Resumen de Analytic difference rings

Thomas Scanlon

• 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� Er�ov 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 alg p ], 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.