Artin-Hasse functions and their invertions in local fields (Q612996)

The Artin-Hasse functions are a convenient tool for describing the arithmetic of the multiplicative group of a local field \(K\) of characteristic zero. These functions were used by \textit{J. Lubin} and \textit{J. Tate} [Ann. Math. (2) 81, 380--387 (1965; Zbl 0128.26501)] to describe explicitly the Hilbert symbol \((\alpha,\beta)\). \textit{S. V. Vostokov} [Math. USSR, Izv. 13, 557--588 (1979; Zbl 0467.12018)] generalized Artin-Hasse functions in the multiplicative case using the Frobenius operator \(\Delta: {\mathcal O}[[X]]\to {\mathcal O}[[X]]\), \(\Delta(\sum a_iX^i)=\sum \sigma(a_i)X^{pi}\), where \({\mathcal O}\) is the ring of integers of the inertia subfield \(N\) of \(K\) and \(\sigma\) is the Frobenius map on \(N\), to the function \[ E_{\Delta}(f(x))=\exp\big(1+\frac{\Delta}{p}+\frac{\Delta^2}{p^2} +\cdots\big)(f(x)). \] The map \(E_{\Delta}\) induces a homomorphism from the additive group \(X{\mathcal O}_K[[X]]\) to the multiplicative group of \(X{\mathcal O}_K[[X]]\) where \({\mathcal O}_K\) is the ring of integers of \(K\). The Artin-Hasse for Honda formal groups were defined by \textit{M. V. Bandarko} and \textit{S. V. Vostokov} [Proc. Steklov Inst. Math. 241, 35--57 (2003); translation from Tr. Mat. Inst. Im. V. A. Steklova 241, 43--67 (2003; Zbl 1125.11357)]. In the present paper the authors extend Artin-Hasse functions and the inverse maps to formal modules in an arbitrary local field.