Analytic morphisms of t -motives

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Let K be a complete valued field, containing the finite field 𝔽q. For abelian t-modules E and E' defined over K, we define an analytic morphism to be a rigid analytic entire homomorphism of the underlying groups which respect the action of t, and we denote the group of such analytic morphisms by Homan(E,E'). On the other hand, denoting the ring of entire functions on K in one variable t by K〈〈t〉〉, the group of analytic morphisms $$$$ between two τ-modules M and M' over K[t] is defined as the group of K〈〈t〉〉-linear homomorphism $$$$ commuting with the respective actions of τ. In this paper, we prove that for pure abelian t-modules E and E' with respective t-motives M(E) and M(E'), there exists a bijection $$$$ In other words, the categories of pure abelian t-modules and pure t-modules, both endowed with analytic morphisms, are antiequivalent. The most prominent instance of this phenomenon is given by the Tate uniformization map of Drinfeld modules at places of bad reduction, which, as we show, induces an interesting analytic description of the associated t-motive.