Primary elements in formal modules

Abstract

In the arithmetic of formal modules constructed on ideals of complete discrete valuation fields by using formal group laws, an essential role is played by the so-called primary elements, which give unramified extensions. In this paper, we obtain primary elements in an arbitrary formal module over a local field with perfect residue field. As formal group laws we consider so-called formal O0-module groups. In Section 1, we construct Hasse-type primary elements, whose definition involves an element of the integer ring of the completion of the maximal unramified extension of the field under consideration. In Section 2, we construct primary elements similar to those first appeared in [1, 2], whose definition involves only elements of the initial field. For this purpose, we expose Honda’s theory for formal O0modules and prove a theorem on the decomposition of the logarithm of a formal O0-module into a product of two factors on the basis of this theory. Then, we construct Artin–Hasse functions for universal formal O0-modules and, finally, prove the main theorems.