Degeneration of the Hilbert pairing in formal groups over local fields

Abstract

For an arbitrary local field K (a finite extension of the field Q _p ) and an arbitrary formal group law F over K, we consider an analog c _F of the classical Hilbert pairing. A theorem by S.V. Vostokov and I.B. Fesenko says that if the pairing c _F has a certain fundamental symbol property for all Lubin–Tate formal groups, then c _F = 0. We generalize the theorem of Vostokov–Fesenko to a wider class of formal groups. Our first result concerns formal groups that are defined over the ring O _K of integers of K and have a fixed ring O ₀ of endomorphisms, where O ₀ is a subring of O _K . We prove that if the symbol c _F has the above-mentioned symbol property, then c _F = 0. Our second result strengthens the first one in the case of Honda formal groups. The paper consists of three sections. After a short introduction in Section 1, we recall basic definitions and facts concerning formal group laws in Section 2. In Section 3, we state and prove two main results of the paper (Theorems 1 and 2). Refs. 8.