# This Week's Finds in Mathematical Physics (Week 216)

by John Baez
Tags: mathematical, physics, week
 P: n/a Zeta functions are wondrous beasts! Thanks for this awesome issue of TWF. John Baez wrote: > May 23, 2005 > This Week's Finds in Mathematical Physics - Week 216 > John Baez > [...] > E) The zeta function of an affine scheme. Given a commutative ring, > we can think of it as the ring of functions on some space. The zeta > function of the commutative ring then counts the points of this space. I realize this wasn't your focus but I feel compelled to mention the Weil conjectures in connection with this. In the related case of an algebraic variety V over a finite field F_q (where q = p^n for some prime p) we have the exponential generating function Z(T) = exp(sum(n positive) #(points of V over F_{q^n}) T^n / n!) which is the so-called local zeta function of the variety. Presumably the word "local" is used because the F_{q^n} are in some sense increasingly local versions of F_q--so the local zeta function counts points of V over increasingly local fields. Part of what Weil conjectured was that Z(T) should always be a rational function. In fact he conjectured a pretty specific form. All this is extremely unexpected! It turns out that it would follow from a Lefschetz-style fixed point formula for a sufficiently rich cohomology theory. But the problem at the time was no such cohomology theory was known--the only natural topology on V is the Zariski topology, which is very pathological. So all the usual ways of setting up cohomology failed to give much useful information. Grothendieck eventually figured out a way around these problems by inventing etale cohomology. Deligne used this to eventually prove the Weil conjectures. I've been trying to learn some of this stuff from J. Milne's lecture notes on etale cohomology at http://www.jmilne.org/math/CourseNotes/math732.html which are excellent but very heavy going (unsurprisingly). The first chapter or two are relatively chatty--they are definitely worth checking out if you are interested in a vague impression of what some of this stuff is about. Per
 P: n/a Dr. Baez et al... Where's the physics application in this post? I read it twice and I'm uncertain how it is applicable to physics, help a dense guy:-). Ken [Moderator's note: Entire bottom-quoted post deleted. Please quote as much of the post as necessary, but as little as possible, and add your reply after that. -P.H.]
P: n/a

## This Week's Finds in Mathematical Physics (Week 216)

Per Vognsen <per.vognsen@gmail.com> wrote:

>Zeta functions are wondrous beasts! Thanks for this awesome issue of
>TWF.

though I know from experience that as soon as I *plan* to
write about something in This Week's Finds, I decide it's
a chore and switch to writing about something else!

>I realize this wasn't your focus but I feel compelled to mention the
>Weil conjectures in connection with this.

Excellent! These were indeed on my mind...

>In the related case of an
>algebraic variety V over a finite field F_q (where q = p^n for some
>prime p) we have the exponential generating function
>
>Z(T) = exp(sum(n positive) #(points of V over F_{q^n}) T^n / n!)
>
>which is the so-called local zeta function of the variety.

Right. Part of what used to bug me immensely is that I didn't
see how this formula was related to other things called zeta functions.
That's why I'm so happy to learn about a very simple idea - the zeta
function of a discrete dynamical system - and to realize that the
above zeta function comes from a dynamical system where the time
evolution map is called the "Frobenius".

I'm sure you understand this already, but just for everyone else:
this is why "week216" I gave a proof that for any discrete dynamical
system

f: X -> X

we can define a function generalizing the one you wrote down:

Z(T) = exp(sum_{n>0} |fix(f^n)| T^n / n)

and then

zeta(s) = Z(p^{-s})

equals what I'd consider the "usual" zeta function of a dynamical
system, namely

zeta(s) = product_P 1/(1 - |P|^{-s})

where P ranges over all periodic orbits, and |P| is the exponential
of the size of the periodic orbit, defined using the base p.

>Presumably the word "local" is used because the F_{q^n} are in some sense
>increasingly local versions of F_q--so the local zeta function counts
>points of V over increasingly local fields.

Right. The fields F_{q^n} are "local fields" in a precise technical
sense, while the rational numbers are a "global field" - but this precise
technical sense is based on a simple beautiful idea, namely that the rational
numbers are functions on a space whose points are just the prime numbers
(together with the "real prime").

So, keeping track of the power of a prime that shows up in a rational number
is a way of studying that number "locally". And, just as global problems in
geometry get easier if we first tackle them locally, the same is true of all
problems in number theory!

>Part of what Weil conjectured was that Z(T) should always be a rational
>function. In fact he conjectured a pretty specific form. All this is
>extremely unexpected!

It was less unexpected to him than to most of us, because he knew
that there was a very simple formula relating the cohomology of
complex Grassmannians to the number of points of these Grassmannians
in "week187" and previous weeks. I knew it was related to Weil's
conjectures. But, only recently did I learn that Weil was motivated
by precisely this (and other facts) to make his conjectures!

 P: n/a In article , baez@galaxy.ucr.edu (John Baez) wrote: > In article <1117079187.198693.76220@z14g2000cwz.googlegroups.com>, > Per Vognsen wrote: > > >In the related case of an > >algebraic variety V over a finite field F_q (where q = p^n for some > >prime p) we have the exponential generating function > > > >Z(T) = exp(sum(n positive) #(points of V over F_{q^n}) T^n / n!) > > > >which is the so-called local zeta function of the variety. > > Right. Part of what used to bug me immensely is that I didn't > see how this formula was related to other things called zeta functions. (I think there shouldn't be a factorial there) It's sort of weird, I think, because for a lot of application it seems like the L-function is much more analogous to the Riemann zeta function. Of course, Hartshorne (Ex C.5.4) has the following (paraphrased) Given a Scheme X of finite type over Spec Z, let Zeta_X(s) = \prod (1 - N(x)^-s)^-1 where the product is taken over all closed points x and N(x) is the number of elements of the residue field k(x)= O_x / m_x. Doing this for X = Spec Z gives you the usual zeta function. Then, if X is of finite type over F_q, we can do the same thing and get Zeta_X(s) = Z(X,q^-s) To define Z(X,t), let N_r be the number of rational points of X x_F_Q \bar{F_q} (the algebraic closure of F_q) over F_q^r. Then stick these into Z(X,t) = Exp(\sum N_r t^r/r) When I'm more awake I might actually try to understand this and relate this definition to the one your state (N_r is also the number of fixed points of the Frobenius automorphism). Aaron
 P: n/a In article , Robert C. Helling wrote: >will there be a part II of this post where you explain the relation to >Gutzwiller's trace formula and zeta functions of operators used to >regularize QFT's? Umm, right now I'm much more fascinated by other things, like trying to understand the network of ideas related to zeta functions and L-functions in number theory - stuff like the Weil hypotheses, the Taniyama-Shimura-Weil conjecture, and the Langlands program. After about a year of battering my head against these, they seem like they're starting to make sense... either that or I've knocked myself silly! I want to write more about this stuff. So, I can't promise to get around to the other aspects of zeta functions that you mention. But, they *are related*, so I should keep them in mind.
 P: n/a In article , Aaron Bergman wrote: >When I'm more awake I might actually try to understand this and relate >this definition to the one you state (N_r is also the number of fixed >points of the Frobenius automorphism). I think I understand this stuff now; the key is that it has absolutely nothing to do with algebraic geometry, schemes, Frobenius automorphisms or all that jazz. It's just a fact about a 1-1 and onto map f: X -> X We can define zeta(s) = product_P 1/(1 - p^{-|P|s}) where P ranges over all periodic orbits of f and |P| is the size of the periodic orbit. We can also define Z(u) = exp(sum_{n>0} |fix(f^n)| u^n / n) where |fix(f^n)| is the number of fixed points of f^n. Then we have zeta(s) = Z(p^{-s}) I gave the proof in "week216" (using slightly different notation). We can then apply this to algebraic geometry by letting f be a Frobenius automorphism, and see that two different-looking definitions of the "zeta function" of a scheme agree. But, we can also apply it to dynamical systems and all sorts of other stuff. ------------------------------------------------------------------------ Michael Berry on the dreamt-of "Riemann dynamical system". He is: "... absolutely sure that ... someone will find a clever way to make it in the lab. Then you'll get the Riemann zeros out just by observing its spectrum." "Finding this system could be the discovery of the century." Berry says. It would become a model system for describing chaotic systems in the same way that the simple harmonic oscillator is used as a model for all kinds of complicated oscillators. It could play a fundamental role in describing all kinds of chaos. The search for this model system could be the holy grail of chaos. Until [it is found] we cannot be sure of its properties, but Berry believes the system is likely to be rather simple, and expects it to lead to totally new physics. It is a tantalising thought." From: Martin Gutzwiller, Quantum Chaos, Scientific American, January 1992, available at http://www.maths.ex.ac.uk/~mwatkins/...ntumchaos.html