Analytic difference rings

Abstract

Generalizing and synthesizing earlier work on the model theory of valued difference fields and on the model theory of valued fields with analytic structure, we prove Ax–Kochen– Eršov style relative completeness and relative quantifier elimination theorems for a theory of valuation rings with analytic and difference structure. Specializing our results to the case of W[F_p_alg], the ring of Witt vectors of the algebraic closure of the field with p elements, given together with the relative Frobenius and the Tate algebras as analytic structure, we develop a model theoretic account of Buium’s p-differential functions. In so doing, we derive a uniform p-adic version of the Manin–Mumford conjecture.