Lautman on Reciprocity

August 14, 2008

Posted by David Corfield

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In ‘Nouvelles Recherches sur la Structure Dialectique des Mathématiques’ (1939), Albert Lautman discusses the use of analysis in number theory. He notes that some have felt uncomfortable with this use and have sought to eliminate it. But Lautman sees no metaphysical necessity for this ‘purification’. Rather than take arithmetic as metaphysically prior to analysis, instead he proposes that we consider them equally as realisations of the same ‘dialectical’ structures.

He gives the example of reciprocal entities. In arithmetic we have quadratic reciprocity, where the Legendre symbols are acting as a kind of inverse to each other.

$\left( \frac{p}{q} \right) \left( \frac{q}{p} \right) = (-1)^{\left(\frac{p-1}{2}\right) \left( \frac{q-1}{2}\right)}$

He goes on to note that it has been possible to generalise reciprocity in two different ways. First, to algebraic integers in any field. Second, to allow more general congruences, not just to a square, but to other powers. This has been achieved algebraically he notes, but he adds that Hecke has also provided analytic means of deriving general quadratic reciprocity results using theta functions.

Here we define

$\theta (\tau) = \sum_{m = - \infty}^{m = + \infty} e^{- \pi \tau m^2},$

noting that singular points are at $\tau = 2 i r$ , $r$ a rational, but that for any such $r$ , $\sqrt \tau \theta (\tau + 2 i r)$ takes a finite value. Then we have

$\theta (1/ \tau) = \sqrt{\tau} \theta (\tau).$

This reciprocity, he claims, is the mainspring of the transcendental proof of quadratic reciprocity. The same dialectical idea is manifesting itself in different branches. He continues

This dialectical idea of reciprocity between elements can be so clearly distinguished from its realisations in arithmetic and in in analysis that it is possible to find a certain number of other mathematical theories in which it realises itself similarly.

He then claims that the dimensions $m$ and $n - m$ in Poincaré duality associated to a manifold of dimension $n$ also stand similarly in an inverse relationship, and further that Weil has shown (1937) a relationship between reciprocity laws and duality theorems.

Do people recognise a useful thought here? Is there a ‘walking’ reciprocity realising itself in different situations?

Posted at August 14, 2008 3:48 PM UTC

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24 Comments & 2 Trackbacks

Re: Lautman on Reciprocity

Just when I had run out of things to think about :-). I found an extended commentary on Lautman, seemingly more philosophical, here.

Posted by: Stephen Harris on August 15, 2008 8:22 AM | Permalink | Reply to this

Re: Lautman on Reciprocity

I think Greg Egan occasionally posts here. He has written a great new SF book “Incandescence”, which appears to introduce an interest in Physics, symmetry, Lautman, and the Langlands Program, unless I’m reading too much into it.

See this.

Lautman -> “the absence of a center of symmetry” and “allusion to the Langlands Program”, seems to show up like a magical Wang’s Carpet :-)!

Posted by: Stephen Harris on August 15, 2008 5:48 PM | Permalink | Reply to this

Re: Lautman on Reciprocity

Stephen Harris wrote:

unless I’m reading too much into it

I’m afraid you are, as I know nothing at all about the Langlands Program. There’s a brief passage in the book describing, in very general terms, the enduring interest of mathematics to any civilisation; I’m glad that many people have endorsed the sentiment, but (apart from a throwaway example of the most well-known exponential map from a Lie algebra to a Lie group) it’s really not about anything specific.

Posted by: Greg Egan on August 16, 2008 3:59 AM | Permalink | Reply to this

Re: Lautman on Reciprocity

Stephen Harris wrote:
unless I’m reading too much into it

Greg: I’m afraid you are as I know nothing at all about the Langlands Program.

SH: Robert Heinlein made a similar disavowal when his fans tried to read too much deep symbolism into “Stranger In A Strange Land”. I was thinking of

“Interesting Truths” referred to a kind of theorem which captured subtle unifying insights between broad classes of mathematical structures.”
[compared with]
‘The Langlands program is a web of far-reaching and influential conjectures that connect number theory and the representation theory of certain groups.’

Posted by: Stephen Harris on August 16, 2008 8:28 PM | Permalink | Reply to this

Re: Lautman on Reciprocity

Regarding people being uncomfortable with analysis being used in number theory, I know you were just using it as a springboard to another point, but I can’t resist saying something about that.

I think most people now would say that analysis (whether real, complex, or p-adic) plays a role in number theory analogous to that of chemistry in biology, or physics in chemistry, or topology in differential geometry, or category theory in mathematics. The whole point of analysis is that rational numbers are much more complicated things than what we need for many purposes, so we complete them to get the reals (or p-adics). This destroys a lot of information, but it keeps information about magnitudes. This is all we really need in many situations, especially in the physical sciences, so people often jump right into the reals instead of worrying about the rationals, and maybe this is what led to the belief that analysis should be separate from number theory. But even in number theory proper, magnitude is still relevant (in fact, extremely important). So you’d expect that sometimes it’s convenient to avail yourself of real analysis and its flexibility even when proving things about the rationals. I might even say that you’re not really doing number theory in its fullest sense until you understand the real/complex, p-adic, and finally global aspects of your question.

This is nicely summed up by the slogan “Real/complex analysis is number theory at infinity”, which is not to suggest that analysis is lesser. While many rich number theoretic structures do become trivial over the reals (indeed, that’s the whole point), the extra flexibility there allows further structures to emerge, which can be very rich themselves.

Posted by: James on August 16, 2008 8:35 AM | Permalink | Reply to this

Re: Lautman on Reciprocity

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Thanks for your comment, James — it’s really illuminating and thought-provoking.

Can you explain the slogan

real/complex analysis is number theory at infinity?

What does it mean? For instance, how should I understand the phrase ‘number theory at X’, and what values might X take other than infinity?

Posted by: Tom Leinster on August 16, 2008 2:55 PM | Permalink | Reply to this

Re: Lautman on Reciprocity

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Is this in the same sense as ‘ $p$ -adic analysis is number theory at $p$ ’?

Posted by: Tim Silverman on August 16, 2008 4:15 PM | Permalink | Reply to this

Re: Lautman on Reciprocity

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That was roughly my guess. I reckon James won’t surface for the next few hours, so let’s assume for now that’s what he meant. In which case, I have two further questions:

Why should number theory at $p$ be $p$ -adic analysis, rather than $p$ -adic number theory? Surely not all of the theory of $p$ -adic numbers is analysis.
How can real/complex analysis be thought of as being like a theory of $\infty$ -adic numbers?

Posted by: Tom Leinster on August 16, 2008 10:26 PM | Permalink | Reply to this

Re: Lautman on Reciprocity

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Not really anything to add to what John and James said, except to expand a little on James’s remark on the justification of the phrase “at $\infty$ ”.

If we look at valuations on $\mathbb{F}_q(t)$ , then for primes $P$ of degree $n$ in $\mathbb{F}_q[t]$ , we factorise general elements $x$ , extract the power $\nu_P(x)$ of $P$ occuring in the factorisation of $x$ , and assign $x$ the absolute value $q^{-n\nu_P(x)}$ . (These are the ‘ $P$ -adic’ absolute values.)

If, instead of a monic irreducible $P\in \mathbb{F}_q[t]$ we pick the rational function $\theta=\frac{1}{t}$ , then a rational function $\frac{a_m t^m+\dots+a_0}{b_n t^n+\dots+b_0}$ gets re-expressed in terms of $\theta$ as $\theta^{n-m}\frac{a_m+\dots+a_0\theta^m}{b_n+\dots+b_0\theta^n}$ . The rational function on the right has no zeros or poles in $\theta$ , so, pretending that $\theta$ is prime, it appears $n-m$ times in the ‘factorisation’ of $z$ . Following the recipe above, since $\theta$ has degree $-1$ , we get an absolute value of $z$ given by $q^{m-n}$ .

However, this is just the absolute value $\vert z\vert_\infty$ on $\mathbb{F}_q(t)$ given by assigning each $z\in\mathbb{F}_q[t]$ the absolute value $Card(\mathbb{F}_q[t]/(z))$ and extending to the field by taking ratios of the valuations of numerator and denominator.

But the latter formula just gives the archimedean absolute value on $\mathbb{Q}$ .

Posted by: Tim Silverman on August 17, 2008 3:14 PM | Permalink | Reply to this

Re: Lautman on Reciprocity

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Tom wrote:

How can real/complex analysis be thought of as being like a theory of $\infty$ -adic numbers?

I’m no expert on this stuff, but there’s been a long-standing battle between ‘ideals’ and ‘valuations’ in number theory, which some trace back to a battle between Kummer and Dedekind. A valuation is a generalization of the usual ‘absolute value’ of a rational number. And the answer to your question seems to rely on thinking of primes as giving, not ideals in the ring of integers, but valuations on the field of rationals.

If we complete the rationals using one of these funny valuations we get the $p$ -adics. But, the ordinary absolute value is also a valuation — and if we complete the rationals using that, we get the real numbers.

A common feature of these completions of the rationals is that they’re complete metric spaces: the valuation gives a notion of distance, and all Cauchy sequences converge. So, we can do analysis in all these completions.

There’s a lot of work in number theory where you try to treat all valuations on an algebraic number field on an equal footing, and study them all simultaneously. One popular way to do this uses ‘adeles’. You can see this idea explained here.

Trying to take seriously the idea that the ordinary absolute value is a deviant ‘prime’ leads us into the mysteries of the real prime, which I would really love to know better.

Posted by: John Baez on August 17, 2008 12:07 AM | Permalink | Reply to this

Re: Lautman on Reciprocity

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Regarding #1 of Tom’s questions, I guess I wasn’t being careful about distinguishing $p$ -adic analysis and $p$ -adic algebra, or real analysis and real algebra. But I might also say that if you’re just doing algebra over the $p$ -adics (or reals or complexes), then you’re not really using all of the $p$ -adics, and you might as well not use them at all. For instance, if you’re doing Galois theory, you might as well work with the algebraic closure of the rationals in the $p$ -adics (or reals or complexes). In fact, all three of these have names—the strict henselization of $\mathbf{Q}$ at $p$ , the real closure of $\mathbf{Q}$ , and the algebraic closure of $\mathbf{Q}$ —and people use them all the time.

I’m not sure I completely believe that myself, though. I might have something more to say if you give me a better idea of what the difference between analysis and algebra means to you. Or it could just be that the slogan could be better.

Regarding #2, I don’t have anything to add to what John said other than to point out that “infinity” is just a meaningless label, probably meant to remind us of the point at infinity of the Riemann sphere, which is the only absolute value on the field $\mathbf{C}(z)$ which doesn’t come from a maximal ideal of $\mathbf{C}[z]$ . (Just as with the usual absolute value, $\mathbf{Q}$ , and maximal ideals of $\mathbf{Z}$ .

Posted by: James on August 17, 2008 3:11 AM | Permalink | Reply to this

Re: Lautman on Reciprocity

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David wrote down formulas for quadratic reciprocity:

$\left( \frac{p}{q} \right) \left( \frac{q}{p} \right) = (-1)^{\left(\frac{p-1}{2}\right) \left( \frac{q-1}{2}\right)}$

and the modular transformation property of the theta function:

$\theta (1/ \tau) = \sqrt{\tau} \theta (\tau)$

Lautman seems to suggest you can derive the former from the latter. Well, actually nothing quite so precise: he just says the latter formula is “the mainspring of the transcendental proof of quadratic reciprocity.” But still: is this really true?

I know that

$\theta (1/ \tau) = \sqrt{\tau} \theta (\tau)$

gives the functional equation for the Riemann zeta function — a kind of symmetry about the magic line $Re(s) = 1/2$ where we think all the nontrivial zeros lie. And, I know there are lots of relations between quadratic reciprocity (and its numerous generalizations) and the zeta functions and $L$ -functions of number fields.

But, I don’t know a direct link between the two equations above. So, either I’m missing something (which is quite possible), or the striking resemblance between these two equations is a bit misleading.

Posted by: John Baez on August 17, 2008 12:36 AM | Permalink | Reply to this

Re: Lautman on Reciprocity

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Let’s see. Lautman claims to be taking this from your friend Hecke, Theorie der algebraiechen Zahlen 1923.

He says that the limit as $\tau \to 0$ of

$\sqrt{\tau} \theta(\tau + 2 i r),$

with $r$ rational, is, up to factors, the Gauss sum $C(-r)$ .

The transformation property of the theta function now tells us that there is a reciprocal relation between $C(r)$ and $C(-1/4 r)$ , from which ordinary quadratic reciprocity follows.

Posted by: David Corfield on August 17, 2008 3:31 PM | Permalink | Reply to this

Re: Lautman on Reciprocity

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Okay, cool! I’ll have to check this someday.

One more piece in the magnificent jigsaw puzzle.

Posted by: John Baez on August 17, 2008 11:14 PM | Permalink | Reply to this

Re: Lautman on Reciprocity

I guess I’m wondering if Lautman had a sense that there was an answer to what John asked for here:

I wish you could summarize your explanation of quadratic reciprocity in one sentence. I’m missing the forest – there are too many trees. I’d like to hear just the moral essence, disregarding all technical details. Something like “??? is periodic because some deck transformation of ??? satisfies ???”

Or, it’s possible that what’s important in your outlook is not quadratic reciprocity but something else, of which quadratic reciprocity is just a spinoff. Then the moral essence of this ‘something else’ is what I’d like to hear.

Is there some primordial idea realising itself in quadratic reciprocity, and elsewhere?

Posted by: David Corfield on August 17, 2008 3:40 PM | Permalink | Reply to this

Re: Lautman on Reciprocity

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David wrote:

Is there some primordial idea realising itself in quadratic reciprocity, and elsewhere?

It’s possible that sometimes a snappy-looking result like quadratic reciprocity may only find a clear conceptual explanation in a much larger framework. It’s an interesting question, how much we need to ‘back off’ from a given result before we see it clearly.

The next time I teach quadratic reciprocity to undergrads, I’ll know much nicer proofs than I used last time, thanks to our discussions here. But these proofs still feel like ‘tricks’ to me. To get a more conceptual understanding of quadratic reciprocity, I feel compelled to do what everyone else does, and learn the vastly more general Artin reciprocity law as part of class field theory.

As far as I can tell, class field theory lets us think of algebraic number fields as being analogous to fields of functions on Riemann surfaces, and thus capable of being studied using homology and cohomology. For Riemann surfaces, homology and cohomology are related by Poincaré duality. Something similar for algebraic number fields gives rise to Artin reciprocity!

Hmm. I need to write a This Week’s Finds and explain this stuff before I completely forget what little I ever knew about it! It’s been a long time since I’ve thought about it. For now, let me say something just a little more precise. It’ll probably be a bit screwed up.

Line bundles over a Riemann surface are classified by elements of a certain first cohomology group. But thanks to Poincaré duality, this is isomorphic to a certain first homology group. So, it can be described in terms of the abelianization of the fundamental group of our Riemann surface.

(All this is a very jargonesque way of summarizing some clear mental pictures, but never mind.)

If we wave the magic wand of analogy and translate all these ideas into the language of algebraic number fields, our ‘line bundles’ become ‘invertible projective modules over the ring of algebraic integers’, which are just ‘ideals’. So, our first cohomology group becomes an ‘ideal class group’ — or better yet, an ‘idèle class group’.

Meanwhile, our ‘fundamental group’ becomes an ‘absolute Galois group’, and our ‘abelianized fundamental group’ becomes the ‘abelianized absolute Galois group’ or ‘Galois group of the maximal abelian exension’.

So, Artin reciprocity is an isomorphism between something like an idele class group, and something like the Galois group of the maximal abelian extension.

But, it’s not that simple, because it’s not just about a single algebraic number field: it’s about two, one an abelian extension of the other. This is like having two Riemann surfaces, one a branched cover of the other, with the group of deck transformations being abelian.

So, if you look up the statement of Artin reciprocity, you’ll see something like this: an isomorphism between the Galois group $G(L/K)$ — where $K$ is an algebraic number field and $L$ is an abelian extension of $K$ — and some other group built from $L$ and $K$ , but using idèles.

Someday before I die I’d like to understand all this well enough to clearly see how quadratic reciprocity is a spinoff of Artin reciprocity, which is itself a close relative of Poincaré duality.

But, I’m nowhere near that point yet.

Posted by: John Baez on August 18, 2008 1:00 AM | Permalink | Reply to this

Re: Lautman on Reciprocity

JB: Someday before I die I’d like to understand all this well enough to clearly see how quadratic reciprocity is a spinoff of Artin reciprocity, …

The issue seems nearly counter-intuitive.

Artin reciprocity and Mersenne primes
H.W. Lenstra, Jr., P. Stevenhagen

“Artin’s reciprocity law does not exhibit any symmetry that would justify the term “reciprocity”. The name derives from the fact that it extends the quadratic reciprocity law, and that its
generalization to number fields extends similar “higher power” reciprocity laws.

Cyclotomic extensions …
Artin’s reciprocity law over Q generalizes the quadratic reciprocity law, and it may be thought that its mysteries lie deeper. Quite the opposite is true: the added generality is the first step on the way to a natural proof. It depends on the study of cyclotomic extensions.”

Posted by: Stephen Harris on August 18, 2008 3:47 PM | Permalink | Reply to this

Re: Lautman on Reciprocity

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The quadratic reciprocity law can be reformulated using the Hilbert symbol.

Consider the set of valuations on $\mathbb{Z}$ : $V=\{prime numbers\}\cup \{\infty\}$ , and write eventually $\mathbb{Q}_\infty=\mathbb{R}$ .

For $v\in V$ , given two non zero numbers $a$ and $b$ in $\mathbb{Q}_v$ , define the Hilbert symbol of $(a,b)$ at $v$ to be:

$(a,b)_v=1$ if the equation $z^2-a x^2-b y^2=0$ has a non trivial solution in $(x,y,z)$ in $\mathbb{Q}^3_v$ , and

$(a,b)_v=-1$ otherwise.

This defines a pairing

$\mathbb{Q}^\times_v/\mathbb{Q}^{\times\, 2}_v\times\mathbb{Q}^\times_v/\mathbb{Q}^{\times\, 2}_v\to\{\pm 1\}.$

There is a close relationship between the Hilbert symbol and the Legendre symbol, and the quadratic reciprocity law implies (and is is essentially equivalent to):

Hilbert’s Theorem. We have $(a,b)_v=1$ , except possibliy for a finite number of valuations $v$ , and we have the equality $\prod_{v\in V}(a,b)_v=1$ .

I recall this version of the quadratic reciprocity law because the Hilbert symbol has a very nice “geometric” interpretation. A first idea is that we can think of the scheme $X=Spec(\mathbb{Z})$ as a kind of $3$ -dimensional sphere, at least as far as etale cohomology (up to $2$ -torsion) is concerned: etale cohomology $H^*(X,\mathbb{G}_m)$ is endowed with a cup product which defines a Poincare duality and makes these cohomology groups formally look like the cohomology of $S^3$ .

If you think of a prime number $p$ as the scheme $K_p=\Spec(\mathbb{Z}/p\mathbb{Z})$ , then $K_p$ is a kind of $1$ -dimensional manifold in $X$ and the (etale) $\pi_1$ of $K_p$ is $\hat{\mathbb{Z}}$ (i.e. something very similar to $\pi_1$ of a circle $S^1$ ). In other words, we can think of $K_p$ as a knot in the “ $3$ -sphere” $X$ . It is then possible to use the Poincare duality on etale cohomology of $X$ to define the linking numbers of two nots $K_p$ and $K_q$ . This linking number will live in $H^3(X,\mathbb{G}_m)=\mathbb{Q}/\mathbb{Z}$ and it can be shown that the linking number of $p$ and $q$ is essentially the collection of the Hilbert symbols $(p,q)_v$ . This point of view is followed seriously in

J.-L. Waldspurger, Entrelacements sur $Spec(Z)$ , Bull. Sci. Math. (2) 100 (1976), no. 2, 113-139.

This might be seen as a rather reenforcement of Lautman’s link between quadratic reciprocity law and Poincare duality.

The analogy “primes vs knots” is part of folklore. For example, Iwasawa Theory is a kind of arithmetical development of the theory of Alexander polynomials associated to knots. The last years have seen this kind of idea be pushed further. For example, the linking number of $2$ knots can be generalized: it is possible to define the linking number of $n$ knots. This is how you get the borromean link: this is the data of three knots $k_1$ , $k_2$ , $k_3$ , such that the linking numbers $(k_i,k_j)$ are trivial for $i\neq j$ , but such that that $(k_1,k_2,k_3)\neq 0$ . These higher linking numbers are known as Milnor numbers, and it is striking to see that the same method allows you to get analogous constructions for prime numbers. If you have a look at

K. Morishita, On certain analogies between knots and primes, J. Reine Angew. Math. 550 (2002), 141-167.

you may meet your first borromean triple of primes!

To come back to “reciprocity and duality”, it is clear that there is a strong link. The zeta function (or, more generaly, the L-functions) associated to schemes over finite fields are indeed good examples: these can be associated to (Weil) cohomology theories, and the good properties of these cohomologies (like Poincare duality) imply a good behaviour of the zeta functions (like the functional equation, which is certainly an instance of “reciprocity”). This principle is one of the main intuition in the construction and study of any kind of motives (in the sense of Grothendieck, Beilinson, Voevodsky…) and of non commutative geometry (Connes, Kontsevich…). All these theories want to produce abstract functorial invariants (cohomologies) which aim is to interpret geometrically invariants defined by analysis (zeta functions, including the Riemann one). There is certainly no wish to reject analysis there! There is even a new analysis associated to motives (developed by Kontsevich, Katz, Loezer, Denef…) which is developed for itself (and have very nice and strong applications to model theory).

Posted by: Denis-Charles Cisinski on August 20, 2008 2:14 AM | Permalink | Reply to this

Re: Lautman on Reciprocity

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Wow! Great! I’ve been trying to understand this stuff for quite a while. I may be forgetting it faster than I’m learning it. You just reminded me of the following passage from week257, on how quadratic reciprocity is related to the symmetry of the linking number of two knots. I think I hadn’t fully integrated how this relates to the Poincaré duality viewpoint on Artin reciprocity! Now I understand it a bit better!

Anyway, I quizzed Minhyong about one of the big mysteries that’s been puzzling me lately. I want to know why the integers resemble a 3-dimensional space - and how prime numbers give something like “knots” in this space!

I made a small step toward explaining this back in “week205”. There I sketched one of the basic ideas of algebraic geometry: every commutative ring, for example the integers or the integers modulo p, has a kind of space associated to it, called its “spectrum”. We can think of elements of the commutative ring as functions on this space. I also explained why the process turning a commutative ring into a space is “contravariant”. This implies that the obvious map from the integers to the integers modulo p

Z → Z/p

gives rise to a map going the other way between spectra:

Spec(Z/p) → Spec(Z)

In “week218” I reviewed an old argument saying that Spec(Z) is analogous to the complex plane, and that Spec(Z/p) is analogous to a point. From this viewpoint, primes gives something like points in a plane.

However, from a different viewpoint, primes give something like circles in a 3d space!

The easy thing to see is how Spec(Z/p) acts more like a circle than a point. In particular, its “étale topology” resembles the topology of a circle. Oversimplifying a bit, the reason is that just as the circle has one n-fold cover for each integer n > 0, so too does Spec(Z/p). To get the n-fold cover of the circle, you just wrap it around itself n times. To get the n-fold cover of Spec(Z/p), we take the spectrum of the field with pⁿ elements, which is called F_pⁿ. Z/p sits inside this larger field:

Z/p → F_pⁿ

so by the contravariance I mentioned, we get a map going the other way:

Spec(F_pⁿ) → Spec(Z/p)

which is our n-fold cover.

I should explain this in much more detail someday - it involves the relation between étale cohomology, Galois theory and covering spaces. I began tackling this in “week213”, but I have a long way to go.

Anyway, the basic idea here is that each prime p gives a “circle” Spec(Z/p) sitting inside Spec(Z). But the really nonobvious part is that according to étale cohomology, Spec(Z) is 3-dimensional - and the different circles corresponding to different primes are linked!

I’ve been fascinated by this ever since I heard about it, but I got even more interested when I saw a draft of a paper by Kapranov and Smirnov. I got it from Thomas Riepe, who got it from Yuri Manin. There’s a version on the web:

7) M. Kapranov and A. Smirnov, Cohomology determinants and reciprocity laws: number field case, available at http://wwwhomes.uni-bielefeld.de/triepe/F1.html

It begins:

The analogies between number fields and function fields have been a long-time source of inspiration in arithmetic. However, one of the most intriguing problems in this approach, namely the problem of the absolute point, is still far from being satisfactorialy understood. The scheme Spec(Z), the final object in the category of schemes, has dimension 1 with respect to the Zariski topology and at least 3 with respect to the etale topology. This has generated a long-standing desire to introduce a more mythical object P, the “absolute point”, with a natural morphism X → P given for any arithmetic scheme X […]

Even though I don’t fully understand this, I can tell something big is afoot here. I think they’re saying that because Spec(Z) is so big and fancy from the viewpoint of étale topology, there should be some mysterious kind of “point” that’s much smaller than Spec(Z) - the “absolute point”.

Anyway, in this paper the authors explain how the Legendre symbol of primes is analogous to the linking number of knots.

The Legendre symbol depends on two primes: it’s 1 or -1 depending on whether or not the first is a square modulo the second. The linking number depends on two knots: it says how many times the first winds around the second.

The linking number stays the same when you switch the two knots. The Legendre symbol has a subtler symmetry when you switch the two primes: this symmetry is called quadratic reciprocity, and it has lots of proofs, starting with a bunch by Gauss - all a bit tricky.

I’d feel very happy if I truly understood why quadratic reciprocity reduces to the symmetry of the linking number when we think of primes as analogous to knots. Unfortunately, I’ll need to think a lot more before I really get the idea. I got into number theory late in life, so I’m pretty slow at it.

This paper studies subtler ways in which primes can be “linked”:

8) Masanori Morishita, Milnor invariants and Massey products for prime numbers, Compositio Mathematica 140 (2004), 69-83.

You may know the Borromean rings, a design where no two rings are linked in isolation, but all three are when taken together. Here the linking numbers are zero, but the linking can be detected by something called the “Massey triple product”. Morishita generalizes this to primes!

But I want to understand the basics…

The secret 3-dimensional nature of the integers and certain other “rings of algebraic integers” seems to go back at least to the work of Artin and Verdier:

9) Michael Artin and Jean-Louis Verdier, Seminar on étale cohomology of number fields, Woods Hole, 1964.

You can see it clearly here, starting in section 2:

10) Barry Mazur, Notes on the étale cohomology of number fields, Annales Scientifiques de l’Ecole Normale Superieure Ser. 4, 6 (1973), 521-552. Also available at http://www.numdam.org/numdam-bin/fitem?id=ASENS_1973_4_6_4_521_0

By now, a big “dictionary” relating knots to primes has been developed by Kapranov, Mazur, Morishita, and Reznikov. This seems like a readable introduction:

11) Adam S. Sikora, Analogies between group actions on 3-manifolds and number fields, available as arXiv:math/0107210.

I need to study it. These might also be good - I haven’t looked at them yet:

12) Masanori Morishita, On certain analogies between knots and primes, J. Reine Angew. Math. 550 (2002), 141-167.

Masanori Morishita, On analogies between knots and primes, Sugaku 58 (2006), 40-63.

Posted by: John Baez on August 20, 2008 7:24 AM | Permalink | Reply to this

Re: Lautman on Reciprocity

Lieven Le Bruyn has made a single file of the Kapranov-Smirnov paper.

Posted by: David Corfield on August 20, 2008 12:14 PM | Permalink | Reply to this

Re: Lautman on Reciprocity

A new expository article: “Analogies between Knots and Primes, 3-Manifolds and Number Rings”.

Posted by: Thomas on April 23, 2009 7:43 AM | Permalink | Reply to this

Re: Lautman on Reciprocity

BTW, where can one read something about the theme of these Atiyah lectures?:

Posted by: Thomas on April 23, 2009 8:12 AM | Permalink | Reply to this

Re: Lautman on Reciprocity

That link doesn’t work. But you can see what Atiyah’s been talking about in the abstact for From Finite Fields to Feynman diagrams? (part 3):

Vector Bundles over algebraic curves are classified by moduli spaces. The rank one case is classical and leads to tori, but the higher rank case is much harder and even calculating the homology groups is a challenge. One approach is via finite fields and counting rational points, while another is by differential geometry inspired by physics. Comparison between the two methods raises intriguing questions about algebraic analogues of Feynman integrals.

Posted by: David Corfield on April 23, 2009 4:28 PM | Permalink | Reply to this

Re: Lautman on Reciprocity

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One small quibble: the cup-product on $H^*(X,G_m)$ takes values in $H^*(X,G_m\otimes G_m)$ , which to my knowledge doesn’t have a trace map. To get the usual duality and reciprocity, you need to cup $H^2(X,Z)$ with $H^1(X,G_m)$ , which lands in $H^3(X,G_m)$ which, as you say, is canonically $Q/Z$ . Then you observe that $Q$ has no higher cohomology in the etale topology, so $H^2(X,Z)$ agrees with $H^1(X,Q/Z)$ , which is the dual of the abelianization of the fundamental group. Therefore the abelianization of the fundamental group is essentially the same as the Picard group $H^1(X,G_m)$ .

Posted by: James on August 20, 2008 10:24 AM | Permalink | Reply to this

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