A $p$-adically entire function with integral values on ${\mathbb Q}_p$ and entire liftings of the $p$-divisible group ${\mathbb Q}_p/{\mathbb Z}_p$

We give a self-contained proof of the fact that, for any prime number $p$, there exists a power series $$\Psi= \Psi_p(T) \in T + T^2\Z[[T]]$$ which trivializes the addition law of the formal group of Witt covectors is $p$-adically entire and assumes values in $\Z_p$ all over $\Q_p$. We actually generalize, following a suggestion of M. Candilera, the previous facts to any fixed unramified extension $\Q_q$ of $\Q_p$ of degree $f$, where $q = p^f$. We show that $\Psi = \Psi_q$ provides a quasi-finite covering of the Berkovich affine line $\A^1_{\Q_p}$ by itself. We prove in section 3 new strong estimates for the growth of $\Psi$, in view of the application to $p$-adic Fourier expansions on $\Q_p$. We locate the zeros of $\Psi$ and to obtain its product expansion. We reconcile the present discussion (for $q =p$) with a previous formal group proof which takes place in the Fr\'echet algebra $\Q_p\{x\}$ of the analytic additive group $\G_{a,\Q_p}$ over $\Q_p$. We show that, for any $\lambda \in \Q_p^\times$, the closure $\sE_\lambda^\circ$ of $\Z_p[\Psi(p^ix/\lambda)\,|\,i=0,1,\dots]$ in $\Q_p\{x\}$ is a Hopf algebra object in the category of Fr\'echet $\Z_p$-algebras. The special fiber of $\sE_\lambda^\circ$ is the affine algebra of the $p$-divisible group $\Q_p/p \lambda \Z_p$ over $\F_p$, while $\sE_\lambda^\circ [1/p]$ is dense in $\Q_p\{x\}$. From $\Z_p[\Psi(\lambda x)\,|\,\lambda \in \Q_p^\times]$ we construct a $p$-adic analog $\AP_{\Q_p}(\Sigma_\rho)$ of the algebra of Dirichlet series holomorphic in a strip $(-\rho, \rho) \times i \R \subset \C$. We start developing this analogy. It turns out that the Banach algebra of almost periodic functions on $\Q_p$ identifies with the topological ring of germs of holomorphic almost periodic functions on strips around $\Q_p$.