Isogeny classes of formal groups over complete discrete valuation fields with arbitrary residue fields

Abstract:

An explicit construction is described for computing representatives in each isogeny class of one-dimensional formal groups over the ring of integers of a complete discrete valuation field of characteristic $0$ with residue field of characteristic $p$. The logarithms of representatives are written out explicitly, and the number of nonisomorphic representatives of the form described in each isogeny class is computed. This result extends and generalizes the result obtained by Laffaile in the case of an algebraically closed residue field. The homomorphisms between the representatives constructed are described completely. The results obtained are applied to computation of the Newton polygon and the “fractional part” of the logarithm for an arbitrary one-dimensional formal group. Moreover, the valuations and the “residues” of the torsion elements of the formal module are calculated. A certain valuation of logarithms of formal groups is introduced and the equivalence of two definitions of the valuation is proved. One of these definitions is in terms of the valuations of the coefficients, and the other is in terms of the valuations of the roots of the logarithm (i.e., of the torsion elements of the formal module). This valuation only depends on the isomorphism class of a formal group, is nonpositive, and equals zero if and only if the formal group in question is isomorphic to one of the representatives considered.

The classification results of M. V. Bondarko and S. V. Vostokov on formal groups are employed, including invariant Cartier–Dieudonné modules and the fractional part invariant for the logarithm of a formal group.