Geometrization of continuous characters of $\mathbb{Z}_p^\times$

TL;DR

This paper introduces a $p$-adic trace for certain local systems on the multiplicative group over $p$-adic numbers, establishing an isomorphism with $ ext{l}$-adic characters of specific depth, unifying Kummer and Artin-Schrier-Witt theories.

Contribution

It defines a new $p$-adic trace linking local systems and $ ext{l}$-adic characters, extending the understanding of their structure and unifying existing theories.

Findings

The $p$-adic trace creates an isomorphism between local systems and $ ext{l}$-adic characters.

This unifies Kummer and Artin-Schrier-Witt theories in a $p$-adic context.

The isomorphism applies to local systems of order dividing $(p-1)p^n$.

Abstract

We define the $p$-adic trace of certain rank-one local systems on the multiplicative group over $p$-adic numbers, using Sekiguchi and Suwa's unification of Kummer and Artin-Schrier-Witt theories. Our main observation is that, for every non-negative integer $n$, the $p$-adic trace defines an isomorphism of abelian groups between local systems whose order divides $(p-1)p^n$ and $\ell$-adic characters of the multiplicative group of $p$-adic integers of depth less than or equal to $n$.