Rank one solvable p-adic differential equations and finite Abelian characters via Lubin–Tate groups

Abstract

We introduce a new class of exponentials of Artin–Hasse type, called π-exponentials. These exponentials depend on the choice of a generator π of the Tate module of a Lubin–Tate group \(\mathfrak{G}\) over \(\mathbb{Z}_p\). They arise naturally as solutions of solvable differential modules over the Robba ring. If \(\mathfrak{G}\) is isomorphic to \(\widehat{\mathbb{G}}_m\) over \(\mathbb{Z}_p\), we develop methods to test their over-convergence, and get in this way a stronger version of the Frobenius structure theorem for differential equations. We define a natural transformation of the Artin–Schreier complex into the Kummer complex. This provides an explicit generator of the Kummer unramified extension of \(\epsilon^\dagger_{K_{\infty}}\), whose residue field is a given Artin–Schreier extension of \(k(\!(t)\!)\), where k is the residue field of K. We then compute explicitly the group, under tensor product, of isomorphism classes of rank one solvable differential equations. Moreover, we get a canonical way to compute the rank one φ-module over \(\epsilon^\dagger_{K_\infty}\) attached to a rank one representation of \(\mathrm{Gal}(k(\!(t)\!)^{\mathrm{sep}}/k(\!(t)\!))\), defined by an Artin–Schreier character.