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

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