I am currently working on level issues in the division central simple algebra case, say $D$ over a local non-archimedean field $F$ (e.g. $\mathbf{Q}_p$). Let say that $\mathcal{O}_D$ and $\mathcal{O}_F$ are the maximal orders of $D$ and $F$, with uniformizers $P_D$ and $P_F = P_D^d$. If $I_D$ is the unique maximal ideal of $\mathcal{O}_D$, which is a valuation ring, then the uniformizer $P_D$ is defined as a generator (minimal valuation element) of $I_D$.

Any conductor $\chi$ of $D^\times$ can be written $\chi = \chi_F \circ n$ where $\chi_F$ is a character of $F^\times$ and $n$ the reduced norm on $D$. I would like to verify some well-known relations between the level of $\chi = \chi_F \circ n$ character of $D^\times$, and the the level of $\chi_F$.

Because of the fact that $\chi(1+x) = \chi_F(1+tr(x)+n(x))$, it seems sufficient to know, for $x \in P_D^k$, where lies its trace and norm. Some answers to those -- probably naive -- questions would be of great help :

The trace is surjective from $\mathcal{O}_D$ onto $\mathcal{O}_F$. Could we say that it goes from $P_D$ to $P_F$, $P_D^2$ to $P_F$, ... $P_D^{d-1}$ to $P_F$, and then $P_D^d$ to $P_F^2$, etc. ? More generaly, do we have $Tr : P_D^k \to P_F^{[k/d]}$ ? And is there surjectivity ?

Similarly, what do we know about the norm ? It seems plausible that the $P_F$-valuation of the norm of $x$ is always greater than the valuation of its trace : is it always true ?

Could we say anything precise about the ramification index $d$ ? For instance in the case of quaternion algebras, there is only one such division algebra $D$, and a unique maximal order $\mathcal{O}_D$ : is $d$ always 2 ?

I am quite not at ease with the effective computations and behaviors of traces and norms in those cases : any idea on one of those questions, or any reference, would be very helpful !

Best regards.

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