Totally ramified solutions to arithmetic differential equations

Arithmetic differential equations are analogues of differential equationsin which functions are replaced by numbers and derivative operators are replacedby Fermat quotient operators. The numbers appearing as solutions to such equations belong  a priori to absolutely unramified  extensions of the p-adic integers.
The talk will explain how some of the main examples of arithmetic differential equations appearing in the theory possess a certain remarkable overconvergence property; this property  allows one to consider solutions of such equations in infinitely ramified extensions of the p-adic integers. The talk is based on joint work with A.Saha and work in progress with L. Miller.