# There is no p-adic 2 pi i

You may not know this, but we can see the search terms people use to find our blog. Yesterday, four people came to our blog using the search string “p-adic 2 pi i”. Presumably, people want to know what the $p$-adic analogue of $2 \pi i$ is.

There isn’t one, and there is a good reason why. I assume that we can all agree that the most important property of $2 \pi i$ is that it is the period of the complex exponential function. Unfortunately, there are no continuous periodic functions on $\mathbb{Q}_p$ except for the locally constant functions. The reason is very simple. Suppose that $f : \mathbb{Q}_p \to \mathbb{Q}_p$ was periodic with some nonzero period $\lambda$. Then we would have $f(k \lambda) = f(0)$ for every integer $k$. But, in the $p$-adic toplogy, the integers are dense in any neighbourhood of the identity. So $f$ would take the value $f(0)$ infinnitely often near $0$, and would thus be constant in a neighbourhood of $0$. The same argument gives that $f$ is constant in a neighborhood of any $a$. (And the same arguement applies if you take $f$ to be complex valued.)

There is an interesting $\mathbb{F}_p[[t]]$-analgoue of $2 \pi i$, related to the Carlitz exponential. But that is a more complicated, and more interesting, subject.

Judging from our searches, what people want to know today is when Ed Witten’s seminar meets. I’m afraid I don’t know that!

Update: This post has drawn some comments which are far smarter than what I wrote. There is a way to make $p$-adic sense of $2 \pi i$, although I don’t understand what it is yet. Come and see our very smart commenters try to explain it to me!

Update: The conversation seems to have stopped for the moment, but I am still trying to understand these period rings, with help from Jay. If I get it, I’ll be sure to post an explanation here.

## 10 thoughts on “There is no p-adic 2 pi i”

1. There is a $p$-adic analogue of $2\pi i$, although for a different definition of $2\p i$; it is also complicated. Here is the basis of the analogy. Consider the variety $X = \mathbb{G}_m$ over $\mathbb{Q}$.

One can attach to $X$ its algebraic de Rham cohomology, whose $H^1$ is a one-dimensional $\mathbb{Q}$-vector space on the canonical basis $dt/t$. (This is a simple algebraic exercise.) Tensoring up to $\mathbb{C}$, Grothendieck’s “algebraic de Rham theorem” shows that one obtains the well-known smooth de Rham cohomology (with complex coefficients) of the complex manifold $X(\mathbb{C})$. (Compare to the fundamental theorem of line integrals, and the residue theorem in complex analysis…)

One can also attach to $X$ the singluar homology of $X(\mathbb{C})$ with coefficients in $\mathbb{Z}$. Since $X(\mathbb{C})$ deformation retracts onto $S^1$, the singular homology is free abelian on one generator; the choice of a square root of $-1$ determines an orientation of $X(\mathbb{C})$ and also a generator $v$ of $H_1$.

The classical (I say: “Archimedean”) de Rham theorem says that “integration” realizes a nondegenerate, $\mathbb{C}$-valued pairing between de Rham $H^1$ and signgular $H_1$. Since these two spaces have rank one, the pairing is determined by the single value $\int_v dt/t$, which, of course, is $2\pi i$. In this sense, $2\pi i$ is be defined to be the “period” of the degree $1$ (co)homology of $X = \mathbb{G}_m$. (Note an internal consistency: $2\pi i$ depends on the choice of $v$, i.e. on a choice of square root of $-1$.)

The $p$-adic analogue of $2\pi i$ comes from knowledge of a non-Archimedean de Rham theorem. It provides a nondegenerate pairing between de Rham $H^1$, which is still generated by $dt/t$, and the $p$-adic etale homology $H_1$, which is a free rank one $\mathbb{Z}_p$-module. Whereas in the singular case $v$ depended on a choice of square root of $-1$, the generator $w$ of the etale homology depends on a choice of compatible system of primitive $p$-power roots of unity. Thus, the $p$-adic $2\pi i$ is the value of $dt/t$ paired with $w$.

A mash of three comments: This sounds straight forward, but didn’t I say earlier that it’s complicated? Wait, you’ve never heard of the non-Archimedean de Rham theorem? In what set did the nondegenerate pairing take its values? Aye, there’s the rub. Something that’s oft-underappreciated by us when we’re young is how arithmetically subtle the topological ring $\mathbb{C}$ is: it is big enough to hold all periods of algebraic varieties. However, $\mathbb{Q}_p$, even $\mathbb{C}_p$ (the $p$-adic completion of an algebraic closure of $\mathbb{Q}_p$), is not big enough! I would love to explain how in more detail, but this comment is already much longer than David’s post.

In short, what Fontaine discovered about 30 years ago is the existence of certain $\mathbb{Q}_p$-algebras, called $B_{crys}^+ \subset B_{dR}^+$, mapping surjectively onto $\mathbb{C}_p$, that do naturally hold all periods of algebraic varieties. This last claim is just a fancy way of saying that the non-Archimedean de Rham theorem pairing, which unfortunately is harder to construct than writing down an integral, takes values in $B_{dR}^+$. (And if a general variety $X$ has good reduction, then the values lie in $B_{crys}^+$.) So the $p$-adic $2\pi i$, whatever it is, is an element of $B_{crys}^+$.

Closing comment: You might quip, “If the $B$-rings surject onto $\mathbb{C}_p$, then why not take the image of $2\pi i$ in $\mathbb{C}_p$?” To this I would reply, its image in $\mathbb{C}_p$ is zero!

2. But there is a p-adic 2-pi-i. It just doesn’t live on Q_p, it lives on Fontaine’s ring of p-adic periods.

3. Jay, just last week you were saying you didn’t know anything that would be suitable for discussion on a math blog! If you can write up an understandable explanation of what $B_{\mathrm{crys}}^+$ is, I’d love to see and link to it.

I’ll do the easy part, by explaining what Jay is getting at when he talks about “a choice of [a] compatible system of primitive $p$-power roots of unity”. Let’s go back to the archimedean case and think about $\mathbb{C}^*$. Consider the path $\gamma$ which winds once around the puncture in $\mathbb{C}^*$. For any positive integer $n$, there is an $n$-fold cover of $\mathbb{C}^*$, which I’ll call $U_n$. Monodromy along the path $\gamma$ gives an automorphism of $U_n$. Explicitly, $U_n \cong \mathbb{C}^*$, with the covering map $u \mapsto u^n$, and the automorphism is of the form $u \mapsto \zeta_n u$. The $\zeta_n$ obey certain compatibilities, namely, $\zeta_b = (\zeta_{ab})^a$.

Of course, this is all a lot of high-falutin’ language, because $\zeta_n$ is nothing other then $e^{2 \pi i/n}$.

But, if I understand Jay correctly, the $p$-adic analogue of $\gamma$ is the data of a choice $\zeta_{p^n} \in \mathbb{C}_p$, one such choice for each power of $p$, such that $\zeta_{p^n}=(\zeta_{p^{n+1}})^p$. And, somehow, there is a way to compute a $p$-adic analogue of $\int_{\gamma} dt/t$.

4. dt says:

Jay, thanks for writing that. This question may be ill-posed, but: what are some p-adic varieties with simple periods, if the periods of G_m are already so complicated? For instance you said that the B rings contain Q_p–is it possible to characterize varieties whose periods belong to Q_p? Alternatively, are there other easy-to-describe varieties whose periods vanish in C_p?

5. Dear dt,

Roughly speaking, zero-dimensional varieties will
have all their periods in Q_p (or finite extensions thereof),
but nothing else will.

As for when the periods vanish in C_p: this is also
pretty common. In the p-adic, as in the classical (archimedean)
setting, there is a Hodge filtration on the de Rham cohomology,
and all the periods which come by “integrating” de Rham
cohomology classes lying in Fil^i for i > 0 will vanish when
pushed down to C_p.

Regards,

Matt

6. James says:

No one said it yet, but the $p$-adic analogue of $2\pi i$ is $\log(\epsilon)$, where $\epsilon$ is the compatible system of $p$-power roots of unity, and where $\log(z)$ is computed using the usual expansion at $z=1$. If you squint, it really does look like the monodromy of the logarithm. Of course, the reason why no one said this is because you have to say what ring you compute the log in and how you view $\epsilon$ as an element of that ring, which takes a bit of work, but I think it’s helpful to say what I did anyway.

7. Thanks James! Just to head off the obvious, if I take $\log \zeta_{p^n}$ in $\mathbb{Q}_p$, the power series will converge, but I get zero. (The baby example of this is $\log -1$ in $\mathbb{Q}_2$. One can easily check that the power series $\sum 2^n/n$ converges $2$-adically; it’s a lot of fun to compute this sum modulo 8 or 16 and watch it vanish)

But there is someplace else that I can take this log and get a nonzero answer? Neato!

8. You can also form a p-adic Carlitz exponential as a power series (with decent radius of convergence), where the denominators are given by a product over monic polynomials of a given degree in 1/p with Teichmüller coefficients. The set of all such (not necessarily monic) polynomials is rather nice – it is multiplicatively closed, and additively closed mod p. This exponential seems to obey similar functional equations as Carlitz, but I don’t know if it’s useful.

9. maxmoo says:

hi, this is my first time commenting on a wordpress blog. i’m a bit scared that there’s no preview button… i hope all my latex works out :-/

anyway i’m just starting to learn this stuff, so i may be making heaps of mistakes, but here’s my take on the $p$-adic rings of periods $B_{dR}^+$ and $B_{cris}$.

before we can define $B_{dR}^+$ we need to talk a little bit about witt vectors. the usual way we think of a p-adic integer is as an infinite sum $a_0 + a_1p + a_2p^2+\cdots$ where $0\leqslant a_i \leqslant p-1$, i.e., $a_i \in \mathbb{F}_p$. in other words, our set of digits for $\mathbb{Z}_p$ is the set $\{0,1,\cdots, p-1\}\subset \mathbb{Z}_p$, where we identify this set with elements of $\mathbb{Z}_p$ using the inclusion $\mathbb{Z}\hookrightarrow \mathbb{Z}_p$. this is similar to our usual set of base-10 digits for $\mathbb{Q}$, where write any rational number (say of norm less than 10) as an infinite sum $a_0+ a_1 10^{-1}+ a_2 10^{-2} + \cdots$ with $a_i \in \mathbb{Z}/10\mathbb{Z}$, and where the $a_i$ are identified with elements of $\mathbb{Q}$ via the inclusion $\mathbb{Z}\hookrightarrow \mathbb{Q}$.

now this is the only natural way that i can think of to include $\mathbb{Z}/10\mathbb{Z}$ as a set of digits in $\mathbb{Q}$. however there is another very natural way of including $\mathbb{F}_p$ in $\mathbb{Z}_p$ called the teichmuller map, which works as follows. let $0\neq a\in \mathbb{F}_p$, $a_n=a^{p^n}$. since $a_{n+1} \equiv a_n (\textrm{mod }p^m)$ for every $m\leqslant n$, we see that the $a_n$ converge to some $[a]\in \mathbb{Z}_p$. (in fact $[a]$ is a $p$-adic $(p-1)$th root of unity.)
we can use these teichmuller representatives as an alternative set of digits for $\mathbb{Z}_p$. that is, any $p$-adic number can be written as a sum $\sum_{n} p^n [x_n]$ with $x_n \in \mathbb{F}_p$.

let $R$ be any perfect ring of characteristic $p$ (perfect means the frobenius map $x\mapsto x^p$ is an automorphism. for example $\mathbb{F}_p$ is perfect, but the ring of polynomials $\mathbb{F}_p[X]$ isn’t.) then there is a unique ring $W(R)$ called the ring of witt vectors, which has characteristic 0, has a unique maximal ideal, and has $\mathrm{Frac}(R)$ as its residue field. this is a generalization of the $p$-adic numbers—if $R =\mathbb{F}_p$, then $W(R) = \mathbb{Z}_p$. any ring of witt vectors has a multiplicative teichmuller map $x\mapsto [x]$, and just as before, any element $x\in W(R)$ can be written in a unique way as $\sum_i [x_i] p^i$ with the $x_i\in R$.

our ring of periods $B_{dR}^+$ is a completion of a ring of witt vectors. let $\widetilde{E}^+$ be the set of all sequences $(x^{(0)},x^{(1)},\ldots)$ in $\mathbb{C}_p$ with $|x^{(i)}|\leqslant 1$ and $(x^{(i+1)})^p= x^{(i)}$. we make this into a ring by the rules $(x+y)^{(i)} = \lim_{j\to \infty} ( x^{(i+j)}+y^{(i+j)})^{p^j}$ and $(xy)^{(i)} = x^{(i)}y^{(i)}$. you can check that this ring has characteristic $p$, and it’s a perfect ring since everything in $\mathbb{C}_p$ has $p$-th roots. define a valuation on $\widetilde{E}^+$ by $v(x) = v_p(x^{(0)})$. let $\widetilde{B}^+ = W(\widetilde{E}^+)[1/p]$. by the teichmuller map, this is the set of all sums $\sum_i [x_i] p^i, x_i\in \widetilde{E}^+$. define a map $\theta: \widetilde{B}^+ \to \mathbb{C}_p$ by $\sum [x_i] p^i\mapsto\sum x_i^{(0)}p^i$. let $\epsilon = (1, \zeta_p, \zeta_{p^n},\ldots)$ be a compatable system of primitive $p^n$-th roots of unity like david was talking about before, considered now as an element of $\widetilde{E}^+$, and let $\omega = ([\epsilon]-1)/([\epsilon^{1/p}]-1) = 1 + [\epsilon^{1/p}] + \cdots + [\epsilon^{(p-1)/p}]$ (you can check this equality using the multiplicative property of the teichmuller map). then $\theta(\omega)= 1 + \zeta_p + \cdots + \zeta_p^{(p-1)} = 0$, and in fact the kernel of $\theta$ is the ideal generated by $\omega$.

define $B_{dR}^+$ as the completion of $\widetilde{B}^+$ with respect to the $\ker(\theta)$-adic topology. that means every element $x\in B_{dR}^+$ can be written as a sum $x = \sum x_n \omega^n$ with $x_n \in \widetilde{B}^+$. finally we can construct our $2\pi i$’ in this ring, as the sum $\log ([\epsilon]) := \sum_n (1-[\epsilon])^n/n$. in the $\ker(\theta)$-adic valuation, we have that $v((1-[\epsilon])^n/n) = v(\omega^n\cdot(1-[\epsilon^{1/p}])^n/n) = n$. since this is strictly increasing as $n\to \infty$, this tells us that there won’t be too much cancellation’ like there was when david tried to take the $p$-adic log of $\zeta_{p^n}$—so our series will converge to something non-zero (of valuation 1).

ok so it sort of seems like this ring $B_{dR}^+$, isn’t really $p$-adic anymore. we’ve taken a ring of witt vectors (a $p$-adic’ thing), and then given it some stupid new topology, just to make our one particular series converge. well this is kind of true. the frobenius automorphism $x\mapsto x^p$ isn’t continous anymore for $B_{dR}^+$ (something that you’d certainly want for a $p$-adic object), so you end up having to take a certain subring $B^+_{cris}\subset B_{dR}^+$ which does have a continuous frobenius.

but we can see that our $\log ([\epsilon])\in B^+_{dR}$ is actually a period for something $p$-adic, so it is in fact already a sort of a $p$-adic $2\pi i$. let $\chi:G= \textrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to \mathbb{Z}_p^\times$ be the cyclotomic character (i.e., $\sigma(\zeta) =\zeta^{\chi(\sigma)}$ for every $p^n$-th root of unity). then for any $\sigma \in G$ we have $\sigma(\log([\epsilon])) = \log([\sigma(1,\zeta_p, \ldots)]) = \log([\epsilon^{\chi(\sigma)}]) = \chi(\sigma) \log([\epsilon]).$ (the so-called ax-sen-tate theorem’ tells us that $\chi$ doesn’t have any periods in $\mathbb{Q}_p$.)

this definition of `period’ might look a bit weird, but you can think of $2\pi i /p = \log(\zeta_p)$ considered as an element of $\mathbb{C}$ as a period for the $p$-th cyclotomic character $\chi_p$ in the same way.

10. The p-adic numbers Qp don’t have a square root of -1, if p=3 mod 4.
So differentiable functions from Qp[i] -> Qp[i] satisfy the Cauchy-Riemann equations.

To what extent would analysis in Qp[i] have the familiar complex analysis
theorems??? You couldn’t prove that Qp[i] is algebraically complete, I
wonder what would block the complex analysis proof of that, that 1/p(x)
would be a bounded entire function if it had no roots.

Laura