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.



On conductors, levels and traces on quaternion algebras
I think I can give you at least a sketch of answer now, but I have not thought carefully about details, so you should check though things carefully. 


Let $K$ be a cyclic extension of $F$ of degree $n$, and $\sigma$ a generator of the Galois group Gal($K/F$). Then we can write $D$ as a cyclic algebra $(K/F, \sigma, a)$ where $a \in F^\times$ has order $n$ modulo $N\_{K/F}(K^\times)$. This is a subalgebra of $M\_n(K)$ generated by elements of the form
$$ \iota(\alpha) = diag(\alpha, \alpha^\sigma, \ldots, \alpha^{\sigma^{n-1}}), \quad \alpha \in K$$
and some element $y$ such that 
$$ y^n = a, \quad \iota(\alpha)y = y\iota(\alpha)^\sigma \quad (\alpha \in K).$$


Then you can consider the order $\mathcal O\_D$ generated by $y$ and $\alpha \in \mathcal O\_K$. I think this should usually (always?) be a maximal order with maximal ideal $I\_D$ generated by $y$ and $\alpha \in \varpi\_K \mathcal O\_K$. (Here you may need $K/F$ is unramified and some assumption on residual/characteristic, but I'm not sure.) When this is the case, you should be able to take for $P\_D$ simply $\iota(\varpi\_K)$. 


* If $K/F$ is unramified, the answer to your 3rd question is that
$d$ should simply be $n$.
* For your first question, the trace of any element will be the trace
of some $\iota(\alpha)$, with $\alpha$ lying in an appropriate ideal
of $K$. Thus the image of the trace on $I\_D$ is simply the image of
trace to $\mathcal O\_F$ on $\varpi\_K \mathcal O\_K$. Probably you can work out an example to see the image is not in $\varpi\_F \mathcal O\_F$.
* For the second question, the reduced norm of an element in $I\_D$ will lie in $\varpi\_F \mathcal O\_F$. However, it is not true that the norm always has greater valuation than the trace. For example, for a quaternion algebra with $F=\mathbb Q\_p$ and $K=F(\delta)$ such that $\sigma(\delta) = -\delta$, consider $x=\iota(p^m+p\delta)$. Generically (avoiding special cases), the norm has valuation 2, but the trace has valuation $m$.


