Last week we visited the geysers of Saturn's moon Enceladus. Afterwards, George Musser pointed me to an article on this subject by Carolyn Porco, leader of the imaging team for the Cassini-Huygens mission - the team that's been taking the photos I showed you. It's a great article, leading up to some intriguing theories about what powers these geysers:
1) Carolyn Porco, Enceladus: secrets of Saturn's strangest moon, Scientific American, November 2008, available at http://www.sciam.com/article.cfm?id=enceladus-secrets
And it's free online! - at least for now. I've criticized the Scientific American before here, but if they keep coming out with articles like this, I'll change my tune. For one thing, it's well-written:
There is obviously a tale writ on the countenance of this little moon that tells of dramatic events in its past, but its present, we were about to find out, is more stunning by far. In its excursion over the outskirts of the south polar terrain, Cassini's dust analyzer picked up tiny particles, apparently coming from the region of the tiger stripes. Two other instruments detected water vapor, and one of them delivered the signature of carbon dioxide, nitrogen and methane. Cassini had passed through a tenuous cloud.
What is more, the thermal infrared imager sensed elevated temperatures along the fractures - possibly as high as 180 kelvins, well above the 70 kelvins that would be expected from simple heating by sunlight. These locales pump out an extraordinary 60 watts per square meter, many times more than the 2.5 watts per square meter of heat arising from Yellowstone's geothermal area. And smaller patches of surface, beyond the resolving power of the infrared instrument, could be even hotter.
For another, it tackles a fascinating mystery. Where does all this power come from? The geysers near the south pole of Enceladus emit about 6 gigawatts of heat. Enceladus is too small to have that much radioactive heating at its core - only about 0.3 gigawatts, probably. The rest must come from tidal heating. This happens when stuff sloshes back and forth in a changing gravitional field: friction converts this motion to heat.
So, what causes tides on Enceladus? It may be important that Enceladus has a 2:1 resonance with Dione: it orbits Saturn twice for each orbit of that larger moon. This sort of resonance is known to cause tidal heating. For example, in "week269", I showed you how Jupiter's moon Io is locked in resonances with Europa and Ganymede. The resulting tidal heat powers its mighty volcanos.
Unfortunately, the resonance with Dione doesn't seem powerful enough to produce the heat we see on Enceladus. Unless something funny is going on, there should only be 0.1 gigawatts of tidal heating - not nearly enough! At least that's what Porco estimated in 2006:
2) Carolyn Porco et al, Cassini observes the active south pole of Enceladus, Science 311 (2006), 1393-1401.
So, we need to dream up a more complicated story.
Here's one: there could be a kind of slow cycle where the orbit of Enceladus gets more eccentric, tidal heating increases, ice beneath its surface melts, more sloshing water causes more tidal heating, and then the release of heat energy damps its eccentric orbit, until it freezes solid and the whole cycle starts over. We could be near the end of such a cycle right now.
Here's another: maybe Enceladus has an sea of liquid water under the frozen surface of its south pole. With enough water sloshing around, there could be a lot more tidal heating than you'd naively guess... and this heating, in turn, could keep the water liquid. The fun thing about this second scenario is that a permanent liquid ocean on Enceladus raises the possibility of life!
Nobody knows for sure what's going on - but Carolyn Porco examines the options in a clear and engaging way. If you like celestial mechanics, also try this paper:
3) Jennifer Meyer, Jack Wisdom, Tidal heating in Enceladus, Icarus 188 (2007), 535-539. Also available at http://groups.csail.mit.edu/mac/users/wisdom/meyerwisdom1.pdf
I wrote about Jack Wisdom's work back in "week107" - it's fascinating stuff. He knows a lot about resonances. For related work on the Jupiter-Saturn resonance, the Neptune-Pluto resonance, and the math of continued fractions, also try the addenda to "week222".
Next I'd like to give you a quick trip through the Earth's history. In "week196" we looked back into the deep past, all the way to the electroweak phase transition 10 picoseconds after the Big Bang. On the other hand, here:
4) John Baez, The end of the universe, http://math.ucr.edu/home/baez/end.html
you can zip forwards into the deep future - for example, 1019 years from now, when the galaxies boil off, shooting dead stars into the the vast night.
But now I'd like to zoom in closer to home and quickly tell the history of the Earth, focusing on an aspect you may never have thought about. You see, Kevin Kelly recently pointed me to this fascinating paper on "mineral evolution":
5) Robert M. Hazen, Dominic Papineau, Wouter Bleeker, Robert T. Downs, John M. Ferry, Timothy J. McCoy, Dmitri A. Sverjensky and Henxiong Yang, Mineral evolution, American Mineralogist 91 (2008), 1693-1720.
Ever since it was formed, the number of different minerals on Earth has kept going up — and ever since life ran wild, it's soared! Some examples are obvious: seashells become limestone, which gets squashed into marble. Some are less so: for example, there wasn't much clay before the advent of life.
Here's a timeline loosely taken from this paper:
Chondrites are stony meteorites that formed early in the history of the solar system. They're made of chondrules — millimeter-sized spheres of olivine, pyroxene and other minerals — together with nuggets called CAIs (calcium-aluminum rich inclusions) and other stuff. These chondrules began life as molten droplets back when the Sun was a T Tauri star, heated only by gravitational collapse.
This is the era when the disk of dust circling the early Sun started forming lumps. As these lumps collided, they got bigger and bigger, eventually forming the asteroids and planets we see today. Some of these proto-planets melted, letting heavier metals sink to their core while lighter material stayed on top. But then some crashed into each other, shattering and forming new kinds of meteorites: iron-nickel meteorites, and stony meteorites called achondrites. Thanks to radioactive dating of these, scientists claim a shockingly precise knowledge of when all this happened: sometime between 4.56 and 4.55 billion years ago.
The Earth's history is divided into four eons: the Hadean, Archean, Proterozoic, Phanerozoic. Back when I was a kid, the Cambrian era seemed really old - but that's just the start of the current eon, when multicellular life emerged: the Phanerozoic. We're digging much deeper now: the Phanerozoic will be end of today's story. The Hadean eon began with a bang: the event that formed the Moon around 4.55 billion years ago! What made the moon? The current most popular explanation is the "giant impact theory" - sometimes called the Big Splat Theory. Dana Mackenzie spends a lot of time writing about math, but he's also written a book about this:
6) Dana Mackenzie, The Big Splat, or How Our Moon Came To Be, Wiley, New York, 2003.
The idea is that another planet formed in one of the "Lagrange points" of Earth's orbit - a stable spot 60 degrees ahead or behind the Earth:
7) John Baez, Lagrange points, http://math.ucr.edu/home/baez/lagrange.html
But when this planet reached about the mass of Mars, it would no longer be stable in this location. So, it gradually drifted toward Earth, and eventually smacked right into us! The impact was incredibly energetic, melting the Earth's entire crust and outer mantle. The iron core of this other planet sank into Earth's core, while about 2% of the outer part formed an orbiting ring of debris. Within a century, about half of this ring formed the Moon we know and love.
It's an amazing story, but most of the evidence seems to support it. The early Moon is known to have been much closer to Earth than it is now - it's been receding ever since. For this and many other reasons, the giant impact theory is sufficiently solid that the hypothetical doomed planet that hit Earth has a name: Theia! You can even watch a simulation of it hitting Earth, produced by Robin Canup:
8) Dana Mackenzie, The Big Splat (animation), http://www.danamackenzie.com/big_splat_animation.htm
Let me quote Mackenzie on this:
The simulation shows the first twenty-four hours after the giant impact. It begins with Theia about to strike the Earth. After the impact, one hemisphere of the Earth is sheared off and flung into space. The remaining part of Earth is very lopsided, and sets up a "gravitational torque" on the debris. This boosts some of the debris into orbit. (Without such a boost, it would all simply fall back down again.)
Within a few hours, the debris has formed an "arm" that smashes spectacularly back into the Earth. This crash is nearly as explosive as the original impact! (The second explosion can be seen much more vividly in the video than in the still frames published in my book.) Notice how the temperature of the Earth has risen, from the blues and greens of the early frames to yellows and reds, indicating more than 2000 degrees Kelvin. Earth has literally become a blast furnace.
Robin Canup's simulation of Earth and Theia, 50 minutes after their initial collision.
As the fateful day continues, the debris gets more uniformly distributed in a disk around the Earth. Notice, though, that this disk is not stable like the rings of Saturn. It develops shock waves that whirl around the Earth, collecting material into spiral arms. According to Alastair Cameron, another modeler the giant impact, these spiral arms also play an important role in the development of the Moon, by "siphoning" debris up from lower orbits into higher ones. Scientists have estimated that a mass at least twice the present mass of the Moon had to be lifted beyond the "Roche limit," roughly twelve thousand miles or three Earth radii above the surface. Any debris that does not make it past the Roche limit will be torn apart by tidal forces, and cannot form a permanent moon.
This simulation stops after 24 hours, a long time before the disk of debris condenses into our Moon. The Moon was not formed in a day! However, it did form much more rapidly than you might expect; current estimates range from 1 to 100 years. This is astounding, compared to ordinary geological time scales. An entire new planet was born within the life span of a single human.
No rocks on Earth are known to survive from before 4.03 billion years ago, so the details of this time period are hotly debated. However, many igneous rocks, especially basalt, must have been formed at this time.
Even after the surface cooled enough to form a crust, volcanoes continued to release steam, carbon dioxide, and ammonia. This led to what is called the Earth's "second atmosphere". The "first atmosphere" was mainly hydrogen and helium; the second was mainly carbon dioxide and water vapor, with some nitrogen but almost no oxygen. This second atmosphere had about 100 times as much gas as today's "third atmosphere"!
As the Earth cooled, oceans formed, and much of the carbon dioxide dissolved into the seawater and later precipitated out as carbonates.
Between 4 and 3.8 billion years ago there was another scary time: the Late Heavy Bombardment. A lot of large craters on the Moon date to this period, so probably the Earth, Venus and Mars got hit too. Why so many impacts after a period of relative calm? One theory is that Jupiter and Saturn locked into their current 2:1 resonance at this time, causing a big shakedown in the asteroid belt and Kuiper belt. Wikipedia has a nice quick review of this and other theories:
9) Wikipedia, Late heavy bombardment, http://en.wikipedia.org/wiki/Late_heavy_bombardment
This time also marked the rise of "cratons". Cratons are a bit like small early "plates" in the sense of plate tectonics: they're ancient tightly-knit pieces of the earth's crust and mantle, many of which survive today. While most cratons only finished forming 2.7 billion years ago, nearly all started growing earlier, in the Eoarchean era.
Cratons are made largely of granitoids. Granitoids are more sophisticated igneous rocks than basalt. Modern granite is one of these. Granite is made in a variety of ways, for example by the remelting of sedimentary rock. Early granitoids were probably simpler.
In the Paleoarchean and Mesoarchean eras, plate tectonics as we know it began. A key aspect of this process is the recycling of the Earth's crust through subduction: oceanic plates slide under continental plates and get pushed down into the mantle. Another feature is underwater volcanism and hydrothermal activity.
The earliest hints of life include some banded iron formations that date back 3.85 billion years. The real fun starts with the rise of photosynthesis leading up to the Great Oxidation Event about 2.5 billion years ago - more on that later. But organisms from the domain Archea can do well in a wide variety of extreme environments without oxygen, and as their name suggests, many of these organisms are very ancient. These organisms gave rise to an active sulfur cycle and deposits of sulfate ores starting in the Paleoarchean era. They also made the atmosophere increasingly rich in methane throughout the Mesoarchean and Neoarchean. So, life was already beginning to affect mineral evolution.
The Archean eon ended and the Proterozoic began with the Great Oxidation Event 2.5 billion years ago. In this event, also known as the Oxygen Catastrophe, photosynthesis put enough oxygen into the atmosphere to make it lethal to most organisms of the time. Luckily evolution found a way out of this impasse. The oyxgen-rich atmosphere in turn led to a wide variety of new minerals.
In the Mesoproterozoic era, increased oxygen levels in the ocean put an end to many anoxic life forms. For example, around 1.85 billion years ago, banded iron formations suddenly ceased. The next gigayear was rather static and dull — if you're mainly interested in new minerals, that is. The term "intermediate ocean" means that during this period, the seawater contained a lot more oxygen than before, but still much less than today.
The Neoproterozoic era probably saw several Snowball Earth events: episodes of runaway glaciation during which most or all the Earth was covered with ice. Since ice reflects sunlight, making the Earth even colder, it's easy to guess how this runaway feedback might happen. The opposite sort of feedback is happening now, as melting ice makes the Earth darker and thus even warmer. The interesting questions are why this instability doesn't keep driving the Earth to extreme temperatures one way or another - and what stopped the Snowball Earth events back then!
Here's a currently popular answer to the second question. Ice sheets slow down the weathering of rock. Weathering of rock is one of the main long-term processes that use up atmospheric carbon dioxide, by converting it into various carbonate minerals. On the other hand, even on an ice-covered Earth, volcanic activity would keep putting carbon dioxide into the atmosphere. So, eventually carbon dioxide would build up, and the greenhouse effect would warm things up again. This process might be very dramatic, with perhaps as much as 13% of the atmosphere being carbon dioxide (350 times what we see today), and temperatures soaring to 50 Celsius! But the details are still the subject of much controversy.
At the end of these glacial cycles, it's believed that oxygen increased from 2% of the atmosphere to 15%. (Now it's 21%.) This may be why multi-celled oxygen-breathing organisms date back to this time. Others argue that the "freeze-fry" cycle imposed tremendous evolutionary pressure on life and led to the rise of multicellular organisms. Both these theories could be true.
During the glacial cycles, few new minerals were formed - unless you count ice. Afterwards, surface rocks were weathered in new ways involving oxidation.
The Phanerozoic eon, beginning with the Cambrian 540 million years ago, marks the rise of of life as we know it. During this time, sea life has given rise to extensive deposits of biominerals such as calcite, aragonite, dolomite, hydroxylapatite, and opal. There has also been increased production of clay and many different types of soil.
Next: Pontryagin duality! Like last week's math topic, I needed to learn more about this for my work on infinite-dimensional representations of 2-groups. And like last week's math topic, it involves a lot of analysis. But it also involves a lot of algebra and category theory.
You may know about Fourier series, which lets you take a sufficiently nice complex-valued function on the circle and write it like this:
f(x) = ∑k gk exp(ikx)
Here n ranges over all integers, so what you're really doing here is taking a function on the circle:
f: S1 → C
and expressing it in terms of a function on the integers:
g: Z → C
More precisely, any L2 function on the circle can be expressed this way for some L2 function on the integers - and conversely. In fact, if we normalize things right, the Fourier series gives a unitary isomorphism between the Hilbert spaces L2(S1) and L2(Z).
You may also know about the Fourier transform, which lets you take a sufficiently nice complex-valued function on the real line and write it like this:
f(x) = ∫ g(k) exp(ikx) dk
Here k also ranges over the real line, so what you're really doing is taking a function on the line:
f: R → C
and expressing it in terms of another function on the line:
g: R → C
In fact, any L2 function f: R → C can be expressed this way for some L2 function g: R → C. And if we normalize things right, the Fourier transform is a unitary isomorphism from L2(R) to itself.
Pontryagin duality is the grand generalization of these two examples! Any locally compact Hausdorff abelian group A has a "dual" A* consisting of all continuous homomorphisms from A to S1. The dual is again a locally compact Hausdorff abelian group - or "LCA group", for short. When you take duals twice, you get back where you started. And the Fourier transform gives a unitary isomorphism between the Hilbert spaces L2(A) and L2(A*).
It's fun to take the Pontryagin duals of specific groups, or specific classes of groups, and see what we get. We've already seen that the dual of S1 is Z, the dual of Z is S1, and the dual of R is R. More generally the dual of the n-dimensional torus is Zn, and vice versa, while the dual of Rn is isomorphic to Rn. What can we glean from these examples?
Well, any discrete abelian group is an LCA group - a good example is Zn. So is any compact Hausdorff abelian group - a good example is the n-dimensional torus. And there's a nice general theorem saying that the dual of any group of the first kind is a group of the second kind, and vice versa!
In particular, if we have an abelian group that's both compact and discrete, its dual must be too. But the only abelian groups like this are the finite abelian groups - products of finite cyclic groups Z/n. So, this collection of groups is closed under Pontryagin duality!
In fact, it's easy to see that for any finite abelian group, A* is isomorphic to A. But not canonically! To get a canonical isomorphism we need to take duals twice: for any LCA group, we get a canonical isomorphism between A and A**. This should remind you of duality for finite-dimensional vector spaces - another famous collection of LCA groups that's closed under Pontryagin duality.
You can take any collection of LCA groups, stare at it through the looking-glass of Pontryagin duality, and see what it looks like. I've mentioned a few examples so far:
A is compact iff A* is discrete.
A is finite iff A* is finite.
A is a finite-dimensional vector space iff A* is a finite-dimensional vector space.
Here are some fancier ones:
A is torsion-free and discrete iff A* is connected and compact.
A is compact and metrizable iff A* is countable.
A is a Lie group iff A* has finite rank.
A is metrizable iff A* is σ-compact.
A is second countable iff A* is second countable.
If you know more snappy results like this, tell me! I'm collecting them - they're sort of addictive.
Because Pontryagin duality turns compact LCA groups into discrete ones - and vice versa - we can use it to turn some topology questions into algebra questions, and vice versa. After all, a discrete abelian group has no more structure than an "abstract" abelian group - one without a topology!
Sometimes this change of viewpoint helps, but sometimes it merely reveals how hard a problem really is.
For example, here's an innocent-sounding question: what are the compact path-connected LCA groups? The obvious example is the circle. More generally, we could take any product of circles - even an infinite product. Are there any others?
It turns out that this question cannot be settled by Zermelo-Fraenkel set theory together with the axiom of choice!
Here's why. An LCA group is compact and path-connected iff its dual is a "Whitehead group". What's that? It's an abelian group A such any short exact sequence of abelian groups like this splits:
0 → Z → B → A → 0
where Z is the integers and B is any abelian group.
We call this sort of short exact sequence an "extension of A by Z". So, if you want to show off your sophistication, you can say that A is a Whitehead group if "Ext(A,Z) = 0".
The obvious examples of Whitehead groups are free abelian groups. Indeed, these are precisely the guys whose Pontryagin dual is a product of circles! So the question is: are there any others? Or is every Whitehead group a free abelian group?
This is a famous old problem, called the Whitehead problem:
10) Wikipedia, Whitehead problem, http://en.wikipedia.org/wiki/Whitehead_problem
In 1971, the logician Saharon Shelah showed the answer to this problem was undecidable using the axioms of ZFC! This was one of the first problems mathematicians really cared about that turned out to be undecidable.
If you want an easy introduction to Pontryagin duality and the structure of LCA groups, you can't beat this:
10) Sidney A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, Cambridge U. Press, Cambridge, 1977.
This classic treatment is still great, too:
14) Lev S. Pontrjagin, Topological Groups, Princeton University Press, Princeton, 1939.
To dig deeper, you need to read this - it's a real mine of information:
15) E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer, Berlin, 1979.
This book has a lot of interesting newer results:
16) David L. Armacost, The Structure of Locally Compact Abelian Groups, Dekker, New York, 1981.
In particular, this is where I learned about path-connected LCA groups and the Whitehead problem.
I'd like to dedicate this issue of This Week's Finds to my father, Peter Baez, who died yesterday around midnight at the age of 87. His health had been failing for a long time, so this did not come as a shock. It's a curious coincidence that I was already writing an issue about minerals, since my dad majored in chemistry and returned to school for a master's in soil science after serving in the Army in World War II. After that he worked in the Blackfeet Nation in Browning Montana, riding around in a jeep, digging up soil samples, and testing them back at the lab for the Army Corps of Engineers. When he found "medicine wheels" - stone circles laid down by the native Americans for ritual reasons - he would report them to his friend the archeologist Tom Kehoe. Later he moved to California, became an editor for the Forest Service, and met my mother.
He got me interested in science at an early age because he was always taking me to museums, bringing me books from the public library, and so on. As a little kid, when I spilled something, he'd say "So you don't believe in the law of gravity?" He liked to joke around. Whenever I said an ungrammatical sentence, he'd tease me for it. "I'm not that hungry." "What do you mean? You're not how hungry?"
I learned a lot of math, physics and chemistry from his 1947 edition of the CRC Handbook of Chemistry and Physics - an edition so old that it listed "mesothorium" among the radioactive isotopes. He brought home the book "From Frege to Gödel" - a sourcebook in mathematical logic - because it was in the math section of the library and he misread "Gödel" as "Googol": he knew I liked large numbers! I didn't understand much of it, but it had a big effect on me.
I owe a lot to him.
Addenda: I thank Michael Barr, Kevin Buzzard and Mike Stay for some interesting comments. Mike Stay pointed out an interesting book on how humans may affect the future of mineral evolution:
17) Jan Zalasiewicz, The Earth After Us: What Legacy Will Humans Leave in the Rocks?, Oxford University Press, Oxford, 2009.
It's not 2009 yet, but the best books about the future are actually published in the future. Here's a quote:
The surface of the Earth is no place to preserve deep history. This is in spite of - and in large part because of - the many events that have taken place on it. The surface of the future Earth, one hundred million years now, will not have preserved evidence of contemporary human activity. One can be quite categorical about this. Whatever arrangement of oceans and continents, or whatever state of cool or warmth will exist then, the Earth's surface will have been wiped clean of human traces.Kevin Buzzard had some interesting comments on Pontryagin duality in number theory:
Thus, one hundred million years from now, nothing will be left of our contemporary human empire at the Earth's surface. Our planet is too active, its surface too energetic, too abrasive, too corrosive, to allow even (say) the Egyptian Pyramids to exist for even a hundredth of that time. Leave a building carved out of solid diamond - were it even to be as big as the Ritz - exposed to the elements for that long and it would be worn away quite inexorably.
So there will be no corroded cities amid the jungle that will, then, cover most of the land surface, no skyscraper remains akin to some future Angkor Wat for future archaeologists to pore over. Structures such as those might survive at the surface for thousands of years, but not for many millions.
I don't know any more general theorems of the form "G has X iff its dual has Y" but, since lecturing on Tate's thesis, I learnt some more nice examples of Pontrjagin duals.Michael Barr wrote:
As you know well, number theorists like to complete fields with respect to norms. The rational numbers are too rich to understand completely, so we choose a norm on them, and then we complete with respect to that norm, and we get either the reals, or, a couple of thousand years later, the p-adics. Now a complete field is a much better gadget to have because there's a chance we'll be able to do analysis on this field. Indeed, for example, it's possible to set up a theory of Banach spaces etc for any complete field, and this isn't just for fun — e.g. Serre showed in the 1960s how to simplify some of Dwork's work on zeta functions of hypersurfaces using standard theorems of analysis of Banach spaces, applied to Banach spaces over the p-adic numbers, and there are oodles of other examples within number theory (leading up to an entire "p-adic Langlands programme" nowadays). But ideally, as well as continuity and differentiation etc, it's nice to be able to do integration as well, and for that you might need a measure, like Haar measure for example, and if you want to use Haar measure then you want your complete field to be locally compact too.
Now, perhaps surprisingly, Weil (I think it was Weil; it might have been earlier though and perhaps I'm doing someone a disservice) managed to completely classify normed fields which were both complete and locally compact. They are: the reals and finite field extensions, the p-adic numbers and finite field extensions, and the fields Fp((t)) and finite field extensions. (Given a complete normed field, there's a unique extension of the norm to any finite field extension and the extension is still complete — for the same sorts of reasons that there's only one vector space norm on Rn up to equivalence and it's complete.)
Tate in his thesis proves the following result (it's not too hard but it's crucial for him): if K is a complete locally compact normed field (considered as a locally compact abelian group under addition), then K is isomorphic to its own Pontrjagin dual. The isomorphism is non-canonical because you have to decide where 1 goes, but after you've made that decision there's a unique natural topological and algebraic isomorphism. So R and C are their own Pontrjagin duals — but the p-adic numbers are also self-dual in this way.
One of the reasons (perhaps historically the main reason) that number theorists are interested in complete fields is that given a more "global" object, like a number field (a finite extension of the rationals), one way of understanding it is by understanding its completions. Before I lectured my class on Tate's thesis, if someone had asked me to motivate the definition of the adeles, I would have said that to study a number field k it's easiest to think about it locally, so we complete with respect to a norm, but we can't choose a natural norm, so we choose all of them, and then we "multiply them all together" so we can get to them all at once. This is not really an ideal answer.
But here's a completely different way of motivating them. Classically, people interested in automorphic forms would typically consider functions on a Lie group G which were invariant, or transformed well, under a well-chosen discrete subgroup: for example one might want to consider smooth functions on R satisfying f(x)=f(x+1) — a very interesting class of functions — or perhaps functions on GL2(R) which are invariant under GL2(Z) (and now you're well on the way to inventing/discovering the theory of modular forms). In fact Z lives in R very nicely:
0 → Z → R→ R/Z → 0
and furthermore there's a bit of magic in this picture: it's self-dual with respect to Pontrjagin duality!
But Z is awkward. It's not a field. You really begin to see the awkwardness if you're Hecke in the 1930s trying to figure out what the correct notion of a Dirichlet character is when working with the integers not of Q but of a finite extension of Q. The problem is that the integers in a general number field aren't a principal ideal domain so it's sometimes hard to get your hands on "local information": prime ideals aren't in general principal so you can't always evaluate a global object at one number (analogous to the prime number p) to get local information.
So let's try and fix this up. Let k be a number field. k is definitely a global object, like Z, and it's also much easier to manage — it's a field rather than just a ring. The question is: what is the analogue of
0 → Z → R → R/Z → 0
if we replace Z by k, a number field? (Let's give k the discrete topology, because it has no natural topology other than this.) Even replacing Z by the rationals Q (with the discrete topology) is an interesting question! "Z is to R as Q is to...what?".
Well, Tate proposes the following: Z is to R as Q is to the adeles of Q! More generally Z is to R as k (a number field) is to its adeles. And the argument he could use to justify this isn't number-theoretic at all, in some sense — it's coming entirely from Pontrjagin duality! Tate shows that the Pontrjagin dual of a number field k is Ak/k, where Ak is the adeles of k, and k is embedded diagonally! Now the analogue of the beautiful self-dual picture 0 → Z → R → R/Z → 0 is going to be
0 → k → ??? → Ak/k → 0
and the natural candidate for ??? is of course now the adeles Ak. These gadgets have appeared "magically" in some sense — the argument seems to me to be topological rather than arithmetic (although of course there is more than one choice for ??? and perhaps the argument that the adeles are the right thing to put in the middle is number-theoretic).
Here's Tate's proof. Pontrjagin duality sends direct sums to direct products and vice-versa. So neither of them in particularly "symmetric" — both get changed. But Tate observes that restricted direct products get sent to restricted direct products! Let k be a number field. Let kv denote a typical completion of k (so if k=Q then kv is either R or Qp). We know kv is complete and it's easy to check it's locally compact (this doesn't use Weil's classification — it's the easy way around). So kv is self-dual. So the restricted product of all the kv (that is, the adeles), is locally compact, and dual to the restricted product of all the (kv)*, and (kv)* is kv again, so the adeles of k are self-dual!! Tate then checks that k (embedded diagonally in Ak) is discrete and equal to its own annihilator (if you choose all the local isomorphisms kv=(kv)* just right), and hence by the "Galois correspondence" between closed subgroups of G and closed subgroups of G* he deduces that Ak/k is the dual of k. In particular
0 → k → Ak → Ak/k → 0
looks like a very natural analogue of 0 → Z → R → R/Z → 0; the quotient is compact, the sub is discrete, and the diagram is self-Pontjragin-dual.
Within about 10 years of Tate's thesis it's visibly clear in the literature that there has been a seismic shift: there seem to be as many people studying G(Q) \ G(adeles) as there are studying G(Z) \ G(R) in the theory of automorphic forms, and the adelic approach has the advantage that, although less concrete, it has "truly local" components, thus motivating the representation theory of p-adic groups, the Langlands programme, and lots of other things.
Did you know that there is a *-autonomous category of topological abelian groups that includes all the LCA groups and whose duality extends that of Pontrjagin? The groups are characterized by the property that among all topological groups on the same underlying abelian group and with the same set of continuous homomorphisms to the circle, these have the finest topology. It is not obvious that such a finest exists, but it does and that is the key.He has a paper on this:
18) Michael Barr, On duality of topological abelian groups, available at ftp://ftp.math.mcgill.ca/pub/barr/pdffiles/abgp.pdf
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© 2008 John Baez