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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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  • $\begingroup$ Can you slow down to explain your question clearly from the beginning? There are too many things undefined, for one. E.g., what are $F$, $\chi_F$, $n$? (The latter 2 can be easily guessed, but your assumptions on $F$ are somewhat ambiguous.) More crucially, how are you defining $P_D$ and the conductor of $\chi$? $\endgroup$
    – Kimball
    Commented Mar 28, 2015 at 15:09
  • $\begingroup$ @Kimball Sorry for so many rush in my explanations, I will edit the original post in order to be more explicite. Here are the details : $F$ is a local field, let's say $\mathbf{Q}_p$. 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 $D$. If $I_D$ is the unique maximal ideal of $\mathcal{O}_D$, the integer (valuation) ring of $D$, then the uniformizer $P_D$ is defined as a generator (minimal valuation element) of $I_D$. Let's forget the conductors, and say I am only talking about levels ;) $\endgroup$
    – Desiderius Severus
    Commented Mar 29, 2015 at 16:19
  • $\begingroup$ @Kimball Maybe should I delete my post and split it more independent and precise questions, in order to be more accessible and swiftly readable ? $\endgroup$
    – Desiderius Severus
    Commented Mar 29, 2015 at 16:25
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    $\begingroup$ I think your definition of $P_D$, $P_F$ and $d$ still has issues. I suppose it may be that no nontrivial power of $P_D$ lies in $F$, so $P_F = P_D^d$ is impossible. And do you know that $d$ does not depend on the choice of $P_D$? If so, please say why. Then I think I can probably answer at least 2 of the questions. $\endgroup$
    – Kimball
    Commented Mar 29, 2015 at 17:17
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    $\begingroup$ One thing that was confusing me was that you define $P_F$ and $P_D$ as uniformizers (elements), then seem to use them to mean ideals later in your question. If you mean ideals, I think $d$ is well-defined (if you write $P_F \mathcal O_D = P_D^d$), and you should be able to determine it using ideal norms. $\endgroup$
    – Kimball
    Commented Mar 31, 2015 at 1:50

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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 $m$.
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  • $\begingroup$ Thanks for those details, it indeed seems a valuable point of view to construct central simple algebras like that, it gives more explicit presentation and elements, computation rules, etc. I will soon be checking in the details (for instance in the case of quaternion algebras). For now, do you have any good reference for this kind of computations on central simple algebras, for instance the fact that is always cyclic of this form, etc. ? $\endgroup$
    – Desiderius Severus
    Commented Mar 30, 2015 at 23:21
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    $\begingroup$ @Dydo That CSAs over local fields are always cyclic is well known and should be in any standard reference on CSAs (see mathoverflow.net/a/198511/6518). Certainly in Reiner or Pierce. I don't know that any of these books do much calculations with cyclic algebras though. But I think once you play around with the presentation, it's not too hard to compute things on your own. $\endgroup$
    – Kimball
    Commented Mar 31, 2015 at 1:43

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