Let $K$ be $\mathbb{Q}_p$ for some prime $p$ (or more generally an unramified extension $W(\mathbb{F}_q)$ of $\mathbb{Q}_p$). If $\xi \in K$, we can write it in a unique way in the form $\sum a_i p^i$ where each $a_i$ is either zero (and must be such for all but finitely many $i<0$) or a root of unity (i.e., a $(q-1)$-th root of unity), collectively known as the Teichmüller representatives. It is now tempting, if $L$ is an arbitrary field containing the $(q-1)$-th roots of unity, to define a (Laurent) power series $f_\xi \in L((t))$ associated to $\xi$ by: $f_\xi = \sum a_i t^i$, assuming we have chosen an isomorphism between the $(q-1)$-th roots of unity in $K$ and those in $L$.

My question is basically whether anything intelligent can be said about $f_\xi$, say, when $\xi$ is a rational, apart from the obvious things like the fact that $f_\xi(p) = \xi$ if we take $L=K$ (with the identity map on roots of unity).

For definiteness, let me concentrate on the first non-trivial case (because already it seems very hard to say anything at all): take $p=5$ and $\xi=2$. Letting $\zeta$ be the $4$-th root of $1$ in $\mathbb{Q}_5$ that is congruent to $2$ mod $5$, we have

$$2 = \zeta - 1 \cdot 5 - \zeta \cdot 5^2 + 0 \cdot 5^3 - 1 \cdot 5^4 + \zeta \cdot 5^5 - 1 \cdot 5^6 - 1 \cdot 5^7 + 1 \cdot 5^8 + \zeta \cdot 5^9 + \zeta \cdot 5^{10} - 1 \cdot 5^{11} + 0 \cdot 5^{12} - \zeta \cdot 5^{13} - 1 \cdot 5^{14} + O(5^{15})$$

and I am asking about the formal series

$$2 + 4 \cdot t + 3 \cdot t^2 + 0 \cdot t^3 + 4 \cdot t^4 + 2 \cdot t^5 + 4 \cdot t^6 + 4 \cdot t^7 + 1 \cdot t^8 + 2 \cdot t^9 + 2 \cdot t^{10} + 4 \cdot t^{11} + 0 \cdot t^{12} + 3 \cdot t^{13} + 4 \cdot t^{14} + \cdots \in \mathbb{F}_5[[t]]$$

(the sequence of coefficients does not appear in the OEIS, which is mildly surprising) or about

$$i - 1 \cdot z - i \cdot z^2 + 0 \cdot z^3 - 1 \cdot z^4 + i \cdot z^5 - 1 \cdot z^6 - 1 \cdot z^7 + 1 \cdot z^8 + i \cdot z^9 + i \cdot z^{10} - 1 \cdot z^{11} + 0 \cdot z^{12} - i \cdot z^{13} - 1 \cdot z^{14} + \cdots \in \mathbb{C}[[z]]$$

Here are some examples of questions which I think are interesting:

  • is the former algebraic/automatic?

  • does the latter satisfy some nontrivial differential equation?

  • can it be extended holomorphically anywhere beyond the unit disk?

  • does it take interesting values at interesting points?

On a related line, we could consider the $5$-adic quantity

$$- \zeta - 1 \cdot 5 + \zeta \cdot 5^2 + 0 \cdot 5^3 - 1 \cdot 5^4 - \zeta \cdot 5^5 + \cdots$$

obtained by exchanging $\pm\zeta$ in the expansion of $2$ above: in other words, it is the value $f_2(5)$ where $f_2$ has been defined using the unique non-identity automorphism of the $4$-th roots of unity; can anything intelligent be said about that quantity? (e.g., is it rational? experimentally, it doesn't look like it is).

Edit: I should have recalled that, for $\xi$ rational, the sequence of coefficients can be computed from Witt polynomials. Namely, assuming $p$ does not divide the denominator of $\xi$, we let $A_0 = \xi$ and by induction $A_n = (\xi - \sum_{i=0}^{n-1} (A_i)^{p^{n-i}}\cdot p^i)/p^n$: then the $A_i$ are rationals and $p$-adic integers, and $a_i$ is (the root of unity which coincides with) $A_i$ mod $p$. Unfortunately, when seen as integers (if $\xi$ is an integer), the $A_n$ grow extremely rapidly in absolute value, and I don't see how that can be made useful.

  • $\begingroup$ Your use of the $\times$ symbol makes this very hard to read. $\endgroup$ – Lubin Jul 10 '14 at 13:21
  • $\begingroup$ @Lubin: You're right. I replaced $\times$ by $\cdot$ everywhere. $\endgroup$ – Gro-Tsen Jul 10 '14 at 15:29
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    $\begingroup$ If you take $K$ to be ramified and play the same game, then as you increase the ramification, your two fields are "more and more isomorphic". This observation of Krasner is the basis for the theory of the "field of norms" and more recently the theory of "perfectoid spaces". Here is what Fontaine says about this in his recent Bourbaki exposé... $\endgroup$ – Laurent Berger Jul 13 '14 at 9:13
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    $\begingroup$ ... "Quiconque s’est intéressé aux corps locaux sait bien qu’une extension très ramifiée du corps $Q_p$ des nombres $p$-adiques ressemble à s’y méprendre à un corps de séries formelles à coefficients dans son corps résiduel. C’est sans doute Marc Krasner qui a tenté le premier de formuler ce phénomène abondamment utilisé depuis en théorie de Hodge p-adique [...]" (Fontaine, Bourbaki 1057). $\endgroup$ – Laurent Berger Jul 13 '14 at 9:14

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