Jump set (awaiting review)

Let $K$ be a $p$-adic field which contains a primitive $p$th root of unity $\zeta_p$. Set $e=v_K(p)$ and $r=v_p(e)$. The jump set associated to $K$ is an increasing sequence of $r+1$ positive integers which determines the structure of the 1-units of $K$ as a filtered $\mathbb{Z}_p$-module. If $K$ does not contain a primitive $p$th root of unity then the jump set of $K$ is not defined.

More precisely, if $\zeta_p \in K$ then Miki [10.1515/crll.1981.328.99] showed that we can write \[\zeta_p=\alpha_0^{p^r}\alpha_1^{p^{r-1}}\dots \alpha_{r-1}^p\alpha_r\] with each $\alpha_i$ satisfying one of the following conditions:

• $v_K(\alpha_i)<pe/(p-1)$ and $p\nmid v_K(\alpha_i)$,
• $v_K(\alpha_i)=pe/(p-1)$ and $\alpha_i$ is not a $p$th power,
• $\alpha_i=1$.

Pagano [10.1016/j.jalgebra.2021.10.038] defines the (extended) jump set associated to $K$ by setting $u_i=v_K(\alpha_i-1)$ for all $1\le i\le r$ such that $\alpha_i\not=1$; these values are independent of the choice of $\alpha_0,\dots,\alpha_r$. Undefined values in the sequence $[u_1,u_2,\dots,u_{r+1}]$ are filled in using the recursion $u_{i+1}=\rho(u_i)$, where $\rho(x)=\min\{px,x+e\}$.