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Let $K$ be a $p$-adic field with slope content $[s_1, \ldots, s_k]_t^u$. The top slope refers to the ramification group $G^{(s)}$ with the largest $s$. So,

  • if $k>0$, then the top slope is $s_k$, which is always greater than $1$
  • otherwise, if $t>1$, then the top slope is $1$
  • otherwise the top slope is $0$

This includes, by convention, that the top slope of $\Q_p$, as an extension of itself, is $0$.

Knowl status:
  • Review status: reviewed
  • Last edited by Andrew Sutherland on 2020-10-24 17:13:57
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