Wednesday, October 29, 2008

Michael Atiyah on the foundations of philosophy, math and physics

Dear Colleagues:
Atiyah has given a very bracing address on "Mind, matter and mathematics". You can obtain a pdf of it at
and can read a report on a similar talk at
My best,

Monday, October 13, 2008

Index theory in Bogotá

Last week there was a meeting on spectral geometry and index theory at the Universidad de Los Andes in Bogotá, Colombia, organized by Alexander Cardona and Jean Carlos Cortissoz. The Encuentro "Geometría Espectral y Teoría del Índice" consisted of four afternoons of lectures and research talks, and this light but effective setup made it a very pleasant meeting. Even more because of the beautiful surroundings and interesting city.

The lectures were organized in three series, and were aimed at an undergraduate/graduate audience. Steven Rosenberg (Boston University) lectured about the Atiyah-Singer index theorem, Alexander Cardona about Index theorem for deformation algebras and I lectured on the Connes-Moscovici index theorem in noncommutative geometry. In his lectures, Cardona gave an overview of Fedosov's deformation quantization of symplectic manifolds, followed by Fedosov's index theorem, connecting this with the b+B-cocycle constructed by Connes, Flato and Sternheimer. The lectures of Rosenberg and myself followed a similar pattern: starting with the minimum but required preliminaries, we arrived at the statement of the two index theorems. Then, after briefly sketching their proofs, we discussed some applications, notably in the computation of the dimension of moduli spaces (both in the commutative and noncommutative case) and to quantum groups.

Besides the lectures, there were three very interesting research talks by Leonardo Cano (Universität Bonn) on "Spectral deformations of the Laplacian on manifolds", Monika Winklmeier (Universidad de Los Andes) "On the spectrum of the Klein-Gordon Operator" and by Andrés Vargas (Universität Bonn) on "Geometry and pinching of spin manifolds". The week ended with the talk by Steven Rosenberg in the mathematics colloquium, on index theorems on loop spaces.

In conclusion, I think that with the many bright master students and the newly started Ph.D. program, the Universidad de Los Andes - and in particular the math department - has still a lot more to offer in the near future!

Friday, October 3, 2008

Update on the field with one element

The paper "Fun with F_un" by Alain, Katia, and Matilde will appear soon in the Journal of Number Theory. An associated video abstract by Alain can be found at
Alain's video is a terrific example of what is possible with these video abstracts!
My best,

Friday, September 5, 2008

calculus and exponentiation in finite characteristic

In this post, I would like to discuss a really beautiful solution of an open issue in the theory of characteristic p L-series. This solution is in the short paper arXiv:0808.4069 by Sangtae Jeong and is a great illustration of the use of calculus in finite characteristic. I have written this post so that, if you choose, you can just copy it and paste it into a latex file and it will compile.

In my previous post of August 4, 2008, I mentioned the domain ${\mathbb S}_\infty={\mathbb C}_\infty^\ast\times {\mathbb Z}_p$ of characteristic $p$ $L$-series. We write $s\in {\mathbb S}_\infty$ as $(x,y)$. Here ${\mathbb Z}_p$ is used in the following fashion: Let $y\in {\mathbb Z}_p$ and let $u$ be a 1-unit in ${\mathbb F}_q((1/T))$; so $u=1+v$ where $v$ has absolute value strictly less than $1$. Then one simply defines $u^y=(1+v)^y$ by using the binomial expansion (which converges since $v$ is small).

The binomial expansion of $u^y$ obviously shows that the function $u \mapsto u^y$ is analytic on the $1$-units. This analyticity is itself crucial for the analytic continuation of the $L$-series of general Drinfeld modules and the like. Indeed, one writes down an integral for these $L$-series of the form
$$\int u^y\, d\mu_x(u)$$
where we integrate over the $1$-units and where $x$ is our parameter. This integral converges absolutely when $x$ is large. In fact, when $x$ is large the integral will converge if $u^y$ is replaced by {\it any} continuous function in $y$. However, the analyticity of $u^y$ gives very powerful a-priori information about the growth of the expansion coefficients of this function (in a suitable polynomial basis for all continuous functions); indeed, the coefficients go to $0$ quite rapidly. On the other hand, for arbitrary $x$ the measures $\mu_x$ blow up rather slowly (in fact, logarithmically). Putting the two facts together allows for the analytic continuation.

The same argument would work for {\it any} locally-analytic endomorphism of the $1$-units and therefore it is quite reasonable to expect that there are no others. This is what Jeong proves in his note.

One can obtain a proof using formal groups. Jeong's proof, however, seems to work only in the case at hand. Its advantage lies in the fact that you can actually watch the $p$-adic integer $y$ arising out of a series of first order, initial value differential equations that naturally arise.

In fact, Jeong's proof is a finite characteristic reflection of that most famous differential equation $y^\prime =y$ that one learns in first year calculus. Please forgive me for recalling how the differential equation is solved: one first learns that $y=e^x$ is a solution of the equation. One then divides any other solution $h$ by $e^x$ to obtain a function whose derivative (by the quotient rule) is identically $0$, and therefore constant, and so all solutions are multiples of $e^x$. In particular, of course, a solution $y$ is nonzero precisely when $y(0)=y^\prime(0)$ is.

Characteristic $p$ calculus presents many challenges; primarily the fact that having an identically vanishing derivative does NOT guarantee that a function is constant. So, first of all, one differentiates a power series in exactly the same fashion as in first year calculus with the same derivation laws. In particular, then, the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$; this vanishes identically if and only if $p$ divides $n$. So a power series $f(x)$ will have identically vanishing derivative if and only if it can be written as $h(x^p)$, where $h(x)$ is another power series, and therefore we are very far from the classical situation. Similarly, the $p$-th derivative of a power series will vanish identically.

The ancients (such as Hasse, Schmidt and Teichmuller) partially compensated for this as follows: Again from first year calculus, one knows that if $f(x)=\sum a_n x^n$ is a convergent power series over the real numbers then $a_n=\frac{D^n}{n!}f(0)$ where $D=\frac{d~}{dx}$; thus Hasse et al shifted the focus from $D^n$ to the operators $f\mapsto a_n$ which are indeed nonzero in finite characteristic. One writes this ({\it formally now}!) as $a_n=\frac{D^n}{n!}f(0)$ and calls these operators "Hasse derivatives", "hyperderivatives," "divided derivatives" etc. They satisfy many formal properties that may be guessed at from classical theory as well as other special properties arising in finite characteristic.

Let me now briefly sketch Jeong's proof and refer you to his paper for the details. You will see echoes in it of the classical theory of $y=y^\prime$ sketched above. So let $f(1+x)$ be our endomorphism of the $1$-units where $f(1+x)=\sum_i a_ix^i$ for $x$ small.

Step 1. The coefficients $a_i$ are in ${\mathbf F}_p$ for all $i$. Indeed, this follows from the fact $f((1+x)^p)=f(1+x^p)=f(1+x)^p$ as $f$ is an endomorphism.

Step 2. $a_i=0$ if and only if $\frac{D^i}{i!}f(1+x)$ vanishes identically. Indeed, Jeong uses the fact that $f(1+x)$ represents an endomorphism, and some algebra, to show that for all $i$
In particular, if the derivative of $f(1+x)$ vanishes at the origin, then $f(1+x)=g(1+x)^p$ for some endomorphism $g(1+x)$ (recall that the coefficients of $f$ are in ${\mathbb F}_p$).

Step 3: Let $j$ be the largest integer such that $f(1+x)=g(1+x)^{p^j}$ for some endomorphism $g(1+x)$; thus by Step 2, $g^\prime (1)\neq 0$. Let $a\in \{0,1,\ldots,p-1\}$ be in the class mod $p$ given by $g^\prime(1)$. Let $h(1+x)=g(1+x)/(1+x)^a$; clearly $h(1+x)$ is again an endomorphism and the quotient rule shows that $h^\prime(1)=0$. In particular, $h(1+x)$ is a $p$-th power and we may repeat the process.

Step 4: Step 3 allows us to inductively create $y\in {\mathbb Z}_p$ so that the power series for $f(1+x)$ IS $(1+x)^y$. As the $1$-units have no torsion (which is easily seen), one concludes that $f(u)=u^y$ forall $1$-units $u$.

Monday, August 4, 2008


In a rather ironical manner the first Higgs mass that is now excluded by the Tevatron latest results is precisely 170 GeV, namely the one that was favored in the NCG interpretation of the Standard Model, from the unification of the quartic Higgs self-coupling with the other gauge couplings and making the "big desert" hypothesis, which assumes that there is no new physics (besides the neutrino mixing) up to the unification scale. My first reaction is of course a profound discouragement, mixed with an enhanced curiosity about what new physics will be discovered at the LHC.
I'll end with these verses of Lucretius:
Suave, mari magno turbantibus aequora ventis,
e terra magnum alterius spectare laborem;
non quia vexari quemquamst jucunda voluptas,
sed quibus ipse malis careas quia cernere suave est.
[Pleasant it is, when over a great sea the winds trouble the waters, to gaze from shore upon another's tribulation: not because any man's troubles are a delectable joy, but because to perceive from what ills you are free yourself is pleasant.]

Group theoretic underpinnings of zeta-phenomenology

This is a follow up to the posts of Alain and Katia on the Vanderbilt
workshop this past May. It was a really great conference and I would again like
to thank the organizers for including me.

After I returned from the conference, I decided to try to write down
what I had talked about. In doing that, I finally was able to glimpse
certain underlying symmetries that I had long been looking for. I
wrote this up in a preprint that can be found here
(a slightly less clean version is in the arXiv...). I would like to explain this
preprint here; I apologize if this post runs a bit long.
Anyway, the upshot is that while classically the functional
equation can be thought of as a Z/(2) action (or a group of order 4 if
you throw in complex conjugation) in characteristic p there is
rather compelling evidence for an associated group which has the
cardinality of the continuum.

Most references not given here can be found in my preprint...

After Drinfeld's great work introducing Drinfeld modules (called
by him "elliptic modules") I began to try to develop the related
arithmetic. I soon learned that L. Carlitz had begun this study four
decades before! What one does is to take a complete, smooth, geometrically
curve X over the finite field Fq and then fix a place "\infty". The global
functions on the affine curve X-\infty is called "A" and it plays the
role of the integers Z in the theory. The domain A is of course a Dedekind
domain and will in general have nontrivial class group.
In particular, Drinfeld (and earlier Carlitz) develops a theory of lattices associated to A and finds that one can obtain Drinfeld modules much like one obtains elliptic curves classically. The Drinfeld modules are algebraic objects and so one can discuss them over finite fields etc. Like elliptic curves, there is also a Frobenius endomorphism with acts on Tate modules (defined in a very natural way). The resulting characteristic polynomial has coefficients in A and one has the local Riemann hypothesis bounds on the absolute values (at \infty) of its roots. So it really makes sense to try to create an associated theory of L-series for Drinfeld modules (and the many generalizations since devised by Drinfeld, G.Anderson, Y. Taguchi, D. Wan, G. Boeckle, R. Pink, M. Papanikolas, etc.)

Now in the 1930's Carlitz developed a very important special
case of Drinfeld modules (called the "Carlitz module") for A=Fq[T]. This is
a rank one object which means that the associated lattice can be
written in the form A\xi where \xi is a certain transcendental element
that looks suspiciously like 2\pi i. Using this \xi Carlitz established
a very beautiful analog of the famous formula of Euler on the values
of the Riemann zeta function at positive even integers. Indeed, he also
developed an excellent (and still quite mysterious) theory of "factorials"
for Fq[T] as well as analogs of Bernoulli numbers which are called
"Bernoulli-Carlitz elements;" they lie in Fq(T). With his incredible
combinatorial power, Carlitz then proceeded to compute the denominator
of these BC elements (all of this is in my paper); this is a "von Staudt" type
result. In particular, he presents TWO conditions for a prime to divide the
denominator. The first condition is very much like the one for classical
Bernoulli numbers. However, the second one involves the sum of the
p-adic digits of the number and seemed extremely strange into just recently.

Let k be the quotient field of A and let k_\infty be the completion
at \infty. So A lies discretely in k_\infty with compact quotient just
as the integers Z lie in the real numbers R. Let C_\infty be the
completion of the algebraic closure of k_\infty equipped with its
canonical topology. So one always views C_\infty as the analog of
the complex numbers except it is NOT locally compact; this is not a great
handicap and one just forges ahead.

In 1977 and 1978 I was at Princeton University (where N. Katz turned
me on to Carlitz's series of papers) and J.-P. Serre was at the Institute.
One knows that having a polynomial be monic is a very good (but
not perfect) substitute
for having an integer be positive (so the product of two monics
is obviously monic but the sum of two monics need not be monic). If
f is a monic polynomial one can clearly raise f to the i-th power where i is any
integer. So in keeping with the spirit of Carlitz and Drinfeld, it made
sense to ask if there were any other elements s so that f^s made sense.
After discussing things with Serre, I came up with the space
S_\infty defined by

S_\infty:=C_\infty^* \times Z_p;

i.e., S_\infty is the product of the nonzero elements in C_\infty
with the p-adic integers.

Let me briefly explain how you can express the operation

f |----> f^i

for a monic polynomial f of degree d and integer i in terms of S_\infty. So we
pick a uniformizer \pi at \infty; for simplicity, let's set
\pi=1/T. Then we have obviously

f^i=(\pi^{-i})^d (\pi^d f)^i

= (T^i)^d (f/T^d)^i

and this corresponds to the point (\pi^{-i}, i) in S_\infty. So in general
for s=(x,y) in S_\infty you define

f^s:=x^d (\pi^d f)^y ;

the point being that (\pi^d f) IS a 1-unit and so can be raised to
a p-adic power by simply using the binomial theorem.

For general A one has the nonclassical problem of having to exponentiate
nonprincipal ideals (as if the integers Z had nontrivial class group!).
It took a while but then (through discussions with Dinesh Thakur) we realized
that the above definitions naturally and easily extended to all fractional
ideals simply because the class group is finite AND the values lie
in C_\infty (as opposed to the complex numbers..).

So one can now proceed easily to define L-series in great generality
by using Euler-products over the primes of A. One always obtains *families* of entire power series in 1/x, where y is the parameter; thus one can certainly talk about the order of zero at a point s in S_\infty, etc. The proof that we obtain such families uses the cohomology of certain "crystals" associated to Drinfeld modules etc., by
G. Boeckle and R. Pink (see e.g., Math. Ann. 323, (2002) 737-795). The idea is thatwhen
y is a negative integer the resulting function in 1/x is a *polynomial* that can be computed cohomologically. Boeckle then shows that the *degree in 1/x* of these polynomials grows *logarithmically* with y (of course logarithmic growth is a standard theme of classical L-series). This, combined with standard and powerful results in nonArchimedean analysis, due to Amice, gives the analytic continuation.

Of course then a reasonable question arises: where is the functional equation?
It turns out that the evidence for *many* functional equations was
there all the time. However, the case A=Fq[T], which is the easiest
to compute with, is misleading (just as the classical zeta function of
the projective line over Fq is misleading; looking only at this function
one might suppose that ALL classical zeta functions of curves/Fq
have no zeroes....). It is only recently that calculations due to Dinesh Thakur and Javier Diaz-Vargas with more general A have given us the correct hints.

Indeed, for general A one writes down the analog of the Riemann zeta
function as

\zeta(s):=\sum_I I^{-s}

for s in S_infty. When A=Fq[T] one has the results of Carlitz mentioned
above at the positive integers i where i is divisible by (q-1). At the negative integers divisible by (q-1) one has "trivial zeroes" which in this case are simple.

So the obvious thing to do is to try to emulate Euler's fabulous discovery of the functional equation of the Riemann zeta function from knowledge only of special values (as in my preprint or, better, the wonderful paper of Ayoub referenced there!). However, this never worked (and one can immediately see problems when q is not 3) and so we were left looking for other ideas.

In retrospect, one reason that a direct translation of Euler's ideas did not work was that at the positive integers, one obtains Bernoulli-Carlitz *ideals* not values. Indeed, as in my paper, Carlitz's notion of factorial makes sense for all A *but* only as an ideal of A, not a value; so when one multiplies by this factorial, one must do it in the group of ideals and we are out of the realm of values alone.

In the mid 1990's, there was some essential progress made by Dinesh Thakur.
Dinesh decided to look at trivial zeroes for more general A than just
Fq[T]. He was able to do some calculations in a few cases; these calculations
were then much more recently extended by Javier Diaz-Vargas. What these
two found intrigued me greatly: If one looks at the values i where the
trivial zero at -i has order strictly greater than the obvious classical looking
lower bound (this is the "non-classical set") one finds that this set appears
to consist of integers with *bounded* sum of q-adic digits!

These inspired calculations of Thakur and Diaz-Vargas thrilled me and
vexed me at the same time! On the one hand, they are so obviously
p-adic that they guarantee we are looking at very new ideas, but on
the other hand I wanted to know just what these ideas might be!

Now first of all, these calculations really do tell us that some sort
of functional equation should be lurking about. Indeed, classically
the order of special values falls out of the functional equation. As
the calculations of Thakur and Diaz-Vargas are only hints; one will need
other techniques to make them truly theorems.

Still I wanted to do better. The set of integers i with bounded sum of
q-adic digits is remarkable. One can take one such i and torture its q-adic
digits in many ways and still stay in the set! It finally dawned on me
that all of these "tortures" really form a group and that this group
replaces the Z/(2) group of classical arithmetic.

So here is the definition of the group S_{(q)}. It consists of
all permutations of the q-adic digits of a p-adic number; you just
reshuffle them in any way you would like! Surprisingly, this shuffling
is continuous p-adically and so we obtain a group of homeomorphisms of
Z_p. This group is obviously huge and indeed its cardinality is that of the
continuum. And, clearly, this group permutes the set of i with bounded sum of q-adic digits etc.

One also sees that these permutations stabilize both the positive
and negative integers and also stabilizes the classes modulo (q-1).

The key point then is a refinement of the observations of
Thakur and Diaz-Vargas:

The order of the trivial zero at -i is an invariant of the action of S_{(q)}.

Again, this is just an observation (which is easily seen to be a theorem
in the A=Fq[T] case as there one only worries about whether i is divisible
by q-1 or not!). But it seems to point the way to deeper structure.
In fact, the special values appear to "know" that they lie on a family of
functions and this large automorphism group may help us control the family...

Finally, this all relates back to Carlitz's von Staudt result: It turns out that
the divisibility of the denominator of Carlitz's von Staudt result is also
an invariant of subgroups of S_{(q)}. This is really mysterious: On the
one had, we have invariants related to zeroes of function and yet on the
other hand we have invariants related to objects made up from special
values. I don't have any good explanation for this at this point. Nor
can I guess, like Euler did, as to the exact form a global statement should take....

There is an associated theory of modular forms on Drinfeld's upper half space. In the past few years, great progress has been made on these forms by Gebhard Boeckle using the techniques mentioned above. There is a great deal of mystery in his results and perhaps these mysteries are related to the huge group of symmetries that now seems to underlie the theory.

Friday, August 1, 2008

F_un days three and fourth: the epilogue....

By AC and K. Consani

The second part of the workshop on the interaction between NCG & the field with one element held at Vanderbilt University this past May 15-18, included the presentation of several new results as well as some speculative discussions with topics ranging from number theory and noncommutative/arithmetic geometry to the most recent developments in Hopf-cyclic theory.

Within the broad area of number-theory/arithmetic geometry we shall review two talks both presented on Saturday May 17th.

In the first talk, David Goss presented his new ideas on ``Zeta-phenomenology in characteristic p''. This great talk has been inspired by the fabulous insights of L. Euler, the first zeta-phenomenologist, as David likes to say. The talk of Goss focussed on how to set up analogs of Bernoulli-Carlitz elements in complete generality. Click here to access his summary in pdf form.

The second talk with a number-theoretic flavor was given on Saturday by Cristian Popescu. Cristian spoke on "1-Motives and Equivariant Iwasawa Theory". The main conjecture in classical Iwasawa theory makes a link between special values of l-adic L-functions and certain Iwasawa modules which are the characteristic zero analogues of the l-adic realizations (Tate modules) of Jacobians of smooth projective curves defined over finite fields of characteristic different from l. From this vantage point, the classical main conjecture is the exact analogue of Weil's Theorem expressing global L-functions in characteristic p (essentially) as characteristic polynomials of the geometric Frobenius morphism acting on the l-adic realizations of the corresponding Jacobian varieties. In the talk, Cristian explained how to extend this analogy in two directions: where on one hand Jacobians are replaced by 1-motives and on the other L-functions are replaced by G-equivariant L-functions.

Among the talks in noncommutative-arithmetic geometry, we will review the talk given by Jorge Plazas-Vargas on "Endomotives, abelian varieties and real multiplication". Jorge explained his new insights on the construction of an algebraic endomotive closely related to the Bost-Connes system which arises by considering iterated powers of a Morita type endomorphism on a real multiplication noncommutative torus. Click here to access his summary in pdf form.

Abhishek Banerjee spoke on "Periodicity in Cyclic Cohomology and Monodromy at Archimedean Infinity". In the talk, Abhishek exposed his recent results on an interpretation of the local monodromy operator for degenerations of arithmetic varieties (both over a disk and at archimedean infinity) in terms of Connes's periodicity operator in cyclic theory. Click here to access his summary in pdf form.

Still on Saturday, Snigdhayan Mahanta gave a speculative interesting talk with the goal to convey his recent ideas that simplicial/cyclic topology should capture the combinatorial aspects of the geometry over the field with one element. In the talk Sniggy discussed some general features of the geometry over F1, mostly highlighting the simplicial structures. Click here to access his summary in pdf form.

The talks of Nadia Larsen "Phase transition in the Bost-Connes C*-dynamical systems from number fields" and Sergey Neshveyev "On von Neumann algebras arising from Bost-Connes type systems" (pdf file here) presented their recent results (in collaboration with Marcelo Laca) on a phase-transition theorem for a quantum statistical mechanical system (due to Ha-Paugam) which generalizes the Bost-Connes system to an arbitrary number field K. Here is the pdf.

Most of the talks on Sunday concentrated on several very recent results in Hopf-cyclic theory.
Masoud Khalkhali talked on "Hopf cyclic Cohomology in Braided Monoidal Categories". In the talk he explained his insights, described by working out several specific examples, on how to develop a Hopf cyclic theory for Hopf algebra objects in an abelian braided monoidal category. Here is his summary in pdf form.

Bahram Rangipour then spoke on "Hopf algebras arising from formal vector fields on the real line and their Hopf cyclic cohomology". Bahram's talk reviewed Hopf cyclic cohomology with coefficients and the powerful results of his recent collaboration with Henri Moscovici. Here is the summary in pdf form.

Finally, Atabey Kaygun spoke on "Products in Hopf-cyclic (co)homology". Atabey derived the structure maps of the cyclic module which defines the Hopf-cyclic cohomology using very basic principles. Here is the pdf of his summary.

As an epilogue of this long-overview articles on the Workshop at Vanderbilt University we would like to thank all the speakers for their spontaneous and generous participation and for sharing their ideas with us about the field with one element and the new connection with NCG. We also would like to thank all the participants for coming to the talks and patiently listening to the discussions which were at times intense and certainly "very alive" and stimulating...

Wednesday, June 4, 2008

F_un : day two...

By A. C. and K. Consani

On the second day of the workshop we presented the main results contained in our recent preprint (joint work with Matilde Marcolli) "Fun with F_1". The pdf file is downloadable here. Our main point is to establish a bridge between F_1 and the quantum statistical mechanical system (known as the BC-system) which gives, after passing to the dual system, a spectral realization of the zeros of the Riemann zeta function, as well as a trace formula interpretation of the Riemann-Weil explicit formulas. While a precise algebraic definition of F_1 is still elusive and premature, the section 13 of the paper "Sur les analogues algébriques des groupes semi-simples complexes" of J. Tits gives the construction of the geometries corresponding to the classical Dynkin diagrams starting from the 4 basic rank two cases: A_1 x A_1, A_2, B_2, G_2 which (over F_1) are associated to the geometries of the polygons with 2, 3, 4 and 6 sides. A striking result of J. Tits is that the groups associated to the geometries over F_1 are the Weyl groups. For instance GL_n(F_1) is the symmetric group S_n of permutations of n letters. Thus, in particular the series A_n suggests to view the category of "finite dimensional vector spaces over F_1" as the category of finite sets (with an additional "0" element). Further unpublished work of Kapranov and Smirnov suggests that, similarly to what happens for F_q, one should have for each integer n an extension F_1^n of degree n of F_1 and the category of "finite dimensional vector spaces over F_1^n" should be that of finite sets with a free action of the cyclic group Z/n of order n (still with a "0" added). This lead Soulé to the following equation :

which provides a definition of the ring obtained by "extension of scalars" from F_1 to Z. Now, for q a prime power, the extensions F_q^n can be organized (after choosing an algebraic closure of F_q) as an inductive system with inclusion of F_q^n in F_q^m when n divides m. For such a pair n,m one has similarly a natural inclusion of the cyclic group Z/n in the cyclic group Z/m which relates (by two adjoint functors) the categories of "finite dimensional vector spaces" over F_1^n and F_1^m. Thus one gets

But the algebra on the right hand side is the abelian part of the BC-system which thus arises naturally by extension of scalars from the field with one element.. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions F_1^n, while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for the BC endomotive, which is related to the original Hecke algebra description. Finally, we described the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra.

The talk by Henri Moscovici "Spectral triples and Q-lattices" constructs the missing type-III (twisted) spectral triple for the GL(2) analogue of the BC-system. Here is his summary:
"The two-part paper: Modular Hecke algebras and their Hopf symmetry; Rankin-Cohen brackets and the Hopf algebra of transverse geometry, (Moscow Math. J. 4 (2004), no. 1) exhibited several striking geometric features of the algebra generated by modular forms and Hecke operators. They all arose from the observation that the Hopf algebra encoding the transverse symmetries of codimension one foliations admits a natural action on this modular Hecke algebra and on its compressions.
The picture that emerged was that of a surprisingly close analogy with the transverse geometry of a codimension 1 foliation. This lecture outlines the construction of a Dirac twisted spectral triple, essentially obtained by replacing the usual Poincaré metric of the canonical bundle over the (connected component of the) Shimura variety with the Ramanujan metric defined in terms of the Dedekind eta function.
This type III spectral triple should provide the missing analytic framework for the underpinning geometry and explain its main attributes: the circle analogy, the arithmetic transgression of the Euler class, and the pseudodifferential calculus underlying the Rankin-Cohen deformations of the modular Hecke algebras."

Gunther Cornelissen's talk "Some Riemann Zeros for Function Fields" explained some computations for the space of toroidal automorphic forms. The following is his summary. "In the 1970s, Don Zagier suggested the following: a formula of Hecke proves that a GL_2 Eisenstein series of weight s integrates over a non-split torus to (almost) the zeta function of the corresponding quadratic field extension. Therefore, if we consider the space T of automorphic forms such that all these integrals vanish, and this space has a unitary structure, then the Riemann hypothesis (and simplicity of the zeros) would follow. I compute this space of so-called toroidal automorphic forms for the function field of three particular elliptic curves over finite fields. Here, the Riemann hypothesis is easily verified using the point-counting interpretation of the zeta function. In the talk, however, no point counting interpretation occured. Instead, only the Tate integral definition of zeta has been used. The condition that the toroidal integral vanishes for the constant extension torus is translated into an infinite system of linear equations in values of the automorphic form at the vertices of a Bruhat-Tits tree (for this, we need to iterate with Hecke operators). By solving this system and simultaneously the eigenvalue equation for the adjacency operator on the tree, we find that only 3 non-cusp eigenforms can occur as possible toroidal eigenforms. Since we know the eigenvalues of Eisenstein series, we can conclude that the Riemann hypothesis holds by comparing these with the possible toroidal eigenvalues.
If one does this computation for a rational function field, one can prove that its zeta function doesn't have any zeros. Now that is cracking a nut with a sledge hammer.
By some more advanced arguments (admissibility theorem of Winnie Li, integrating over other specific tori, multiplicity one for cusp forms, higher degree Hecke operators), one can even prove that in these three cases, the space of toroidal forms is one dimensional, spanned by the Eisenstein series of weight a zero of the corresponding zeta function.
- paper is on arxiv:0710.2994
- thesis of Oliver Lorscheid, available from (in this thesis, the space of toroidal forms is studied in general, and specifically computed for elliptic curves, but by a method that is a bit different, and in particular does not allow one to prove the Riemann hypothesis)."

Finally, in his talk: "Algebraic Continuation of multiple zeta values by Renormalization" Li Guo gave a remarkable application of the algebraic Birkhoff decomposition method of renormalization to the study of multiple zeta values outside the natural domain of convergence of the multiple series involved. His talk is available here.

Saturday, May 24, 2008

NCG and F_un

(by A.C. and K. Consani)

Right after the end of the Sixth Annual Spring School/Conference on Noncommutative Geometry and Operator Algebras, a second meeting took place at Vanderbilt University, on May 15-18. This workshop has been dedicated to explore some aspects of several emerging relations linking Noncommutative Geometry and the geometry over the field with one element.
In the next weeks, we expect to post a more elaborate and thoughtful overview on this new interesting development in noncommutative/arithmetic geometry. In the meanwhile, the following is a first outline of the topics covered in several of the main talks. Some of the speakers have supplied us with the files of their transparencies and/or with an abstract of their presentation. When available, we added the abstracts within quotation marks "... ". The pdf files presently available have also been included here below.

The first day of the meeting was dedicated to the review and the discussion of the following 5 papers whose main subject has an evident connection with the field with one element.

1) Lisa Carbone gave in her talk "Kac-Moody groups, finite fields and Tits geometries" an overview of the seminal paper by Jacques Tits "Sur les analogues algebriques des groupes semi-simples complexes" (Colloque d'Algebre superieure, 1956, Bruxelles). The following is her review.
"Motivated by trying to find a "geometric" interpretation of a finite dimensional simple Lie group G in contrast to the "algebraic" version of G proposed by Chevalley the previous year, Tits introduced a "geometry" X which has G as its automorphism group. Tits' geometry X also had the mysterious property that when constructed over a finite field F_q , one could take the limit q --> 1 in which the group G tends to the discrete subgroup W (the Weyl group of G) and the geometry X tends to the geometry of W.
Tits' examples are quite sketchy in the 1956 paper. I would like to propose however that this was the seed of a deep circle of ideas that Tits cultivated and developed over three decades. The 1956 paper seems to have been a precursor to the notion of a Bruhat-Tits building for a Chevalley group over a finite field, or a simple algebraic group over a nonarchimedean local field. These constructions also evolve naturally into the notion of a Tits building for a Kac-Moody group over a finite field associated to the Tits functor for Kac-Moody groups.
In all of these subsequent constructions of Tits, the notion of a "field with 1 element" is present, both on the group level, and in the associated Tits geometry.
In my talk, I attempted to give an overview of examples from each of the classes described above, and to indicate what happens as we try to take a limit F_q --> F_1. This viewpoint has been very useful in my work, and I indicated a number of things I have been able to prove using the Tits geometries over F_1."

2) Christophe Soule review in his talk "Algebraic varieties over F_1" the main aspects developed in his paper "Les varietes sur le corps a un element" (Mosc. Math. J. 4 (2004), no. 1, 217--244, 312). An abstract of his presentation is downloadble here and in clear below:
Define a gadget over F_1 to be the pair X=(X, A_X) where X is a covariant functor from the category F of finite abelian groups to the category of sets, and A_X is a complex algebra. Given a finite abelian group G, a point x in X(G), and a character s of G, we assume given a character, e_{x,s} of A_X. If f: G-> G' is a morphism and y belongs to X (G'), the following equality is supposed to be satisfied : e_{f(y),s}= e_{y , s \circ f} for any character s.
An affine variety V over Z defines a gadget X over F_1 by letting X(G) be the set of points of V in the group algebra of G and by defining the algebra A_X to be the ring of regular functions on the complex points of V (with the obvious evaluation maps).

A morphism X -> Y, between two gadgets over F_1 consists of a natural transformation from the functor X to Y and a morphism of algebras from A_Y to A_X compatible with evaluation maps. It is called an immersion when both maps are injective.
An affine variety over F_1 is a gadget X such that
- For every G the set X(G) is finite;
- The complex algebra A_X is a commutative Banach algebra;
- There exists an affine variety X_Z over Z and an immersion i: X -> X_Z of gadgets satisfying the following property:
for any affine variety V over Z and any morphism of gadgets h: X -> V, there exists a unique algebraic morphism h_Z: X_Z -> V such that h equals h_Z composed with i.

Examples of varieties X_Z, where X is an affine variety over F_1, include smooth toric varieties and the algebraic group-schemes GL_2 and GL_3.

3) Niranjan Ramachandran gave in his talk "Zeta functions and motives (d'apres Manin)" an overview of the paper by Y. Manin "Lectures on zeta functions and motives (according to Deninger and Kurokawa)" (Columbia University Number-Theory Seminar, New-York 1992, Asterisque No. 228 (1995), 4, 121--163).
The following is his review.
"The aim of my talk was to provide a brief introduction to the beautiful paper of Yuri Manin (Lectures on motives and zeta functions - to be found on Katia's website on the fascinating ideas of Christopher Deninger and Nobushige Kurokawa on zeta functions and F1.
The basic analogy between number fields and function fields has driven much of 20th century arithmetic geometry. This leads one to the desire to view Spec Z as a curve, but over which field? Of course, F1.
Deninger has expressed the completed Riemann zeta as R divided by s.(s-1)
where R is a regularized determinant to be viewed as infinite-dimensional analogue of a determinant of an endomorphism of a finite dimensional vector space. Compare with the zeta function of a smooth projective curve (of genus g) over a finite field F_q: a polynomial of degree 2g divided by (1-t) (1-qt) where t is the variable q^{-s}.
Manin provides an overview of the theory of motives over a finite field. He comments that even though we may not be able to define F1 or the category of varieties or motives over F1, we can certainly discuss zeta functions of motives over F1. The discussion strongly suggests that the only zeta functions that one obtains are generated by (s-n) for an integer n. Classical groups G are supposed to define varieties over F1 (original insight of Jacques Tits that G(F1) = W_G the Weyl group of G) as are projective spaces P^n; the zeta function of P^n over F1 is supposed to be s.(s-1)....(s-n). Thus the denominator in Deninger's expression for the completed Riemann zeta function is the zeta function of P1 over F1 which exactly parallels the function field case.
Manin also points out that the stable homotopy groups of spheres should be viewed as the algebraic K-theory of F_1 and the classical map J in algebraic topology from the stable homotopy groups to the algebraic K-theory of the integers is the one induced by the map F1 --> Z. This is a very important observation. The order of the image of J involves Bernoulli numbers (and hence zeta values!).
Manin also discusses the Kurokawa product of zeta functions and provides many examples from arithmetic and geometry (Selberg zeta, multiple gamma functions, ..) which could not be covered in this lecture. In particular, Kurokawa has defined the zeta function of (Spec Z) x_{Spec F1} (Spec Z) even though a mathematical definition of the fibre product is still lacking."

4) Jack Morava presented in his talk "K-theory of ring objects in homotopy theory" some relevant aspects of the paper by D. Quillen "On the cohomology and K-theory of the general linear groups over a finite field" (Annals of Mathematics, 2nd Ser., Vol.96, No.3, 1972, 552--586). An overview of his presentation is downladable here. Here is his abstract:

"Direct sum gives the category of finitely generated projective modules over a ring R (together with their isomorphisms) a symmetric monoidal structure. In 1972, Quillen defined the algebraic K-theory of R in terms of the best approximation to the geometric realization of this category by an abelian object in the homotopy category: an infinite loop-space or, in topologists' contemporary language, a spectrum.
These ideas have been vastly extended in the four decades since, in particular to general symmetric monoidal categories (Segal, eg finite sets) or to `categories with cofibrations and weak equivalences' (Waldhausen, eg finite cell complexes). Relatively recent developments (eg the theory of symmetric spectra) in our understanding of commutative ring objects in homotopy theory provide a unified approach to these generalizations and to related constructions (eg `topological' Hochschild and cyclic (co)homology).
My talk was basically historical; I tried to sketch the development of this language, and to use it to compare the category of vector spaces over a finite field and the category of finite sets. I wanted to clarify the extent to which the K-theory of the latter can be viewed as a limit, as q->1, of the K-theory of F_q."

5) Eugene Ha lectured on the main parts of the theory developed by Nikolai Durov in his preprint "New Approach to Arakelov Geometry" (arXiv:0704.2030). Here is the Eugene's review:
"Arakelov geometry is an amalgam of scheme-theoretic algebraic geometry and complex differential geometry that allows one to do intersection theory on models of algebraic varieties over the "compactification" of Spec(Z). However, missing from Arakelov's theory is a direct definition of the fiber at archimedean infinity, and in particular, of the notion of an "archimedean valuation ring." Recently, this situation has been rectified by Nikolai Durov who has created a full-fledged theory of generalized (commutative) rings and schemes that provides a framework for treating compactified arithmetic varieties in an uniform scheme-theoretic manner [A New Approach to ArakelovGeometry, 2007]. The "classical" algebraic geometry of Grothendieck is preserved in Durov's theory since the category of generalized rings contains the classical commutative rings as a full subcategory.
To develop a theory of "rings," like the "localization of Z at the infinity prime," that are monoidal but not additive, one can try to first frame classical ring theory in categorical terms,which has the advantage of allowing one to think of additivity as a monoidal structure. (For example, a ring is simply a monoid in the monoidal category of abelian groups.) It is well-known that the categorical notion that enables this transition is that of a monad in sets, i.e., a monoid in the monoidal category of endofunctors of sets. For a (classical) commutative ring R, the monad M_R attached to R is the functor that maps a set S to the set underlying the free R-module generated by S. The category of R-modules is then the category of modules of the monad M_R, and the ring R itself can be recovered from this category in the usual way (take the center of the endomorphism ring of the identity functor of R-modules).
This motivates the definition of a generalized commutative ring as a monad in sets which is moreover algebraic (commutes with filtered inductive limits of sets) and commutative (an algebraic monad in sets determines a family of n-ary operations on its modules, and commutativity for the monad means, roughly, that all these n-ary operations commute).
To see how one might arrive at a "correct" notion of the "local ring of Z at infinity" (or rather of its completion) suitable for a scheme-theoretic Arakelov geometry, Durov considers the notion of a "Z_infinity-lattice" in a real vector space. In the p-adic case, Z_p-lattices in a finite-dimensional p-adic vector space V correspond (up to similitude) to the maximal compact submonoids of End(V). This leads to the definition of Z_infinity-lattices (again, up to similitude) in a finite-dimensional real vector space E as the compact convex symmetric bodies in E. Further comparison with the p-adic case leads to the definition of the set underlying the free Z_infinity-module with basis S as the standard octahedron in R^{(S)}, and hence to the definition of Z_infinity as the generalized ring corresponding to this endofunctor.
In particular, Z_infinity is a generalized subring (i.e., algebraic submonad) of the real numbers R, as is Z_+, the generalized ring that maps a set S to the set of formal finite non-negative-integral linear combinations of elements of S. Thus one can take the intersection of Z_+ and Z_infinity: this is Durov's definition of F_1, the so-called field of one element. One can also describe F_1 as the free algebraic monad in sets with a single 0-"arity" generator. Modules over F_1 are simply sets with a marked point.
Going far beyond generalized commutative algebra, Durov has also developed a rather complete theory of spectra and generalized schemes. In his theory the "affine line" Spec(Z) is an affine scheme defined over F_1, and the compactification of Spec(Z) is a pro-generalized scheme.
Finally, while many of the motivations and constructionsof Durov's theory are very natural, the results of some of his computations differ from various widely-held expectations. Forexample, Durov has computed the Picard group of the compactification of Spec(Z) and has found that it is the multiplicative group of positive rational numbers, whereas the function field-number field analogy suggests that it should be be the positive real numbers. Moreover, the product S of Spec(Z) with itself over F_1 is shown to be Spec(Z) in Durov's theory, which is inconsistent with Kurokawa's definition of the zeta function of S."

Friday, May 16, 2008

Vanderbilt Lectures

in Vanderbilt University wrapped up yesterday. Some of the lectures and lecture series are already available online here.

Wednesday, April 9, 2008

On Gelfand-Naimark Theorems

There seems to be some inconsistency, mainly in the Internet, in naming one of the main theorems proved by Gelfand and Naimark in their foundational 1943 paper:

  • On the imbedding of normed rings into the ring of operators in Hilbert space. Rec. Math. [Mat. Sbornik] N.S. 12(54), (1943). 197--213


I have seen the result in question referred to as Gelfand-Naimark' or Gelfand's theorem. Also, talking to younger people I get a sense of confusion as to how it should be called. Before getting to the theorem in question, let me indicate why this paper is so important. This paper is fundamental for the following 4 reasons:

1) C*-algebras were abstractly defined in this paper for the first time ever (in their main axioms they had two extra conditions, which as authors themselves indicated, but were unable to prove, were redundant. It took some 17 years to reach to the current concise formulation of the main axiom, called the C*-identity-see below. This needs another post to explain and I hope we can get to that in another time. This book gives a detailed account of this circle of ideas). Together with Murray-von Neumann's series of papers on Rings of Operators, or what later came to be called von Neumann algebras (1936-1943), the Gelfand-Naimark paper formed the foundaton stone of operator algebras and, eventually, noncommutative geometry.

2) Commutative C*-algebras were fully characterized in this paper as algebras of continuous functions on compact spaces. This is the theorem that concerns us in this post and we shall get to that later.

3) General (i.e. not necessarily commutative) C*-algebras were shown to admit a faithful embedding in the algebra of bounded operators on a Hilbert space.

4) The notion of state on a C*-algebra was introduced (but not under its current name) and used in the proof of 3) . This was later streamlined by I. Segal in 1947 and today we talk of the GNS (Gelfand-Naimark-Segal) construction.

Now the theorem in item 2) above (let us call it CGNT for `commutatve Gelfand-Naimark theorem') only appears as a Lemma in the paper, in page 3, and was not even mentioned in the introduction! For sure it was needed for the proof of the noncommutative theorem, but they also mention that it is of independent interst as well. Obviously the authors felt, as reflected in their title, that the noncommutative result in the main theorem of the paper. I have seen CGNT referred to as the Gelfand-Naimark theorem or Gelfand's isomorphism theorem. Operator algebra books correctly call it the commutative Gelfand-Naimark theorem. Wickepdia calls it Gelfand's reprsenation theorem and reseves Gelfand-Naimark for the noncommutative theorem in item 3) above.

The proof of the CGNT is based on Gelfand's theory of commutative Banach algebras and in fact is one of its landmark applications. Another, earlier, major success of the theory was Gelfand's surprizingly short and elegant proof of Wiener's 1/f theorem: if a function f has an absolutely convergent Fourier series and is nowhere zero then its inverse 1/f has an absolutely convergent Fourier series as well.

A (complex, unital) Banach algebra is a complex algebra, equipped with a complete normed vector space structure. Futhermore, the norm and the multipcative structure are related by the identity . A basic commutative example to keep in mind is the algebra C(X) of complex valued continuous functions on a compact Hausdorff space under pointwise addition and multiplication and the sup norm. But there are many other commutative examples of different natures. Important notions introduced in Gelfand's theory were that of the spectrum of an element and the fact that it is always non-empty; the spectrum of the algebra; and the Gelfand transform. The spectrum Spec (A) consists of multiplicative linear maps A....> C (charcaters or C-points of A). Being a subset of the dual space of A, it inherits a natural compact Hausdorff topology. Spec (A) can also be described as the set of maximal ideals of A: to a character associate its kernel...... The Gelfand transform is the more or less tautological map:

It is clearly an algebra map and is contractive (norm decreasing), but it need not be faithful. In the special case when A is the group algebra of an abelian group it reduces to Fourier transform. Finding the right class of commutative Banach algebras for which is an isometric isomorphism is what is acheived by CGNT. A C*-algebra is an involutive Banach algebra which satisfies the C*-identity

It is hard to exagerate the importance of the C*-identity. It has many implications, e.g. the uniqueness of the C*- norm, continuity of involutive algebra maps,....... Typically, for an involutive Banach algebra we have just an inequality x*x\leq x^2. The C*-identity puts
C*-algebras in a very special place among all Banach algebras, rather similar to the privileged position of Hilbert spaces among all Banach spaces. The world of Banach spaces is wild, but the Hilbertian universe is tame!

A few comments are in order here:

i) These days everything must be `categorical' (I am afraid this is utterly out of date now and I should say `categorified'!-but let us be pedantic). In fact the CGNT goes a long way towards establishing an equivalence between the categories of commutative unital C*-algebras and compact Hausdorff spaces. Let us call these categories (with appropriate notion of morphism in each case) A and S. We have two functors
Spec: A^o .......>S and C: S ^o..............>A

(o means the dual or opposite category), assigning the spectrum and the algebra of complex valued continuous functions, respectively. These are equivalences of categories, (quasi) inverse of each other. In fact the composition C Spec: A .....> A is just the Gelfand transform and by the CGNT we know that it is isomorphic to the identity functor. This is the hard part. To show that the functor Spec C: S .......>S is isomorhic to the identity functor is much easier and is elementary. You just have to show that the Spec (C(X))=X for any X.

Now this way of thinking about the CGNT makes it very similar to other duality theorems in mathematics that puts in duality a category of spaces with a category of commutative algebras. A grand example of this is Hilbert's Nullstellensatz which implies that: the category of affine algebraic varieties over an algebraically closed field is equivalent to the dual of the category of finitely generated commutative reduced algebras. (reduced means there are no nilpotent elements). I think, comparing the two theorems, this reduced condition should be compared with the C*-identity. Here is a question that has puzzled me for some time and is for experts in noncommutative algebraic geometry: what is the right notion of a noncommutative affine algebraic variety sugested by Nullstellensatz? We know that in NCG the category of C*-algebras is in many ways a good category of noncommutative spaces. In other words we keep the C*-identity. In the algebraic case shall we keep this reduced condition?

ii) The spectrum of a commutative C* -algebra may not be `visible' at first sight, unless the algebra is already in the form C (X). This often leads to interesting results. For eaxmple, the algebra C_b (R) of bounded continuous functions on the real line R is a unital commutative
C*-algebra in a natural way. So, by CGNT , we know that C_b (R) = C ( X), where X is a compact Hausdorff space. What is X and how is it related to R? It is easy to see that is in fact the Stone-Cech compactification of R. More generally, for a locally compact Hausdorff space X the spectrum of C_b (X) can be shown to be homeomorphic to , the Stone-Cech compactification of X. For an example of a different flavour, let X be a topological space which is manifestly non-Hausdorff and let A=C (X). Then the spectrum of A has the effect of turning X into a Hausdorff space and is in some sense the `Hausdorffization' of X. The reader should try to describe the spectrum of .