exists in $\mathscr{D}$ and any continuous function $f: \mathbb{Q}_p \to \mathbb{Q}_p$ admits the following generalized Amice-Fourier expansion \[f (-) = \sum_{q \in S} ( \int_{\mathbb{Q}_p} f \, \mu_{\mathrm{can}}^q ) G_q(-) \;,\] where the series converges as above.