Gauss-Manin connection and local fields

After a reminder on Gauss-Manin connections and higher local fields we will explain how higher-dimensional residues of differential forms and higher- dimensional decompositions of rational functions into series lead to solutions of Picard-Fuchs equations, even in the absence of the equation in an explicit form. As an application we will discuss a so-called main period formula, which is useful in mirror symmetry. The talk is based on a joint work with V. V. Przyjalkowski.