Classification of formal \(A\)-modules in the case of small ramification
Let \(K_0\) be a local field with residue field of finite order \(q\) and \(A\) the ring of integers of \(K_0\). Let \(K\) be a finite extension of \(K_0\) with residue field of characteristic \(p>2\) and \(\mathcal{O}_K\) be its ring of integers. Let \(\pi_0\) (resp. \(\pi\)) be a prime element of \(A\) (resp. \(\mathcal{O}_K\)). Let \(T\) denote the inertia subfield in \(K/K_0\) and \(\mathcal{O}_T\) be its ring of integers. Let \(\mathcal{O}_T\Delta\) be the noncommutative ring of power series in which the multiplication is given by \(\Delta a = \sigma(a) \Delta\) and let \(\Delta\) be the operator that acts on the series by the rule \(\Delta(cX^k) = cX^{qk}\). In this paper, an \(A\)-module is an \(m\)-dimensional formal group law \(F\) over \(\mathcal{O}_K\) such that the embedding \([\cdot]:A \to \text{End}_{\mathcal{O}_K}F\) satisfies \([a](X) \equiv aI_m\) modulo degree \(2\) terms. Let \(\lambda\) be the logarithm of the formal group \(F\). Every formal \(A\)-module over an \(A\)-algebra \(B\) is strictly isomorphic over \(B\) as a formal \(A\)-module to an \(A\)-typical formal \(A\)-module. Since the homomorphism \(B \to B \otimes_A K\) is injective, the logarithm of a formal \(A\)-module has the form \(\lambda(X) = \Lambda(\Delta)(X)\). The main result of this paper is to show that \(\lambda(X) = \Lambda(\Delta)(X)\) is the logarithm of an \(A\)-typical formal \(A\)-module over \(\mathcal{O}_K\) if and only if \(\lambda = vu^{-1}\) where \(u \in \mathcal{O}_T\Delta\), \(u \equiv \pi_0\) mod \(\Delta\), and \(v \in \pi_0+\pi\mathcal{O}_K\Delta\Delta\). Any formal \(A\)-module over \(\mathcal{O}_K\) is isomorphic to an \(A\)-typical one with such logarithm. The authors also give criteria for testing when two \(A\)-typical formal \(A\)-modules are strictly isomorphic in terms of conditions on logarithms.
scientific article; zbMATH DE number 6676844