Galois equivariant functions on Galois orbits in large $p$-adic fields

Abstract

Given a prime number p let \mathbb{C}_p be the topological completion of the algebraic closure of the field of p -adic numbers. Let O(T) be the Galois orbit of a transcendental element T of \mathbb{C}_p with respect to the absolute Galois group. Our aim is to study the class of Galois equivariant functions defined on O(T) with values in \mathbb{C}_p . We show that each function from this class is continuous and we characterize the class of Lipschitz functions, respectively the class of differentiable functions, with respect to a new orthonormal basis. Then we discuss some aspects related to analytic continuation for the functions of this class.