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Set $q=p^r$ for the finite field $\Bbb F_q$. In the formal power series ring $\Bbb F_q[[x]]$, there is a notion of convergence given by the underlying $(x)$-adic topology. If $\alpha=\sum_{n\ge0}a_np^n$ is a $p$-adic integer and we let $S_n=a_0+\cdots+a_np^n$ denote partial sums, then $f(x)^{S_n}$ should converge if $f(x)$ is contained in the translated ideal $L:=1+(x)$ (i.e. $x\mid(f-1)$). We use this to define $p$-adic powers on $L$.

The binomial series also allows us to define fractional powers on $L$ so long as the fraction's denominators are not divisible by $p$. These fractions are precisely the elements of $U:=\Bbb Z_p\cap \Bbb Q$.

Do these two definitions of powers of $U$ on $L$ agree with each other? Is there a high-altitude reason why we might expect them to agree? (Also, the first paragraph above reminds me of something called "Witt vectors" I heard about; do they fit into the picture I described above somehow?)

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Nice question +1 –  Babak S. Nov 17 '12 at 5:48

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I guess the "high-altitude" reason is that $L$, the group of principal units of the discrete valuation ring $\mathbf{F}_q[[x]]$, is an abelian pro-$p$ group. If $A$ is any abelian pro-$p$ group, then there is a unique structure of $\mathbf{Z}_p$-module on $A$ which is continuous in the sense that the map $\mathbf{Z}_p\times A\rightarrow A$ is continuous and which extends the given $\mathbf{Z}$-module structure $(n,a)\mapsto a^n$. It is defined as in your first definition: take $f\in L$ and $z\in\mathbf{Z}_p$, and write $z=\lim_nz_n$ for integers $z_n$. Then the sequence $f^{z_n}$ converges, and we define $f^z$ to be the limit. This turns out to be independent of the choice of sequence converging to $z$.

Regarding Witt vectors, the ring $\mathbf{F}_q[[t]]$ is an equicharacteristic $p$ discrete valuation ring, whereas $p$-typical Witt vectors over perfect fields of characteristic $p$ are mixed characteristic discrete valuation rings. Namely, $W(\mathbf{F}_q)$ is a complete discrete valuation ring of characteristic $0$ with uniformizer $p$ and residue field $\mathbf{F}_q$. Again $1+pW(\mathbf{F}_q)$ is an abelian pro-$p$ group, and so again it is canonically a $\mathbf{Z}_p$-module.

Now, the binomial series definition makes sense in characteristic zero, because $\mathbf{Z}_p\subseteq W(\mathbf{F}_q)$, and one can make sense of the $\tbinom{z}{k}$ for $z\in\mathbf{Z}_p$ and $k\in\mathbf{Z}_{\geq 0}$, but I'm not sure if this continues to work in characteristic $p$. If it does, then it can be used to define $\mathbf{Z}_p$-exponentiation in $A$, and then, if you wanted to prove that this definition agreed with the one described above (your first description), you could show that both extended the usual $\mathbf{Z}$-exponentiation and that both gave $A$ the structure of continuous $\mathbf{Z}_p$-module. Then uniqueness (really density of $\mathbf{Z}$ in $\mathbf{Z}_p$) would imply that they coincide.

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Right. The definition by way of binomial numbers does work, but I don’t know, offhand, of any use that’s been made of the fact. –  Lubin Nov 17 '12 at 8:00
    
Thanks for the background. I think to show the fractional binomial series definition extends it to $\mathbf{Z}_p$-module structure, we'd need to show that for $f\in L$, sequences $f^{z_n}$ converge when $z_n\in U$ is Cauchy in the $p$-adic metric, plus establish independence of the resulting value with respect to sequences $z_n$ (modulo null sequences). And I guess then show continuity. –  blue Nov 17 '12 at 16:07

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