Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?)

share|improve this question
Nice question +1 –  B. S. Nov 17 '12 at 5:48
add comment

1 Answer 1

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.

share|improve this answer
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
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.