On a question of Goss

TL;DR

This paper proves that the group of locally analytic endomorphisms on the 1-units of a locally compact field of characteristic p>0 is isomorphic to the p-adic integers, answering a question posed by Goss.

Contribution

It establishes an isomorphism between locally analytic endomorphisms and p-adic integers, clarifying the structure of these endomorphisms in characteristic p>0.

Findings

The group of locally analytic endomorphisms is isomorphic to the p-adic integers.

Provides a complete characterization of these endomorphisms.

Answers a previously open question by Goss.

Abstract

In this note we answer the question raised by D. Goss in [Applications of non-Archimedean integration to the $L$-series of $\tau$-sheaves, {\em J. Number Theory,} 110 (2005), no. 1, 83--113] by proving that the group of locally analytic endomorphisms on the 1-units of a locally compact field of characteristic $p>0$ is isomorphic to the $p$-adic integers.