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.