- #1
John Baez
Also available as http://math.ucr.edu/home/baez/week218.html
June 5, 2005
This Week's Finds in Mathematical Physics - Week 218
John Baez
Classes are over! Summer is here! Now I can finally get some work done!
I'll be traveling to Sydney, Canberra, Beijing, Chengdu and Calgary, but
mainly I want to finish writing some papers.
First, though, I need to recover from a hard quarter. I need to goof off a
bit! I spent most of yesterday lying in bed reading. Now I want to talk
some more about number theory.
Let's see, where were we? I had just begun to introduce the theme of
L-functions and their corresponding automorphic forms. My ultimate goal
is to understand the Langlands Conjectures well enough to give a decent
explanation of what they say. Instead of simply stating them, I'd like
to really make them plausible, and this will take quite an elaborate warmup.
So, this Week I want to talk about some background.
Actually, this reminds me of sometime Feynman wrote: whenever he worked on a
problem, he needed the feeling he had some "inside track" - some insight or
trick up his sleeve that nobody else had. Most of us will never be as good
as Feynman at choosing an "inside track". But I think we all need one to
convert what would otherwise be a dutiful and doomed struggle to catch up
with the experts into something more hopeful and exciting: a personal quest!
For anyone with a physics background, a good "inside track" on almost any math
problem is to convert it into some kind of crazy physics problem. It doesn't
need to be realistic physics, just anything you can apply physics intuition
to! This is part of why string theorists have been so successful in cracking
math problems. It also underlies Alain Connes' attempt to prove the Riemann
Hypothesis:
1) Alain Connes, Noncommutative Geometry, Trace Formulas, and the Zeros of
the Riemann Zeta Function. Ohio State course notes and videos at
http://www.math.ohio-state.edu/lectures/connes/Connes.html
Alain Connes, Trace Formula in Noncommutative Geometry and the Zeros of
the Riemann Zeta function, available as math.NT/9811068.
2) Mathilde Marcolli, Noncommutative Geometry and Number Theory,
available at http://www.math.fsu.edu/~marcolli/ncgntE.pdf
Of course, Connes also has another "inside track", namely his theory of
noncommutative geometry.
By the way: a number theorist I know says he thinks Connes has essentially
proved the Riemann Hypothesis, in the same way that Riemann "essentially"
proved the Prime Number Theorem. Namely, he has reduced it to some facts that
seem obviously true! Of course, it took about 40 years, from 1859 to 1896,
for Riemann's plan to be fulfilled by Hadamard and De La Vallee Poussin. So,
even if Connes' insights are correct, it may be a while before the Riemann
Hypothesis is actually proved.
For anyone with a background in geometry, a good "inside track" on almost any
math problem is to convert it into a geometry problem. In the case of number
theory this trick is old news, but still very much worth knowing. It's based
on an analogy which I began discussing in "week198".
The analogy starts out like this:
NUMBER THEORY COMPLEX GEOMETRY
Integers, Z Polynomial functions on the complex plane, C[z]
Rational numbers, Q Rational functions on the complex plane, C(z)
Prime numbers, P Points in the complex plane, C
Why is this analogy good? Well, for starters:
Every rational number is a ratio of integers.
Every rational function is a ratio of polynomials.
Better yet:
Every integer can be uniquely factored into primes
(modulo invertible integers, namely +1 and -1).
Every complex polynomial can be uniquely factored into linear polynomials
(modulo invertible polynomials, namely nonzero constants).
There's one linear polynomial z-a for each point a in the complex plane,
so PRIMES are like POINTS in the complex plane.
We can make this precise using the concept of "spectrum", which I defined in
"week199". Ignoring a certain little sublety which is discussed there:
The spectrum of Z is the set of prime numbers.
The spectrum of C[z] is the complex plane.
This way of thinking let's us treat the spectrum of any algebraic extension of
the integers, like the Gaussian integers, as a "covering space" of the set of
prime numbers. I've already drawn this picture:2+i 3+2i
--- 1+i --- 3 --- --- 7 --- 11 --- --- GAUSSIAN INTEGERS
2-i 3-2i-----2------3------5------7-----11------13----- INTEGERSBut, now I'm saying that the "line" down below really acts like the
complex *plane*. Taking this strange idea seriously leads to all sorts
of amazing insights.
For example, if you poke a hole in this "plane" at some prime, there's
something like a little *loop* that goes around this hole! In other words,
there's a sense in which the spectrum of Z has a nontrivial "fundamental
group", which contains an element for each prime. Technically this group
is called the Galois group Gal(Qbar/Q), and we get an element in it for
each prime, called the "Frobenius automorphism" for that prime.
Another cool thing is that we can study integers "locally", one prime at a
time, just like we study complex functions locally. We can analyze functions
at a point using Taylor series and Laurent series. And, we can stretch our
analogy to include these concepts:
NUMBER THEORY COMPLEX GEOMETRY
Integers, Z Polynomial functions on the complex plane, C[z]
Rational numbers, Q Rational functions on the complex plane, C(z)
Prime numbers, P Points a in the complex plane, C
Integers mod p^n, Z/p^n (n-1)st-order Taylor series, C[z]/(z-a)^n
p-adic integers, Z_p Taylor series, C[[z-a]]
p-adic numbers, Q_p Laurent series, C((z-a))
All the weird symbols are just the standard notations for these gadgets.
The analogy goes as follows:
To study a polynomial "at a point" a in the complex plane,
we can look at its value modulo (z-a), or more generally mod (z-a)^n.
To study an integer "at a prime" p,
we can look at its value modulo p, or more generally mod p^n.
This is nice because the value of a polynomial modulo (z-a)^n is just its
Taylor series at the point a, where we keep terms up to order n-1.
We can also also take the limit as n -> infinity. If we do this to the
integers mod p^n we get a ring called the "p-adic integers". For example,
a typical 3-adic integer, written in base 3, looks like this:
...21001102020110102012102201
They're just like natural numbers in base 3, except they go on forever to the
left! We add and multiply them in the obvious way, for example:
...21001102020110102012102201
+ ...10201101012201201122010012
-----------------------------------
...01202210110012010211112220
If we take the same sort of limit for Taylor series, we get Taylor series
that go on forever - in other words, formal power series.
We can also ratios of p-adic integers, which are called p-adic numbers,
and ratios of Taylor series, which are called Laurent series. A typical
3-adic number, written in base 3, looks like this:
...121010010012121201201201011.2102122020101022102011022...
Laurent series can be used to describe functions that have a pole at some
point, like rational functions. Similarly, p-adic numbers can be used
to describe rational numbers. Using more math jargon:
For any point a in C, there's a homomorphism
from the field of rational functions
to the field of Laurent series,
which sends polynomials to Taylor series.
For any prime p, there's a homomorphism
from the field of rational numbers
to the field of p-adic numbers,
which sends integers to p-adic integers.
This let's us study rational numbers "locally" at the prime p using p-adic
numbers, just as we can study a rational function locally at a point using
its Laurent series. This technique can be quite useful. For example, a
polynomial equation can have rational solutions only if it has p-adic
solution for all primes p.
We might hope for the converse, but then we would be ignoring a funny extra
"prime" besides the usual ones... something called the "real prime"!
The point is, besides being able to embed the rational numbers in the p-adics
for any prime p, we can also embed them in the real numbers! This embedding
is a bit different than the rest: it's based on a weird thing called an
"Archimedean valuation", while the usual primes correspond to non-Archimedean
valuations.
I'm sort of joking here, since if you're more used to real numbers than
p-adics, you'll probably find Archimedean valuations to be *less* weird than
non-Archimedean ones. The Archimedean valuation on the rational numbers is
just the usual absolute value, while the non-Archimedean ones are other
concepts of "absolute value", one for each prime p. If we take limits of
rational numbers that converge using the usual distance function |x-y|, we
get real numbers; if we take limits that converge using one of the
non-Archimedean versions of this distance function, we get p-adic numbers.
But from the viewpoint of number theory, it's the Archimedean valuation
that's the odd man out! It indeed does act very weird and different than
all the rest. That's why someone wrote this book:
3) M. J. Shai Haran, The Mysteries of the Real Prime, Oxford
U. Press, Oxford, 2001.
... which you will see is deeply connected to mathematical physics.
If we take this weird "real prime" into account, things work better.
We sometimes get results saying that some kind of polynomial equations
have a rational solution if they have p-adic solutions for all primes p
and also a real solution. For example, Hasse proved this was true for
systems of quadratic equations in many variables.
Results like this are called "local-to-global" results, since they're
analogous to constructing a function from local information, like its
Laurent series at all different points.
In 1950, in his famous PhD thesis, John Tate came up with a clever way to
formalize this "Laurent series at all different points" idea in the context
of number theory. To do this, he formed a ring called the "adeles".
Indeed, this is what my whole discussion so far has been leading up to!
Adeles are a really nice formalism, and you pretty much need to understand
them to follow what people are doing in work on the Langlands Conjectures,
or even simpler things, like class field theory. But, adeles seem like
an arbitrary construction until you see them as an inevitable outgrowth
of our desire to study integers "locally" at all different primes, including
the real prime.
The definition is simple. An adele consists of a p-adic number for each
prime p, together with a real number... but where all but finitely many
of the p-adic numbers are p-adic integers!
This is the number-theoretic analogue of a Laurent series for each point in
the complex plane, including the point at infinity... but with poles at only
finitely many points! We could call such a thing an "adele for the rational
functions".
Any rational function gives such a thing, just as any rational number gives
an adele. And, we don't lose any information this way:
There's a one-to-one (but not onto) homomorphism
from the rational functions to the adeles for the rational functions.
There's a one-to-one (but not onto) homomorphism
from the rational numbers to the adeles for the rational numbers.
So, our table now looks like this. For good measure, I'll combine it with
the related table in "week205":
NUMBER THEORY COMPLEX GEOMETRY
Integers Polynomial functions on the complex plane
Rational numbers Rational functions on the complex plane
Prime numbers Points in the complex plane
Integers mod p^n (n-1)st-order Taylor series
p-adic integers Taylor series
p-adic numbers Laurent series
Adeles for the rationals Adeles for the rational functions
Fields One-point spaces
Homomorphisms to fields Maps from one-point spaces
Algebraic number fields Branched covering spaces of the complex plane
There's a *lot* more to say about this analogy, but I think this is enough
for now. Again, one of my secret goals was to start getting you comfy with
adeles and the idea of studying number theory "locally".
For more on the geometrical side of number theory, I again recommend these:
4) Juergen Neukirch, Algebraic Number Theory, trans. Norbert Schappacher,
Springer, Berlin, 1986.
5) Dino Lorenzini, An Invitation to Arithmetic Geometry, American
Mathematical Society, Providence, Rhode Island, 1996.
But now, back to the subject of "inside tracks" - sneaky ways to get the
beneficial feeling that you have secret insights into some problem.
For anyone with a background in categories, a good "inside track" on
almost any math problem is to categorify it: to see that people are using
sets where they could, and therefore *should*, be using categories or
n-categories.
I've already hinted that zeta functions are an example of
"decategorification". Now I'd like to make this more precise.
Let's think about the zeta function of a set X equipped with a one-to-one
and onto function
f: X -> X
If you're a physicist, you might call this a "discrete dynamical system",
with f describing one step of "time evolution". If you're a mathematician,
you might call this a "Z-set". After all, for any group G, a "G-set" is a
set equipped with an action of G. If G = Z (the additive group of integers),
this amounts to a one-to-one and onto function from some set to itself.
No matter what you call them, these are fundamental things. So, let's
look at the *category* of Z-sets! Here the objects are Z-sets and the
morphisms are functions that commute with time evolution.
As explained near the end of "week216", we can define a kind of zeta
function for a Z-set as follows:
Z(x) = exp(sum_{n>0} |fix(f^x)| x^n / n)
where |fix(f^n)| is the number of fixed points of f^n. Of course, this
only makes sense if all these numbers are finite; henceforth I'll assume
my Z-sets are "finite" in this special sense.
It turns out that you know a finite Z-set up to isomorphism if you know its
zeta function. So, a zeta function is just a sneaky way of talking about
an ISOMORPHISM CLASS of finite Z-sets.
This is a fancy example of something we all learn as kids: counting! When
we "count" a finite set, assigning a natural number to it, we are really
determining its isomorphism class. Two finite sets are isomorphic if and
only if they have the same number of elements. Operations on finite sets,
like disjoint union and Cartesian product, are what give rise to operations
on natural numbers, like addition and multiplication.
Summarizing this, we have the following motto, suitable for making into
a bumper sticker:
THE SET OF NATURAL NUMBERS IS THE DECATEGORIFICATION OF
THE CATEGORY OF FINITE SETS
Similarly, this is what we're seeing now:
THE SET OF ZETA FUNCTIONS IS THE DECATEGORIFICATION OF
THE CATEGORY OF FINITE Z-SETS
Beware: here I'm only talking about zeta functions of the above form. There
are lots of other things people call zeta functions. So, don't read too
much into this statement. But don't read too little into it, either! With
an extra twist we can get most of the zeta functions showing up in number
theory. In number theory, we typically get a Z-set for each prime p,
coming from the "Frobenius" for that prime. We thus get a bunch of "local"
zeta functions Z_p(x), one for each prime. We then multiply these to get
one big fat "global" zeta function:
zeta(s) = product_p Z(p^{-s))
Each local zeta function is a formal power series, while this global zeta
function is a Dirichlet series. As I mentioned in "week217", formal power
series live in the monoid algebra of (N,+,0), while Dirichlet series live
in the monoid algebra of (N,x,1). (N,+,0) is the free commutative monoid
on one generator, while (N,x,1) is the free commutative monoid on countably
many generators - the primes! Everything fits together sweetly.
So, it's a good first step to think about the zeta function of a single
Z-set.
Now, there's another motto along the lines of the above two, which I've
talked about before:
THE SET OF GENERATING FUNCTIONS IS A DECATEGORIFICATION OF
THE CATEGORY OF STRUCTURE TYPES
I explained this in "week185", "week190", and "week202". I've even taught a
whole course on structure types (also known as "species") and the
combinatorics of Feynman diagrams. The course notes by Derek Wise are
available online:
6) John Baez and Derek Wise, Quantization and Categorification, available at:
http://math.ucr.edu/home/baez/qg-fall2003/
http://math.ucr.edu/home/baez/qg-winter2004/
http://math.ucr.edu/home/baez/qg-spring2004/
So, I think this third example of decategorification is great. But, I'm not
going to explain it in much detail here - just enough to say how it's related
to zeta functions!
A stucture type F is a gadget that gives a set F_n for each n = 0,1,2,...
We think of the elements of F_n as "structures of type F" on an n-element
set - for example, orderings, or cyclic orderings, or n-colorings, or
whatever type of structure you like. We only require that permutations of
the n-element set act on this set of structures. But for what we're doing
now, let's also assume this set F_n is finite.
Any structure type has a "generating function", which is a formal
power series |F| given by
|F_n|
|F|(x) = sum ------- x^n
n!
Isomorphic structure types have the same generating function. However,
structure types with the same generating function can fail to be isomorphic.
This is why I said generating functions are "a" decategorification of
structure types, instead of "the" decategorification.
Despite this defect, generating functions are still very useful in
combinatorics. So, when we see a zeta function like
Z(x) = exp(sum_{n>0} |fix(f^n)| x^n / n)
as a trick for decategorifying Z-sets, we should instantly wonder if it's
a generating function in disguise. And of course, it is!
Actually it's easiest to leave out the exponential at first. This power
series:
sum_{n>0} |fix(f^n)| x^n / n
is the generating function for the structure type "being cyclically
ordered and equipped with a morphism to the Z-set X".
Huh?
We "cyclically order" a finite set by drawing it as a little circle of dots
with arrows pointing clockwise from each dot to the next. A cyclically
ordered set is automatically a Z-set in an obvious way. So, here's a type
of structure you can put on a finite set: cyclically ordering it and
equipping the resulting Z-set with a morphism to the Z-set X.
And, if you work out the generating function of this structure type, you get
sum_{n>0} |fix(f^n)| x^n / n
Check it and see!
What about the exponential? Luckily, there's a standard way to take the
exponential of a structure type: to put an exp(F)-structure on a finite set
S, we chop S into disjoint parts and put an F-structure on each part. So,
the zeta function
Z(x) = exp(sum_{n>0} |fix(f^n)| x^n / n)
is the generating function for "being chopped up into cyclically ordered
parts, each equipped with a morphism to the Z-set X".
But this is just a long way of saying: "being made into a Z-set and equipped
with a morphism to the Z-set X".
Or, in category theory jargon, "being a Z-set over X".
So:
THE ZETA FUNCTION OF THE Z-SET X IS THE GENERATING FUNCTION OF
"BEING A Z-SET OVER X"
By the way, this is the kind of thing you could do with *any* structure type
F. Given an F-structured set X, we get a new structure type "being an
F-structured set equipped with a morphism to X". Or, in category theory
jargon, "being an F-structured set over X". The generating function of this
could be called the "zeta function" of our F-structured set X. I have no
idea how important this is...
... but I want to keep gnawing away on the connection between zeta functions
and the generating functions of combinatorics, because to understand number
theory, I need all the "inside tracks" I can get!
Quote of the week, brought to me by David Corfield:
The scientific life of mathematicians can be pictured as a trip inside the
geography of the "mathematical reality" which they unveil gradually in their
own private mental frame.
It often begins by an act of rebellion with respect to the existing dogmatic
description of that reality that one will find in existing books. The young
"to be mathematician" realize in their own mind that their perception of the
mathematical world captures some features which do not fit with the existing
dogma. This first act is often due in most cases to ignorance but it allows
one to free oneself from the reverence to authority by relying on one's
intuition provided it is backed by actual proofs. Once mathematicians get to
really know, in an original and "personal" manner, a small part of the
mathematical world, as esoteric as it can look at first, their trip can
really start. It is of course vital not to break the "fil d'arianne" which
allows one to constantly keep a fresh eye on whatever one will encounter
along the way, and also to go back to the source if one feels lost at times...
It is also vital to always keep moving. The risk otherwise is to confine
oneself in a relatively small area of extreme technical specialization, thus
shrinking one's perception of the mathematical world and its bewildering
diversity.
The really fundamental point in that respect is that while so many
mathematicians have been spending their entire life exploring that world they
all agree on its contours and on its connexity: whatever the origin of one's
itinerary, one day or another if one walks long enough, one is bound to reach
a well known town i.e. for instance to meet elliptic functions, modular
forms, zeta functions. "All roads lead to Rome" and the mathematical world
is "connected".
In other words there is just "one" mathematical world, whose exploration is
the task of all mathematicians, and they are all in the same boat somehow.
- Alain Connes
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
June 5, 2005
This Week's Finds in Mathematical Physics - Week 218
John Baez
Classes are over! Summer is here! Now I can finally get some work done!
I'll be traveling to Sydney, Canberra, Beijing, Chengdu and Calgary, but
mainly I want to finish writing some papers.
First, though, I need to recover from a hard quarter. I need to goof off a
bit! I spent most of yesterday lying in bed reading. Now I want to talk
some more about number theory.
Let's see, where were we? I had just begun to introduce the theme of
L-functions and their corresponding automorphic forms. My ultimate goal
is to understand the Langlands Conjectures well enough to give a decent
explanation of what they say. Instead of simply stating them, I'd like
to really make them plausible, and this will take quite an elaborate warmup.
So, this Week I want to talk about some background.
Actually, this reminds me of sometime Feynman wrote: whenever he worked on a
problem, he needed the feeling he had some "inside track" - some insight or
trick up his sleeve that nobody else had. Most of us will never be as good
as Feynman at choosing an "inside track". But I think we all need one to
convert what would otherwise be a dutiful and doomed struggle to catch up
with the experts into something more hopeful and exciting: a personal quest!
For anyone with a physics background, a good "inside track" on almost any math
problem is to convert it into some kind of crazy physics problem. It doesn't
need to be realistic physics, just anything you can apply physics intuition
to! This is part of why string theorists have been so successful in cracking
math problems. It also underlies Alain Connes' attempt to prove the Riemann
Hypothesis:
1) Alain Connes, Noncommutative Geometry, Trace Formulas, and the Zeros of
the Riemann Zeta Function. Ohio State course notes and videos at
http://www.math.ohio-state.edu/lectures/connes/Connes.html
Alain Connes, Trace Formula in Noncommutative Geometry and the Zeros of
the Riemann Zeta function, available as math.NT/9811068.
2) Mathilde Marcolli, Noncommutative Geometry and Number Theory,
available at http://www.math.fsu.edu/~marcolli/ncgntE.pdf
Of course, Connes also has another "inside track", namely his theory of
noncommutative geometry.
By the way: a number theorist I know says he thinks Connes has essentially
proved the Riemann Hypothesis, in the same way that Riemann "essentially"
proved the Prime Number Theorem. Namely, he has reduced it to some facts that
seem obviously true! Of course, it took about 40 years, from 1859 to 1896,
for Riemann's plan to be fulfilled by Hadamard and De La Vallee Poussin. So,
even if Connes' insights are correct, it may be a while before the Riemann
Hypothesis is actually proved.
For anyone with a background in geometry, a good "inside track" on almost any
math problem is to convert it into a geometry problem. In the case of number
theory this trick is old news, but still very much worth knowing. It's based
on an analogy which I began discussing in "week198".
The analogy starts out like this:
NUMBER THEORY COMPLEX GEOMETRY
Integers, Z Polynomial functions on the complex plane, C[z]
Rational numbers, Q Rational functions on the complex plane, C(z)
Prime numbers, P Points in the complex plane, C
Why is this analogy good? Well, for starters:
Every rational number is a ratio of integers.
Every rational function is a ratio of polynomials.
Better yet:
Every integer can be uniquely factored into primes
(modulo invertible integers, namely +1 and -1).
Every complex polynomial can be uniquely factored into linear polynomials
(modulo invertible polynomials, namely nonzero constants).
There's one linear polynomial z-a for each point a in the complex plane,
so PRIMES are like POINTS in the complex plane.
We can make this precise using the concept of "spectrum", which I defined in
"week199". Ignoring a certain little sublety which is discussed there:
The spectrum of Z is the set of prime numbers.
The spectrum of C[z] is the complex plane.
This way of thinking let's us treat the spectrum of any algebraic extension of
the integers, like the Gaussian integers, as a "covering space" of the set of
prime numbers. I've already drawn this picture:2+i 3+2i
--- 1+i --- 3 --- --- 7 --- 11 --- --- GAUSSIAN INTEGERS
2-i 3-2i-----2------3------5------7-----11------13----- INTEGERSBut, now I'm saying that the "line" down below really acts like the
complex *plane*. Taking this strange idea seriously leads to all sorts
of amazing insights.
For example, if you poke a hole in this "plane" at some prime, there's
something like a little *loop* that goes around this hole! In other words,
there's a sense in which the spectrum of Z has a nontrivial "fundamental
group", which contains an element for each prime. Technically this group
is called the Galois group Gal(Qbar/Q), and we get an element in it for
each prime, called the "Frobenius automorphism" for that prime.
Another cool thing is that we can study integers "locally", one prime at a
time, just like we study complex functions locally. We can analyze functions
at a point using Taylor series and Laurent series. And, we can stretch our
analogy to include these concepts:
NUMBER THEORY COMPLEX GEOMETRY
Integers, Z Polynomial functions on the complex plane, C[z]
Rational numbers, Q Rational functions on the complex plane, C(z)
Prime numbers, P Points a in the complex plane, C
Integers mod p^n, Z/p^n (n-1)st-order Taylor series, C[z]/(z-a)^n
p-adic integers, Z_p Taylor series, C[[z-a]]
p-adic numbers, Q_p Laurent series, C((z-a))
All the weird symbols are just the standard notations for these gadgets.
The analogy goes as follows:
To study a polynomial "at a point" a in the complex plane,
we can look at its value modulo (z-a), or more generally mod (z-a)^n.
To study an integer "at a prime" p,
we can look at its value modulo p, or more generally mod p^n.
This is nice because the value of a polynomial modulo (z-a)^n is just its
Taylor series at the point a, where we keep terms up to order n-1.
We can also also take the limit as n -> infinity. If we do this to the
integers mod p^n we get a ring called the "p-adic integers". For example,
a typical 3-adic integer, written in base 3, looks like this:
...21001102020110102012102201
They're just like natural numbers in base 3, except they go on forever to the
left! We add and multiply them in the obvious way, for example:
...21001102020110102012102201
+ ...10201101012201201122010012
-----------------------------------
...01202210110012010211112220
If we take the same sort of limit for Taylor series, we get Taylor series
that go on forever - in other words, formal power series.
We can also ratios of p-adic integers, which are called p-adic numbers,
and ratios of Taylor series, which are called Laurent series. A typical
3-adic number, written in base 3, looks like this:
...121010010012121201201201011.2102122020101022102011022...
Laurent series can be used to describe functions that have a pole at some
point, like rational functions. Similarly, p-adic numbers can be used
to describe rational numbers. Using more math jargon:
For any point a in C, there's a homomorphism
from the field of rational functions
to the field of Laurent series,
which sends polynomials to Taylor series.
For any prime p, there's a homomorphism
from the field of rational numbers
to the field of p-adic numbers,
which sends integers to p-adic integers.
This let's us study rational numbers "locally" at the prime p using p-adic
numbers, just as we can study a rational function locally at a point using
its Laurent series. This technique can be quite useful. For example, a
polynomial equation can have rational solutions only if it has p-adic
solution for all primes p.
We might hope for the converse, but then we would be ignoring a funny extra
"prime" besides the usual ones... something called the "real prime"!
The point is, besides being able to embed the rational numbers in the p-adics
for any prime p, we can also embed them in the real numbers! This embedding
is a bit different than the rest: it's based on a weird thing called an
"Archimedean valuation", while the usual primes correspond to non-Archimedean
valuations.
I'm sort of joking here, since if you're more used to real numbers than
p-adics, you'll probably find Archimedean valuations to be *less* weird than
non-Archimedean ones. The Archimedean valuation on the rational numbers is
just the usual absolute value, while the non-Archimedean ones are other
concepts of "absolute value", one for each prime p. If we take limits of
rational numbers that converge using the usual distance function |x-y|, we
get real numbers; if we take limits that converge using one of the
non-Archimedean versions of this distance function, we get p-adic numbers.
But from the viewpoint of number theory, it's the Archimedean valuation
that's the odd man out! It indeed does act very weird and different than
all the rest. That's why someone wrote this book:
3) M. J. Shai Haran, The Mysteries of the Real Prime, Oxford
U. Press, Oxford, 2001.
... which you will see is deeply connected to mathematical physics.
If we take this weird "real prime" into account, things work better.
We sometimes get results saying that some kind of polynomial equations
have a rational solution if they have p-adic solutions for all primes p
and also a real solution. For example, Hasse proved this was true for
systems of quadratic equations in many variables.
Results like this are called "local-to-global" results, since they're
analogous to constructing a function from local information, like its
Laurent series at all different points.
In 1950, in his famous PhD thesis, John Tate came up with a clever way to
formalize this "Laurent series at all different points" idea in the context
of number theory. To do this, he formed a ring called the "adeles".
Indeed, this is what my whole discussion so far has been leading up to!
Adeles are a really nice formalism, and you pretty much need to understand
them to follow what people are doing in work on the Langlands Conjectures,
or even simpler things, like class field theory. But, adeles seem like
an arbitrary construction until you see them as an inevitable outgrowth
of our desire to study integers "locally" at all different primes, including
the real prime.
The definition is simple. An adele consists of a p-adic number for each
prime p, together with a real number... but where all but finitely many
of the p-adic numbers are p-adic integers!
This is the number-theoretic analogue of a Laurent series for each point in
the complex plane, including the point at infinity... but with poles at only
finitely many points! We could call such a thing an "adele for the rational
functions".
Any rational function gives such a thing, just as any rational number gives
an adele. And, we don't lose any information this way:
There's a one-to-one (but not onto) homomorphism
from the rational functions to the adeles for the rational functions.
There's a one-to-one (but not onto) homomorphism
from the rational numbers to the adeles for the rational numbers.
So, our table now looks like this. For good measure, I'll combine it with
the related table in "week205":
NUMBER THEORY COMPLEX GEOMETRY
Integers Polynomial functions on the complex plane
Rational numbers Rational functions on the complex plane
Prime numbers Points in the complex plane
Integers mod p^n (n-1)st-order Taylor series
p-adic integers Taylor series
p-adic numbers Laurent series
Adeles for the rationals Adeles for the rational functions
Fields One-point spaces
Homomorphisms to fields Maps from one-point spaces
Algebraic number fields Branched covering spaces of the complex plane
There's a *lot* more to say about this analogy, but I think this is enough
for now. Again, one of my secret goals was to start getting you comfy with
adeles and the idea of studying number theory "locally".
For more on the geometrical side of number theory, I again recommend these:
4) Juergen Neukirch, Algebraic Number Theory, trans. Norbert Schappacher,
Springer, Berlin, 1986.
5) Dino Lorenzini, An Invitation to Arithmetic Geometry, American
Mathematical Society, Providence, Rhode Island, 1996.
But now, back to the subject of "inside tracks" - sneaky ways to get the
beneficial feeling that you have secret insights into some problem.
For anyone with a background in categories, a good "inside track" on
almost any math problem is to categorify it: to see that people are using
sets where they could, and therefore *should*, be using categories or
n-categories.
I've already hinted that zeta functions are an example of
"decategorification". Now I'd like to make this more precise.
Let's think about the zeta function of a set X equipped with a one-to-one
and onto function
f: X -> X
If you're a physicist, you might call this a "discrete dynamical system",
with f describing one step of "time evolution". If you're a mathematician,
you might call this a "Z-set". After all, for any group G, a "G-set" is a
set equipped with an action of G. If G = Z (the additive group of integers),
this amounts to a one-to-one and onto function from some set to itself.
No matter what you call them, these are fundamental things. So, let's
look at the *category* of Z-sets! Here the objects are Z-sets and the
morphisms are functions that commute with time evolution.
As explained near the end of "week216", we can define a kind of zeta
function for a Z-set as follows:
Z(x) = exp(sum_{n>0} |fix(f^x)| x^n / n)
where |fix(f^n)| is the number of fixed points of f^n. Of course, this
only makes sense if all these numbers are finite; henceforth I'll assume
my Z-sets are "finite" in this special sense.
It turns out that you know a finite Z-set up to isomorphism if you know its
zeta function. So, a zeta function is just a sneaky way of talking about
an ISOMORPHISM CLASS of finite Z-sets.
This is a fancy example of something we all learn as kids: counting! When
we "count" a finite set, assigning a natural number to it, we are really
determining its isomorphism class. Two finite sets are isomorphic if and
only if they have the same number of elements. Operations on finite sets,
like disjoint union and Cartesian product, are what give rise to operations
on natural numbers, like addition and multiplication.
Summarizing this, we have the following motto, suitable for making into
a bumper sticker:
THE SET OF NATURAL NUMBERS IS THE DECATEGORIFICATION OF
THE CATEGORY OF FINITE SETS
Similarly, this is what we're seeing now:
THE SET OF ZETA FUNCTIONS IS THE DECATEGORIFICATION OF
THE CATEGORY OF FINITE Z-SETS
Beware: here I'm only talking about zeta functions of the above form. There
are lots of other things people call zeta functions. So, don't read too
much into this statement. But don't read too little into it, either! With
an extra twist we can get most of the zeta functions showing up in number
theory. In number theory, we typically get a Z-set for each prime p,
coming from the "Frobenius" for that prime. We thus get a bunch of "local"
zeta functions Z_p(x), one for each prime. We then multiply these to get
one big fat "global" zeta function:
zeta(s) = product_p Z(p^{-s))
Each local zeta function is a formal power series, while this global zeta
function is a Dirichlet series. As I mentioned in "week217", formal power
series live in the monoid algebra of (N,+,0), while Dirichlet series live
in the monoid algebra of (N,x,1). (N,+,0) is the free commutative monoid
on one generator, while (N,x,1) is the free commutative monoid on countably
many generators - the primes! Everything fits together sweetly.
So, it's a good first step to think about the zeta function of a single
Z-set.
Now, there's another motto along the lines of the above two, which I've
talked about before:
THE SET OF GENERATING FUNCTIONS IS A DECATEGORIFICATION OF
THE CATEGORY OF STRUCTURE TYPES
I explained this in "week185", "week190", and "week202". I've even taught a
whole course on structure types (also known as "species") and the
combinatorics of Feynman diagrams. The course notes by Derek Wise are
available online:
6) John Baez and Derek Wise, Quantization and Categorification, available at:
http://math.ucr.edu/home/baez/qg-fall2003/
http://math.ucr.edu/home/baez/qg-winter2004/
http://math.ucr.edu/home/baez/qg-spring2004/
So, I think this third example of decategorification is great. But, I'm not
going to explain it in much detail here - just enough to say how it's related
to zeta functions!
A stucture type F is a gadget that gives a set F_n for each n = 0,1,2,...
We think of the elements of F_n as "structures of type F" on an n-element
set - for example, orderings, or cyclic orderings, or n-colorings, or
whatever type of structure you like. We only require that permutations of
the n-element set act on this set of structures. But for what we're doing
now, let's also assume this set F_n is finite.
Any structure type has a "generating function", which is a formal
power series |F| given by
|F_n|
|F|(x) = sum ------- x^n
n!
Isomorphic structure types have the same generating function. However,
structure types with the same generating function can fail to be isomorphic.
This is why I said generating functions are "a" decategorification of
structure types, instead of "the" decategorification.
Despite this defect, generating functions are still very useful in
combinatorics. So, when we see a zeta function like
Z(x) = exp(sum_{n>0} |fix(f^n)| x^n / n)
as a trick for decategorifying Z-sets, we should instantly wonder if it's
a generating function in disguise. And of course, it is!
Actually it's easiest to leave out the exponential at first. This power
series:
sum_{n>0} |fix(f^n)| x^n / n
is the generating function for the structure type "being cyclically
ordered and equipped with a morphism to the Z-set X".
Huh?
We "cyclically order" a finite set by drawing it as a little circle of dots
with arrows pointing clockwise from each dot to the next. A cyclically
ordered set is automatically a Z-set in an obvious way. So, here's a type
of structure you can put on a finite set: cyclically ordering it and
equipping the resulting Z-set with a morphism to the Z-set X.
And, if you work out the generating function of this structure type, you get
sum_{n>0} |fix(f^n)| x^n / n
Check it and see!
What about the exponential? Luckily, there's a standard way to take the
exponential of a structure type: to put an exp(F)-structure on a finite set
S, we chop S into disjoint parts and put an F-structure on each part. So,
the zeta function
Z(x) = exp(sum_{n>0} |fix(f^n)| x^n / n)
is the generating function for "being chopped up into cyclically ordered
parts, each equipped with a morphism to the Z-set X".
But this is just a long way of saying: "being made into a Z-set and equipped
with a morphism to the Z-set X".
Or, in category theory jargon, "being a Z-set over X".
So:
THE ZETA FUNCTION OF THE Z-SET X IS THE GENERATING FUNCTION OF
"BEING A Z-SET OVER X"
By the way, this is the kind of thing you could do with *any* structure type
F. Given an F-structured set X, we get a new structure type "being an
F-structured set equipped with a morphism to X". Or, in category theory
jargon, "being an F-structured set over X". The generating function of this
could be called the "zeta function" of our F-structured set X. I have no
idea how important this is...
... but I want to keep gnawing away on the connection between zeta functions
and the generating functions of combinatorics, because to understand number
theory, I need all the "inside tracks" I can get!
Quote of the week, brought to me by David Corfield:
The scientific life of mathematicians can be pictured as a trip inside the
geography of the "mathematical reality" which they unveil gradually in their
own private mental frame.
It often begins by an act of rebellion with respect to the existing dogmatic
description of that reality that one will find in existing books. The young
"to be mathematician" realize in their own mind that their perception of the
mathematical world captures some features which do not fit with the existing
dogma. This first act is often due in most cases to ignorance but it allows
one to free oneself from the reverence to authority by relying on one's
intuition provided it is backed by actual proofs. Once mathematicians get to
really know, in an original and "personal" manner, a small part of the
mathematical world, as esoteric as it can look at first, their trip can
really start. It is of course vital not to break the "fil d'arianne" which
allows one to constantly keep a fresh eye on whatever one will encounter
along the way, and also to go back to the source if one feels lost at times...
It is also vital to always keep moving. The risk otherwise is to confine
oneself in a relatively small area of extreme technical specialization, thus
shrinking one's perception of the mathematical world and its bewildering
diversity.
The really fundamental point in that respect is that while so many
mathematicians have been spending their entire life exploring that world they
all agree on its contours and on its connexity: whatever the origin of one's
itinerary, one day or another if one walks long enough, one is bound to reach
a well known town i.e. for instance to meet elliptic functions, modular
forms, zeta functions. "All roads lead to Rome" and the mathematical world
is "connected".
In other words there is just "one" mathematical world, whose exploration is
the task of all mathematicians, and they are all in the same boat somehow.
- Alain Connes
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