convergent series representation for p-adic complex numbers

The field $\mathbb{C}_p$ of $p$-adic complex numbers is the completion of the algebraic closure of $\mathbb{Q}_p$ with the corresponding extension of the usual non-Archimedean valuation $|\;\;|_p$.

Every $x\in\mathbb{Q}_p$ has a unique representation of the form $\sum_{i=m}^\infty a_ip^i$, where $m\in\mathbb{Z}$ and the $a_i$'s are representatives of the classes in $\mathbb{Z}/p\mathbb{Z}$. Does $y\in\mathbb{C}_p$ have a similar representation as a generalized power series? Any reference?

• Not only does every element of ${\bf Q}_p$ have such a representation, but every such series represents an element of ${\bf Q}_p$. So there's no room left for non-elements of ${\bf Q}_p$ to be represented as convergent power series in $p$. – Gerry Myerson Nov 17 '16 at 1:33
• but what about rational exponents? – Chilote Nov 17 '16 at 1:34
• Then you're talking about something other than a power series. – Gerry Myerson Nov 17 '16 at 1:35
• There is a paper by David Lampert related to this: deepblue.lib.umich.edu/bitstream/handle/2027.42/26390/… – KConrad Nov 17 '16 at 2:45
• @KConrad, in the spirit of preventing link rot, I mention that your paper is Lampert, Algebraic $p$-adic expansions, J. Number Theory 23 (1986), no. 3, 279–284 (MR). – LSpice Nov 17 '16 at 17:14

The blog post http://sbseminar.wordpress.com/2007/08/21/p-adic-fields-for-beginners gives a good overview of the situation.

To briefly summarize (extracted from the above post — any errors are probably mine):

• The elements of $\mathbb{Q}_p$ are exactly those represented by series of the form $\sum_{r \in S} a_r p^r$, where $S$ is a bounded-below subset of the integers and each $a_r$ is a multiplicative lift (i.e., a "Teichmüller lift") of an element of $\mathbb{F}_p$.
• The elements of $\mathbb{Q}_p^{nr}$ (the maximal unramified extension of $\mathbb{Q}_p$) are exactly represented by series of the form $\sum_{r \in S} a_r p^r$, where $S$ is again a bounded-below subset of the integers, but the $a_r$ can now be lifts of any element of $\mathbb{F}_q$, where $q$ is a fixed power of $p$. (As explained in the comments below, the $a_r$ cannot be lifts of any elements of $\bar{\mathbb{F}}_p$; they must all be contained in a common finite extension of $\mathbb{F}_p$. This constraint is removed in the completion of $\mathbb{Q}_p^{nr}$.)
• The elements of $\bar{\mathbb{Q}}_p$ and its completion $\mathbb{C}_p$ are represented by certain series of the form $\sum_{r \in S} a_r p^r$, where the $a_r$ are again lifts of elements of $\bar{\mathbb{F}}_p$, but now $S$ can be a more general well-ordered subset of $\mathbb{Q}$. Which such series give rise to elements of $\mathbb{C}_p$ is discussed in Kedlaya, Power series and $p$-adic algebraic closures (MR).
• The elements of $\Omega_p$, the spherical completion of $\mathbb{C}_p$, are represented exactly by series of the form $\sum_{r \in S} a_r p^r$, where the $a_r$ are as before, but now $S$ can be any well-ordered subset of $\mathbb{Q}$ (with no other restrictions). This is attributed to the undergraduate thesis of Bjorn Poonen, written up in Poonen, Maximally complete fields (MR).

And this is in some sense as far as we can go, since $\Omega_p$ is "maximally complete": we can't extend it any further without "adding geometry", in the sense that it's the unique largest field of characteristic zero with value group $\mathbb{Q}$ and residue field the algebraic closure of $\mathbb{F}_p$.

• I think that your description of $\mathbb Q_p^{\text{nr}}$ is not quite right (I haven't checked the rest). For each number individually, the coefficients $a_r$ should probably all come from some fixed finite extension of $\mathbb F_p$, i.e., the field $\mathbb F_p[a_r : r \in S]$ should be a finite extension of $\mathbb F_p$ (although, of course, the extension can be arbitrarily large as we consider different elements of $\mathbb Q_p^{\text{nr}}$). – LSpice Nov 17 '16 at 17:20
• @LSpice that's right. The ring $\mathbf Z_p^{\rm nr}$ is not the same thing as the Witt vectors of $\overline{\mathbf F}_p$: the former require coefficients for each element to come from a finite set of possibilities (depending on the element, as you say), while in the latter there is no such restriction. For more or less the same reason, $\mathbf Z_p^{\rm nr}$ is not complete and its completion is the Witt vectors of $\overline{\mathbf F}_p$. – KConrad Nov 17 '16 at 19:09
• Thanks for the correction; I fixed the description of $\mathbb{Q}_p^{nr}$. – Daniel Hast Nov 18 '16 at 0:23