A p-adically entire function with integral values on Qp and entire liftings of the p-divisible group Qp/Zp

We give a self-contained proof of the fact, discovered in [1] and proven in [2] with the methods of [6], that, for any prime number p, there exists a power seriesPsi = Psi(p) (T) is an element of T + T-2 Z[[T]],which trivializes the addition law of the formal group of Witt covectors [6], [15, II.4], 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(Qp)(1) by itself. We prove in Section 4 new strong estimates for the growth of 'If, in view of the application [3] to p-adic Fourier expansions on Q(p). We refer to [3] for the proof of a technical corollary (Proposition 4.13) which we apply here to locate the zeros of Psi and to obtain its product expansion (Corollary 4.16). We reconcile the present discussion (for q = p) with the formal group proof given in [2], which takes place in the Frechet algebra Q(p) {x} of the analytic additive group G(a), Q(p) over Q(p). We show that, for any lambda is an element of Q(p)(X) , the closure epsilon(lambda)degrees: of Z(p) [Psi(p(i)x/lambda) vertical bar i = 0, 1, ...] in Q(p) {x} is a Hopf algebra object in the category of Frechet Z(p)-algebras. The special fiber of epsilon(lambda)degrees is the affine algebra of the p-divisible group Q(p)/p lambda Z(p) over F-p, while epsilon(lambda)degrees[1/p] is dense in Q(p) {x}. From Z(p) [Psi(lambda x) vertical bar lambda is an element of Q(p)(x) ] we construct a p-adic analog AP(Qp) (Sigma(rho)) of the algebra of Dirichlet series holomorphic in a strip (-rho, rho) x iR subset of 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).