Logarithms of formal $A$-modules in the case of small ramification

Abstract:

Formal $\mathcal {O}_0$-modules over the ring of integers $\mathcal {O}$ of a local field, i.e., formal groups over $\mathcal {O}$ with endomorphism ring including a fixed ring $\mathcal {O}_0$ are studied. A complete description of the logarithms of all such modules is obtained in the case of small ramification. Earlier it was shown that in the case of small ramification ($e(\mathcal {O}/\mathcal {O}_0)<q$), any $\mathcal {O}_0$-module is strictly isomorphic to an $\mathcal {O}_0$-module the logarithm of which can be represented in the form $vu^{-1}(X)$, where $u$ and $v$ are certain matrices over the ring of operators described in the paper. The result obtained in the present paper enables one to determine the type ($u$ and $v$) of a formal $\mathcal {O}_0$-module by the form of its logarithm, and provides a way for constructing all formal $\mathcal {O}_0$-modules.