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gave p-adic complex numbers an entry
Why mention ’$\geq 2$’ in
For $p \geq 2$ a prime number ?
No need to, of course.
This analogy seems less complete than stated.
$\mathbb{Q}_p$ isn't analogous to $\mathbb{Q}$, it's analogous to $\mathbb{R}$. Everything starts with $\mathbb{Q}$, just with different metrics.
To construct $\mathbb{C}_p$ from $\mathbb{Q}$, we go through $3$ steps:
But to construct $\mathbb{C}$, we can skip the first step (or alternatively, skip the last step).
Do I have this right?
Hm, not sure maybe I am missing something. Or maybe I’d rather think of $\mathbb{Q}_p$ primarily as a formal neighbourhood in $\mathbb{Q}$ than as an analogue of $\mathbb{R}$.
Maybe I have this wrong. But in the entry the desired analogy is explained, following the PlanethMath entry here. Are you technically objecting to the analogy as discussed there, or do you rather mean to prefer a different analogy?
I agree with Toby: $\mathbb{Q}_p$ is the completion of the field of fractions of the local ring (of $\mathbb{Z}$) at the prime $p$, so for $p = \infty$, we should really identify it with $\mathbb{R}$, not $\mathbb{Q}$.
Of course it’s also true that $\overline{\mathbb{Q}_p}$ is already algebraically isomorphic to $\mathbb{C}$, by the Steinitz characterization of algebraically closed fields in terms of characteristic and transcendence degree.
Oh, I see the Idea-sentence had $\mathbb{Q}_p$ and $\mathbb{C}$ swapped. It’s supposed to summarize what follows in the Definition section (is there any disagreement about the Definition section?) so it should read
For $p$ a prime number, the field of complex $p$-adic numbers $\mathbb{C}_p$ is to the p-adic rational numbers $\mathbb{Q}_p$ as the complex numbers $\mathbb{C}$ are to the rational numbers.
Changing the idea section makes no difference, $a : b :\!: c : d$ means the same as $a : c :\!: b : d$. I (and Zhen) are saying that $d$ should be $\mathbb{R}$ rather than $\mathbb{Q}$.
The definition as you wrote it is correct, but it's not morally correct; it's a special feature of $p = \infty$ that we can take this shortcut.
@Todd #7: Even so, the topology is different.
Yep. The topology of $\mathbb{C}_p$ is also different from that of $\mathbb{C}$.
I see which story you all tell, but I don’t see yet why the other story is, as Toby thankfully clarified now, deemed morally wrong.
Here is a story without reals:
We want to understand geometry over Z. That’s hard, so we employ a hierarchy of approximations that stagewise make this look more and more like familiar geometry.
First stage: look at the local model geometry of each point, that means passing to Z_p for each p.
Second stage: to apply ordinary (“non-arithmetic”) algebraic geometry, first turn rings into fields, that gives Q_p, then into algebraically closed fields, that gives barQ_p.
Third stage, the algebraic geometry may still be too hard, so we pass further to analytic geometry. That yields C_p
The above message was from my phone, therefore not nicely formatted. With better typesetting the “story” I have in mind reads as what I have now dropped into the Sandbox.
OK, now that I've read your story, I see how $\mathbb{Q}_p$ is analogous to $\mathbb{Q}$. It's not so much that you tell this story without $\mathbb{C}$ (because your story doesn't treat that case at all) as that you tell it without $\mathbb{Q}$. If we skip the first stage, then $\mathbb{Q}$ appears in place of $\mathbb{Q}_p$, as you want, and $\mathbb{C}$ appears in place of $\mathbb{C}_p$, as we all want. (ETA: Yes, this is exactly what your Sandbox version has.)
So what is analogous to what depends on where you start.
Is it about where we start, or is it about which order of steps we take?
I gather in the other story the difference is that we first “pass to analytic geometry” and then algebraically compete, whereas I am thinking of it the other way around.
I have a reason for thinking this way: I believe a lot of conceptual confusion has resulted from thinking of the $\mathbb{Q}_p$ as being analogous to the real numbers. That’s what causes all these people to speculate about whether “the universe might be p-adic” etc.,following the thought that all of standard geometry and physics might be re-done with $\mathbb{R}$ replaced by $\mathbb{Q}_p$.
But it seems to me by what we actually see happening in those parts of math and physics where p-adics appear “by themselves” (in some strong theorems, because it just works, and not because one feels it should work) is that they appear as the local approximations to arithmetic geometry. From this perspective it is clear that we are not to “replace $\mathbb{R}$ by $\mathbb{Q}_p$” at all.
A good (I think) example of this is the constrution of the Witten genus via the string-orientation of tmf: one starts out wanting to construct something in complex analytic/algebraic geometry. In doing so, one finds that one needs to refine all the way to artihmetic geometry (namely pass from the moduli of complex elliptic curves to that of arithmetic elliptic curves). That problem however is hard to attack face-on, but the fracture theorems say that it decomposes into the p-local stories. So one discusses all those separately and in the end glues togtether to indeed find reproduced the complex-analytic Witten genus.
Here $\mathbb{R}$ never appears anywhere.
For the record, here's Urs's table from the Sandbox. (I had to make a couple of changes because apparently there's a bug in how the Forum parses mixed pipelinks and tables; perhaps that was why Urs was having trouble making it here before.)
arithmetic geometry | -field of fractions$\to$ | algebraic geometry | -algebraic closure$\to$ | absolute Galois theory | -analytic completion$\to$ | analytic geometry | |
---|---|---|---|---|---|---|---|
globally | $\mathbb{Z}$ (integers) | $\mathbb{Q}$ (rational numbers) | $\overline{\mathbb{Q}}$ | $\mathbb{C}$ (complex numbers) | |||
formal geometry at $p$ | $\mathbb{Z}_p$ (p-adic integers) | $\mathbb{Q}_p$ (p-adic rational numbers) | $\overline{\mathbb{Q}_p}$ | $\mathbb{C}_p$ (p-adic complex numbers) |
Is it about where we start, or is it about which order of steps we take?
Well, it depends, because there are many options. So one's view of the proper analogy will depend on all of these. I believe that this diagram is accurate:
$\array { & & \mathbb{N} & \overset{\text{localize at }\,p}\rightarrow \\ & & \text{complete to ring}\downarrow & & \mathbb{Z}_p \\ & & \mathbb{Z} & \overset{\text{localize at }\,p}\rightarrow \\ & & \text{complete to field}\downarrow & & \text{complete to field}\downarrow \\ \mathbb{R} & \overset{\text{complete in euclidean metric}}\leftarrow & \mathbb{Q} & \overset{\text{complete in }\,p\text{-adic metric}}\rightarrow & \mathbb{Q}_p \\ \text{algebraic completion}\downarrow & & \text{algebraic completion}\downarrow & & \text{algebraic completion}\downarrow \\ \mathbb{C} & \overset{\text{complete in euclidean metric}}\leftarrow & \overline{\mathbb{Q}} & & \overline{\mathbb{Q}_p} & \overset{\text{complete in }\,p\text{-adic metric}}\rightarrow & \mathbb{C}_p }$There is clearly an analogy between $\mathbb{Q}_p$ and $\mathbb{R}$, and there is clearly a different analogy between $\mathbb{Q}_p$ and $\mathbb{Q}$. I (and I suppose Zhen) are more familiar with the former analogy, but you meant the latter.
@Toby, where you’ve written “localize at $p$” I would have said something like “algebraically complete at $p$” (although I guess that’s a bit confusing since we’re also talking about “algebraic completion”). Isn’t “localizing $\mathbb{Z}$ at $p$” rather what gives us $\mathbb{Z}_{(p)}$? Geometrically, both are looking at “local” information in some sense, but I think algebraically “localize” generally has the one meaning and not the other.
Also is there not an “$\mathbb{N}_p$” that we can get by completing at $p$ in the world of semirings?
Toby is thinking of the fact that the $p$-adic integers are the function algebra on the formal neighbourhood of the point determined by $p$. So it’s localization in this geometric sense. But of course disambiguating and clarifying this point is always good.
Is there a standard “geometric” term for this kind of “localization” that distinguishes it from the standard algebraic meaning of “localization” (which is also geometrically motivated, but just looks at a “larger infinitesimal neighborhood”)?
The standard term refers to formal neighbourhoods. Perhaps one could say “formal-localisation”?
I imagine there might be some site for which these are literally the local rings. It’s almost true for the étale site.
Urs kept saying ‘local’, so I wrote ‘localize’, not thinking about it too much. You're right that this is not the proper meaning of the term. One could certainly just say ‘complete in $p$-adic metric’ again.
As for $\mathbb{N}_p$, I take this to be the same as $\mathbb{Z}_p$, because $\mathbb{N}/p$ is the same as $\mathbb{Z}/p$, and these are what we take the limit of. Or looking at it topologically, $\mathbb{N}$ is dense in $\mathbb{Z}_p$, so the completion is all of $\mathbb{Z}_p$. Explicitly, $\ldots666_7$ is $-1$ in $\mathbb{N}_7$ (for example), and it is both the limit (in the categorial sense) of $\cdots \mapsto [666_7]_{7^3} \mapsto [66_7]_{7^2} \mapsto [6]_{7} \mapsto [0]_1$ and the limit (in the topological sense) of $(0, 6, 66_7, 666_7, \ldots)$. (Edit: missing ‘${}_7$’s.)
You’re right about $\mathbb{N}_p$.
It’s curious that for the $p$-adic metric, completion commutes with the field of fractions, but not for the Euclidean metric.
It’s curious that for the $p$-adic metric, completion commutes with the field of fractions, but not for the Euclidean metric.
And conversely, [topological] completion commutes with algebraic completion for the euclidean metric, but not for the $p$-adic metric. (Well, actually, I don't know how to put the $p$-adic metric on $\overline{\mathbb{Q}}$ in the first place, but even if it can be done, it couldn't commute, since $\overline{\mathbb{Q}_p}$ isn't topologically complete at all.)
But $\overline{\mathbb{Q}}$ does have a Euclidean metric? Is there any way to get it other than by picking a random embedding into $\mathbb{R}$? Couldn’t you similarly get a $p$-adic metric on it by picking an embedding into $\overline{\mathbb{Q}_p}$ or $\mathbb{C}_p$?
Re #25: The theorem on the page real closed field indicates that the unique (order-preserving) field map $\mathbb{Q} \to \mathbb{R}$ extends uniquely to a field homomorphism from the real closure of $\mathbb{Q}$ into $\mathbb{R}$ (and that the “the” in “the real closure” is justified). So given a choice of real subfield $\mathbb{R}$ of $\mathbb{C}$ (such a choice is determined from a Euclidean metric on $\mathbb{C}$ by taking norms), we get a canonical embedding $\widebar{\mathbb{Q}} \to \mathbb{C}$, unless I’m confused.
Yes, $\overline{\mathbb{Q}}$ has many euclidean metrics, equivalently many embeddings into $\mathbb{C}$, but what matters is that it has at least one (and they're all topologically equivalent). (I'm not going to worry about constructive math here; then we'd have to go back and think about algebraic closure even means, and I don't want to get into that. Similarly if we wanted to worry about functoriality.) But does it have any embedding into $\mathbb{C}_p$? I don't see right away why it does.
@Todd: But isn’t there a lot of choice involved in embedding the real-closure of $\mathbb{Q}$ into its algebraic closure? The former is ordered by definition, while the latter is not a priori. E.g. $\overline{\mathbb{Q}}$ contains two square roots of $2$, but it doesn’t “know” which of them is positive and which is negative. And then there ought also to be a choice regarding which square root of $-1$ in $\overline{\mathbb{Q}}$ goes to which square root of $-1$ in $\mathbb{C}$.
@Toby: What do you mean by “topologically equivalent”? They don’t induce the same topology on $\overline{\mathbb{Q}}$, do they? Do you mean the induced topologies differ by an automorphism of $\overline{\mathbb{Q}}$?
As for embedding $\overline{\mathbb{Q}}$ in $\mathbb{C}_p$, I thought that if $K\subset L$ is a field extension with $L$ algebraically closed, then it could be extended to also embed the algebraic closure of $K$ in $L$. Is that not true?
But isn’t there a lot of choice involved in embedding the real-closure of ℚ\mathbb{Q} into its algebraic closure?
That was exactly the point I was making when I mentioned the words “given a choice of real subfield $\mathbb{R}$ of $\mathbb{C}$” (emphasis now added). But each Euclidean metric on $\mathbb{C}$, of which there are many, gives such a choice (use the real subfield that contains the norms of that metric).
Okay, interesting. I tend to think of the field denoted “$\mathbb{C}$” as coming with a Euclidean metric, just like the field denoted “$\mathbb{R}$”. Is there some way to construct it in such a way that it doesn’t come with a canonical Euclidean metric that you’d have to forget?
Anyway, I thought #26 was disagreeing with #25, but it seems you were really agreeing that $\overline{\mathbb{Q}}$ has many Euclidean metrics. (I sloppily said “embedding into $\mathbb{R}$” when I should have said “embedding into $\mathbb{C}$”.)
We know that a field is isomorphic to $\mathbb{C}$ iff it is algebraically closed and has continuum-cardinality transcendence basis over $\mathbb{Q}$. So abstractly one can construct, however one does, the algebraic closure of $\mathbb{Q}(X)$ with ${|X|} = c$, without any metric around to have to forget.
What do you mean by “topologically equivalent”? They don’t induce the same topology on $\overline{\mathbb{Q}}$, do they? Do you mean the induced topologies differ by an automorphism of $\overline{\mathbb{Q}}$?
Yes, I think that that's what I meant. Which is not the usual meaning of two metrics' being topologically equivalent, so I shouldn't have put it that way. But the point is that you can take any one of these, ask for the completion under this metric, and get the correct result. (Of course, if that wasn't true, then we wouldn't count it as ‘one of these’, so probably I wasn't really saying anything of significance.)
As for embedding $\overline{\mathbb{Q}}$ in $\mathbb{C}_p$, I thought that if $K\subset L$ is a field extension with $L$ algebraically closed, then it could be extended to also embed the algebraic closure of $K$ in $L$. Is that not true?
Well, I don't know! Let me check my algebra textbooks. … Yes, of course! The elements of $\overline{\mathbb{Q}_p}$ that are algebraic over $\mathbb{Q}$ comprise an algebraic closure of $\mathbb{Q}$ (Dummit & Foote, 1991; Proposition 13.31), which has an isomorphism with $\overline{\mathbb{Q}}$. So any choice of $\overline{\mathbb{Q}_p}$ (actually, we could fix one for all) and choice of embedding $\overline{\mathbb{Q}} \hookrightarrow \overline{\mathbb{Q}_p}$ will induce a $p$-adic metric^{1} on $\overline{\mathbb{Q}}$. Is the topological completion of this all of $\mathbb{C}_p$? I don't know.
I've been assuming that $\overline{\mathbb{Q}_p}$ comes equipped with a natural $p$-adic metric, but it occurs to me that I'm not sure about that either. ↩
Any finite extension of $\mathbb{Q}_p$ has a $p$-adic metric, and $\overline{\mathbb{Q}_p}$ is a directed union of finite extensions. So $\overline{\mathbb{Q}_p}$ has a well-defined topology, at least.
OK, good! That means that we know what it means to complete it to get $\mathbb{C}_p$, as we should.
If anybody can figure out whether $\overline{\mathbb{Q}}$ is dense in $\mathbb{C}_p$, then I'll be happy.
I have added a pointer at p-adic complex number to prop. 3.4.1.3 of
for the statement that the completion of the algebraic closure of a complete normed field is still algebraically closed.
Above we had some discussion about what seems to be two different perspectives on the conceptual role of the p-adic numbers. Maybe I am missing something, but it seems to me that the fact that there are these two perspectives is an important point which is not duely dealt with in existing literature.
I may be all wrong here, but I want to reduce my internal entropy on this point.. That’s why I’d enjoy to chat about it, if anyone still has the energy. Feel invited to tell me that I am all mixed up, if that’s the case, but tell me why.
So the problem I see is that there are these two incompatible perspectives on the fact that $\mathbb{R} = \mathbb{Q}_{p = \infty}$
(perspective I) We say: where we developed geometry modeled on the real line $\mathbb{R}^1$ (or the complex plane $\mathbb{C}$) and hence on Cartesian spaces $\mathbb{R}^n$ (or complex polydisks), we should try to find an analog with $\mathbb{R}$ replaced by $\mathbb{Q}_p$.
That’s actually what most of the authors have in mind who speculate about p-adic physics (as cited there).
(perspective II) We say instead: by the function field analogy the rings $\mathbb{Q}_p$ are analogous to the rings $\mathbb{C}((x-t))$ of Laurent series around a point $t$ in the complex plane, with that point being “$p$”. From this perspective it makes no sense to think of $\mathbb{Q}_p$ as the analog of the continuum real line on whose products we want to model our geometry. Instead, from this perspective $\mathbb{R} = \mathbb{Q}_{p = \infty}$ is instead to be thought of as a formal power series ring, too, the one “around infinity”. From this perspective, the model space on which we should model our geometry is instead $Spec(\mathbb{Z})$.
It seems to me that the second perspective is the one that is actually conceptually “correct”.
What I am still trying to better understand is to which extent rigid/Berkovich/Huber-style analytic geometry is truthful to the second perspective, if it is.
For instance, from the second perspective the topology involved is “just” there to get the ring completions, which one could alternatively also define purely algebraically. Is it actually conceptually correct, in perspective II, to try to treat the topology on rings of p-adic numbers in analogy with the topology in standard real/complex geometry?
I saw a comment about algebraic topology (let us say, to the level and technology of Lurie, but I can’t remember who) in a presentation, that working at various primes, that is, with various generalised cohomologies (e.g. $BP$, $E$-theories etc) at a fixed prime $p$, is really just because the item of real interest is global (over $Spec(\mathbb{Z})$), but is too hard to understand, so work with the fibres over the various points (=primes).
I don’t know if this lines up properly with your second perspective, but it feels similar.
Yes, that is exactly another incarnation of the same phenomenon. In discussion of fracture theorems people tend to essentially admit this, though I am not sure if many texts make it as explicit as Emiliy Riehl’s recent text which I have now cited there.
The adic decomposition of the moduli stack of elliptic curves appears for instance prominently in the construction of tmf, as mentioned in that entry, and so the spectrm is built “one prime at a time”, as usual in algebraic topology.
However, maybe ironically, in the analytic geometry literature proper, this idea that working over the p-adics should really mean working in an infinitesimal neighbourhood around some point in some something is by and large not considered. Maybe in the global analytic geometry literature, I’d need to further check that out.
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