Arithmetic of hyperbolic formal modules

Abstract

The paper considers hyperbolic formal groups which come out from the elliptic curve theory from the point of view of the theory of formal modules. The first part of the paper considers characteristics of hyperbolic formal groups, certain formulas for the formal logarithm and exponent are given, and convergence is scrutinized. Part two gives us the definition for p-typical logarithm; isogeny, its height and kernel are found. Further on Artin—Hasse function and Vostokov function were constructed, both correctness and their main features were checked. Using these constructions primary elements in this kind of formal modules can be easily built, which are useful for constructing the Shafarevich basis. As a natural way of evolution, pairing on the hyperbolic formal modules can be explored. To sum up, this is the very first step of deriving explicit formula for Hilbert pairing on these kind of formal modules. The research is to be continued in this particular direction. Refs 4.