The arithmetic of hyperbolic formal modules

Abstract

This paper considers hyperbolic formal groups, which come from the elliptic curve theory, in the context of the theory of formal modules. In the first part of the paper, the characteristics of hyperbolic formal groups are considered, i.e., the explicit formulas for the formal logarithm and exponent; their convergence is studied. In the second part, the isogeny and its kernel and height are found; a p-typical logarithm is defined. The Artin–Hasse and Vostokov functions are then constructed; their correctness and main properties are evaluated.