Formal groups over sub-rings of the ring of integers of a multidimensional local field

Abstract

In the work we construct so-called convergence rings for the ring of integers of a multidi-mensional local field. The convergence ring is a sub-ring of the ring of integers having the property that any power series with coefficients from the sub-ring converges when replacing a variable on an arbitrary element of the maximum ideal. The properties of convergence rings and an explicit formula for their construction are derived. Note that the multidi-mensional case is fundamentally different from the case of the classical (one-dimensional) local field, where the convergence ring is the whole ring of integers. Next, we consider a multidimensional local field with zero characteristic of the penultimate residue field. For each convergence ring of such a field, we introduce a homomorphism that allows for a power series with coefficients from the ring to construct a formal group over the same ring with a logarithm having coefficients from the field, and we give an explicit formula for the coefficients. In addition, we constructe a generalization of the formal Lubin-Tate group over this ring, study the endomorphisms of these formal groups and derive a criterion for their isomorphism. We prove a one-to-one correspondence between formal groups created by ring homomorphism and by isogeny. Also for any finite extension of a multidimensional local field with zero characteristic of the penultimate residue field, we considere the point group generated by the corresponding Lubin-Tate formal group.