Winding quotients and some variants of Fermat's Last Theorem.
Let be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian -adic Lie extensions , where is a local field with residue field , and a category whose objects are pairs , where and is an abelian -adic Lie subgroup of . In this paper we extend this equivalence to allow and to be arbitrary abelian pro- groups.
We prove an inequality linking the growth of a generalized Wronskian of m p-adic power series to the growth of the ordinary Wronskian of these m power series. A consequence is that if the Wronskian of m entire p-adic functions is a non-zero polynomial, then all these functions are polynomials. As an application, we prove that if a linear differential equation with coefficients in ℂₚ[x] has a complete system of solutions meromorphic in all ℂₚ, then all the solutions of the differential equation are...