Formal group exponentials and Galois modules in Lubin-Tate extensions

TL;DR

This paper extends the theory of formal group exponentials and Lubin-Tate extensions to provide explicit descriptions of Galois module generators in wildly ramified p-adic extensions, advancing understanding of their algebraic structure.

Contribution

It generalizes Pulita's power series and Pickett's constructions using formal group exponentials and Lubin-Tate theory to explicitly describe Galois module generators in all abelian wildly ramified extensions.

Findings

Explicit Galois module generators for inverse different in all abelian wildly ramified extensions.

Generalized Pulita's power series to Lubin-Tate extensions.

Provided new analytic representations of normal basis generators.

Abstract

Explicit descriptions of local integral Galois module generators in certain extensions of $p$-adic fields due to Pickett have recently been used to make progress with open questions on integral Galois module structure in wildly ramified extensions of number fields. In parallel, Pulita has generalised the theory of Dwork's power series to a set of power series with coefficients in Lubin-Tate extensions of $\Q_p$ to establish a structure theorem for rank one solvable p-adic differential equations. In this paper we first generalise Pulita's power series using the theories of formal group exponentials and ramified Witt vectors. Using these results and Lubin-Tate theory, we then generalise Pickett's constructions in order to give an analytic representation of integral normal basis generators for the square root of the inverse different in all abelian totally, weakly and wildly ramified extensions of a p-adic field. Other applications are also exposed.