Basic Noncommutative Geometry

Well, finally BNCG is published now. You can check its cover here. The book is published by the European Mathematical Society Publishing House.

Happy holidays to everyone!

I.M. Gelfand, 1913-2009

The noncommutative geometry blog is deeply saddened to inform its readers of the passing of Israil Moiseevic Gelfand, one of the most influential mathematicians of the 20th century, on October 5 at the age of 96. Gelfand's monumental contributions to mathematics and science covered many areas including commutative normed rings and functional analysis, representation theory of Lie groups and Lie algebras, generalized functions, mathematical physics, partial differential equations, and theoretical biology. His celebrated structure theorems for commutative and noncommutative C*-algebras are among the pillars of operator algebras and NCG. We hope to cover Gelfand' s work and impact more extensively but meanwhile you can check the AMS web site for more on Gelfand's work.

And here is a NYT obituary, and an Interview from six years ago.

A DAY OF JAMI MEETING AT JHU

I was in Baltimore last week attending the Jami Conference at JHU which was followed by a two days long workshop on F_1. Both events were coordinated mostly by Katia Consani, with the help of S. Mahanta (and myself). I went to all lectures and feel ready to make some comments for those talks which I had the impression to understand. For now I'll just talk about the first day.

The first talk was by Paul Baum, with his abstract and title:

Morita Equivalence Revisited

The essence of this talk, in Paul's very pleasant lecturing style, is that in their joint work with A.M.Aubert and R.J.Plymen, on representation theory of reductive p-adic groups, the authors deal with algebras which are finite extensions of commutative algebras but to which the tools of NCG, such as cyclic homology, apply succesfully. In order to formulate their conjectured geometric description of the primitive ideal space in the representation theory of reductive p-adic groups, the need for a suitable weakening of Morita equivalence of algebras has emerged. The new notion is defined in the general algebraic set-up and time will tell if it provides a useful comparison for algebras. One basic difficulty seems to be that the relation is defined by iterating noncommuting basic steps, so that the corresponding maps of cyclic cohomology groups depend upon the chain of steps implementing the equivalence of two objects. Thus in a way, it seems that they are defining a new category of algebras, where the objects are the same but the morphisms are obtained as composition of the various allowed steps in their equivalence relation.

The second talk was by Masoud Khalkhali, with his abstract and title

Holomorphic Structures on the Quantum Projective Line
The three authors of this work, Masoud Khalkhali, Gianni Landi and Walter Van Suijlekom, have started very recently this joint work and are in an "exploratory" stage with this q-deformation of the two sphere. It is a very interesting concrete case to discover how much of the miraculous structure of the two sphere as a complex curve, or equivalently as a conformal manifold, actually survive the q-deformation. Many tools of NCG are available there, including the abstract perturbation of conformal structures by Beltrami differentials as explained in Chapter 4, section 4, pages 339-346 of the ncg book. A great challenge is to prove an analogue of the measurable Riemann mapping theorem in the q-deformed case. The formalism of q-groups allows one to set-up a simple algebraic framework but the real challenge resides in the analysis.

That was all for the talks of the morning session. The first talk in the afternoon was given by David Goss with abstract and title

The group S(q) and indications of functional equations in finite characteristic

In this brilliant talk congruences were flying all over and from my own "amateur" point of view I learnt something I should have known for years, namely the congruence due to Lucas on binomial coefficients. It says that modulo a prime p, the binomial coefficient (n choose k) is the product of the binomial coefficients (n_j choose k_j) of the respective digits of n and k in base p. In particular this expression is invariant under the operation of arbitrarily permuting the digits. David explained in his talk how to define a group S(q) of homeomorphisms of the compact space Zp of p-adic integers. His group has the cardinality of the continuum and involves arbitrary infinite permutations of the p-adic digits. The computational evidence shows that this group should be involved in a functional equation for characteristic p zeta functions. Since David has written a post in this blog on this precise topic I will just refer to his explanations.

The next talk was by Sacha Goncharov with abstract and title

The quantum dilogarithm and quantization of cluster varieties The subject of the talk is the joint work of Sacha Goncharov with V. V. Fock. The talk was excellent but it created a quite uneasy feeling in me which I had a hard time to identify. At first I thought it was due to the usual difficulty I have to hear a talk on something called the "quantum torus" and which looked like a reincarnation of the work I had done in 1980 on the representation of the noncommutative torus in L^2(R) and on the duality I had discovered there between the torus for theta and 1/theta (cf line 10 of page 8 of the english translation of the note). But in fact this "reincarnation" was appearing in a very strange way, with a factor of i=square root(-1) in the exponents of the operators acting on L^2(R) and that was the real reason why I felt disturbed. What I have found since then, and checked in an email exchange with Sacha, is that the commutation relations of these self-adjoint operators are only "formal" and hold on a dense domain but these operators actually do not commute. If you take the simplest case where q=1 then the presentation of the "quantum torus" is simply:

------------------------------A=A*, B=B* and AB=BA---------------------------------

Thus you should get two commuting self-adjoint operators in L^2(R) given by Af(s) = exp(s) f(s) and Bf(s)= f(s+2\pi i). Now it is true that A and B are self-adjoint, since A is a multiplication operator by the (positive) function exp(s) and B is similar in Fourier. But the trouble is that they do not commute, even though they commute on a dense domain. Thus when you exponentiate to the corresponding one-parameter groups exp(itA) and exp(isB) just do not commute. This can be seen because the function s-> exp(s) is injective from R to C and hence the operator A generates the algebra which is maximal abelian in L^2(R). This is the algebra of all multiplication operators. Similarly the operator B generates the algebra of all translation operators.. and of course these two algebras not only do not commute but form an irrep in L^2(R). Another way to understand why the formal commutation on the dense domain is not enough is that the vectors in the dense domain are not analytic vectors and for instance the L^2 norm of A^n f for f(x)=exp(-x^2/2) grows like exp(n^2).... One could argue that the case q=1 is special and that requiring exact commutation is too demanding but when one considers the "dual torus" (corresponding to 1/theta) it is quite reasonable to require that it exactly commutes with the initial torus (and in fact generates its commutant as in my 1980 note). However, for the same reason as in the case q=1, this will just fail with the above representation of the "quantum torus"...

I checked with Sacha, who told me that he knew that the operators do not commute, that this is not a problem for what they are doing, namely the unitary group representation. But for the "algebra" it clearly indicates a kind of dichotomy between "formal computations" of deformation quantization and real Hilbert space stuff.

The final talk of this first day of the meeting was by Patrick Brosnan, with abstract and title

Essential dimension

I knew the author from his work with PRAKASH BELKALE on the conjecture of Kontsevich. In fact it was pretty useful for me to hear also Marc Levine playing the role of the "motivic expert" throughout the conference. The talk of Patrick Brosnan started by defining in a very simple and natural manner the notion of dimension using degree of transcendence of field extensions, as a classical way to count the "number of parameters". His talk ended with a beautiful result which I had the impression to understand, namely the evaluation of the essential dimension of the spinor group Spin_n which is the totally split form of the spin group over a field k. The result says, roughly, that this essential dimension is between 2^((n-1)/2) -n(n-1)/2 and 2^((n-1)/2) and hence has an "exponential" demand of new parameters. This is quite striking given that the previously known lower bound was linear in n.....

A very simple example

In my previous contributions to this blog , I have mentioned how the calculations of Dinesh Thakur and Javier Diaz-Vargas suggested that the nonclassical trivial zeroes of characteristic ${\displaystyle p}$ zeta functions associated to ${\displaystyle {\backslash bfF}_q\left[t\right]}$ should have the following two properties (where nonclassical means that the actual order is higher than what one would expect from classical theory):

1. If a nonclassical trivial zero occurs at ${\displaystyle -i}$ then the sum of the ${\displaystyle p}$-adic digits of ${\displaystyle i}$ must be bounded.

2. The orders of the trivial zeroes should be an invariant of the action of the group ${\displaystyle S_{\left(q\right)}}$ of homeomorphisms of ${\displaystyle Z_p}$ which permute the ${\displaystyle q}$-adic digits of a ${\displaystyle p}$-adic integer.

In my last entry, I discussed Dinesh's remarkable result on valuations of certain basic sums in this game; one key point is that the valuations for arbitrary ${\displaystyle d}$ iteratively reduced to valuations just involving sums of monics of degree one. Here I want to again use monics of degree one to give a very simple example with properties very similar to 1 and 2 above. We will then draw some conclusions for the relevant theory of nonArchimedean measures.

The example presented here was first mentioned by Warren Sinnott, in the ${\displaystyle q=p}$ case in Warren's paper "Dirichlet Series in function fields" (J. Number Th. 128 (2008) 1893-1899). The ${\displaystyle L}$-functions that occur in the theory of Drinfeld modules and the like are functions of two
variables ${\displaystyle (x,y)}$. If one fixes ${\displaystyle x}$, the functions in ${\displaystyle y\in Z_p}$ that one obtains are uniform limits of finite sums of exponentials ${\displaystyle u^y}$ where ${\displaystyle u}$ is a ${\displaystyle 1}$-unit. In his paper Warren studies such functions and shows that if ${\displaystyle f\left(y\right)}$ is a nonzero such function, its zero set *cannot* contain an open set (unlike arbitrary continuous functions such as step-functions).

In what follows ALL binomial coefficients are considered modulo ${\displaystyle p}$ so that the basic lemma of Lucas holds for them.

Lemma: 1. Let ${\displaystyle \sigma \in S_{\left(q\right)}}$. Let ${\displaystyle y\in Z_p}$ and ${\displaystyle k}$ a nonnegative integer. Then

${\displaystyle {\displaystyle \left(\genfrac{}{}{0ex}{}{y}{k}\right)=\sigma \left(y\left(\genfrac{}{}{0ex}{}{)}{\sigma }\right)\right(k).}}$

2. Let ${\displaystyle i,j}$ be two nonnegative integers. Then

${\displaystyle {\displaystyle i+\left(\genfrac{}{}{0ex}{}{j}{j}\right)=\sigma \left(i\right)+\sigma \left(j\left(\genfrac{}{}{0ex}{}{)}{\sigma }\right)\right(j).}}$

Proof: 1 is simply ${\displaystyle q}$-Lucas. For 2 note that if there is carry over of digits in the addition for ${\displaystyle i+j}$ then there is also in the sum for ${\displaystyle \sigma \left(i\right)+\sigma \left(j\right)}$, and vice versa; in this case, both sides are ${\displaystyle 0}$. If there is no carry over the result follows from ${\displaystyle q}$-Lucas again.     □

As before, let ${\displaystyle q=p^m}$ and let ${\displaystyle y\in Z_p}$. Let ${\displaystyle A=Fq\left[t\right]}$ and let ${\displaystyle \pi =1/t}$; so ${\displaystyle \pi }$ is a positive uniformizer at the place ${\displaystyle \infty }$ of ${\displaystyle {\backslash bfF}_q\left(t\right)}$. Define

${\displaystyle {\displaystyle f\left(y\right):=\sum _{g\in A^{+}\left(1\right)}(\pi g{)}^y;}}$

where ${\displaystyle A^{+}\left(1\right)}$ is just the set of monic polynomials of degree ${\displaystyle 1}$. The sum can clearly be rewritten as

${\displaystyle {\displaystyle f\left(y\right)=\sum _{\alpha \in \backslash Fq}(1+\alpha \pi {)}^y.}}$

Upon expanding out via the binomial theorem, and summing over ${\displaystyle \alpha }$, we find

}${\displaystyle }$

f(y)= -\sum_{k \in I} {y \choose k} \pi^k

${\displaystyle {\displaystyle }}$

where ${\displaystyle I}$ is the set of positive integers divisible by ${\displaystyle q-1}$.

Let ${\displaystyle X\subset Z_p}$ be the zeroes of ${\displaystyle f\left(y\right)}$; it is obviously closed. When ${\displaystyle q=p}$, Warren (in his paper and in personal communication) showed that ${\displaystyle X}$ consists pricisely of those non-negative integers ${\displaystyle i}$ such that the sum of the ${\displaystyle p}$-adic digits of ${\displaystyle i}$ is less than ${\displaystyle p}$.

Now, in order to show that ${\displaystyle f\left(y\right)\ne 0}$, for a given ${\displaystyle y}$ in ${\displaystyle Z_p}$, it is necessary and sufficient to simply show that there is ONE ${\displaystyle k\in I}$ such that ${\displaystyle \left(\genfrac{}{}{0ex}{}{y}{k}\right)}$ is nonzero in ${\displaystyle {\backslash bfF}_p}$. When ${\displaystyle q=p}$, this is readily accomplished.

However, when ${\displaystyle q}$ is general it gets much more subtle to make sure that the reduced binomial coefficient is non-zero.

Proposition: The set ${\displaystyle X}$ is stable under ${\displaystyle S_{\left(q\right)}}$. Moreover, there is an explicit constant ${\displaystyle C}$ (which depends on ${\displaystyle q}$) such that the elements of ${\displaystyle X}$ have their sum of ${\displaystyle q}$-adic coefficients less than ${\displaystyle C}$.

(As Warren has remarked, the Proposition then reduces the problem of finding the zero set to checking *finitely many* orbits!)

Proof:

Let ${\displaystyle \sigma \in S_{\left(q\right)}}$. The first part follows immediately from the first part of the Lemma and the fact that ${\displaystyle I}$ is stable under ${\displaystyle S_{\left(q\right)}}$.

To see the second part, let ${\displaystyle C:=(q-2)(1+2+\cdots +q-1)=(q-2)(q-1)q/2}$. Let ${\displaystyle y}$ be any ${\displaystyle p}$-adic integer with the property that its sum of ${\displaystyle q}$-adic digits is greater than ${\displaystyle C}$. Then there must be at least one ${\displaystyle e}$ with ${\displaystyle e}$ between ${\displaystyle 0}$ and ${\displaystyle q-1}$ such that ${\displaystyle e}$ occurs at least ${\displaystyle q-1}$ times in the expansion of ${\displaystyle y}$. It is then easy to find ${\displaystyle k}$ such that the reduction of ${\displaystyle \left(\genfrac{}{}{0ex}{}{y}{k}\right)}$ is nonzero.     □

There are other important results that arise from the first part of the Lemma. Indeed, upon replacing ${\displaystyle k}$ with ${\displaystyle {\sigma }^{-1}\left(t\right)}$, we obtain

}${\displaystyle }$

{y \choose \sigma^{-1}(t)}= {\sigma(y) \choose t\,.

${\displaystyle {\displaystyle (*)}}$

This immediately gives the action of ${\displaystyle S_{\left(q\right)}}$ on the Mahler expansion of a continuous function from ${\displaystyle Z_p}$ to characteristic ${\displaystyle p}$. One also obviously has

}${\displaystyle }$

\sum_k {\sigma y \choose k} x^k=
\sum_k {\sigma(y) \choose \sigma (k)}x^{\sigma(k)\,.

${\displaystyle {\displaystyle }}$

But, by the first part of the Lemma, this then equals

}${\displaystyle }$

\sum {y \choose k}x^{\sigma k}\,,

${\displaystyle {\displaystyle }}$

which is a sort of change of variable formula.

As the action of ${\displaystyle S_{\left(q\right)}}$ is continuous on ${\displaystyle Z_p}$ there is a dual action on measures; if the measures are characteristic ${\displaystyle p}$ valued, then this action is easy to compute from (*) above.

However, there is ALSO a highly mysterious action of ${\displaystyle S_{\left(q\right)}}$ on the *convolution algebra* of characteristic ${\displaystyle p}$ valued measures on the maximal compact subrings in the completions of ${\displaystyle F_q\left(T\right)}$ at its places of degree ${\displaystyle 1}$ (e.g, the place at ${\displaystyle \infty }$ or associated to ${\displaystyle \left(t\right)}$, if the place has higher degree one replaces ${\displaystyle S_{\left(q\right)}}$ with the appropriate subgroup). Indeed, given a Banach basis for the space of ${\displaystyle Fq}$-linear continuous functions from that local ring to itself, the "digit expansion principle"gives a basis for ALL continuous functions of the ring to itself (see, e.g., Keith Conrad, "The Digit Principle", J. Number Theory 84(2000) 230-257). In the 1980's Greg Anderson and I realized that this gives an isomorphism of the associated convolution algebra of measures with the ring of formal *divided power series* over the local ring.

But let ${\displaystyle \sigma \in S_{\left(q\right)}}$ and define

}${\displaystyle }$

\sigma (z^i/i!):= z^{\sigma (i)}/\sigma(i)! \.

${\displaystyle {\displaystyle }}$

The content of the second part of the Lemma is precisely that this definition gives rise to an algebra automorphism of the ring of formal divided power series.

Dinesh Thakur's remarkable recursion formla

In this blog entry, I would like to highlight a remarkable formula due to Dinesh Thakur in the arithmetic of function fields over finite fields. This formula appears in page 5 of his preprint "Power sums with applications to multizeta values and zeta zeros" which can be downloaded at

http://math.arizona.edu/~thakur/power.pdf

Before presenting Dinesh's formula, I will present a little history. Early on in the theory of characteristic ${\displaystyle p}$ zeta functions, I used a simple lemma to obtain strong enough estimates to establish that such functions, and their interpolations at finite primes, are indeed "entire" (which, in this case, means a family of entire power series ${\displaystyle \zeta (x,y)}$ in ${\displaystyle x^{-1}}$ where the parameter ${\displaystyle y}$ lies in the ${\displaystyle p}$-adic integers). In the middle of the 90's, I discovered some old formulas of Carlitz gave much better (exponential) estimates for some special values of ${\displaystyle y}$. At that point, Daqing Wan and Yuichiro Taguchi were visiting me to discuss applications of Dwork theory to general ${\displaystyle L}$-series of Drinfeld modules. So I asked Daqing if he could use their theory to obtain such exponential estimates. The next day he came and showed me his elementary calculations for the Newton polygons for ${\displaystyle \zeta (x,y)}$ where he worked in the simplest possible case of ${\displaystyle {\backslash bfF}_p\left[t\right]}$. It was quite a shock when he stated that these calculations showed that the zeroes of ${\displaystyle \zeta (x,y)}$ were simple and in the field ${\displaystyle {\backslash bfF}_p\left(\right(1/t\left)\right)}$ (indeed there was at most ${\displaystyle 1}$ zero, with multiplicity, of a given absolute value); in other words, all the zeroes lie "on the line" given by ${\displaystyle {\backslash bfF}_p\left(\right(1/t\left)\right)}$ itself. Clearly this was a form of the Riemann hypothesis for these functions and Wan's results marked the first indication that these characteristic ${\displaystyle p}$ functions possess a profound theory of their zeroes.

In the characteristic ${\displaystyle p}$ theory, the theory for ${\displaystyle {\backslash bfF}_p\left[t\right]}$ and general ${\displaystyle {\backslash bfF}_q\left[t\right]}$ (${\displaystyle q=p^m}$, ${\displaystyle m}$ arbitrary) should be the same; so one wanted to know whether the Newton polygons associated to ${\displaystyle {\backslash bfF}_q\left[t\right]}$ also had the same simple form as given in the ${\displaystyle q=p}$ case. This was finally proved by Jeff Sheats based on some ideas of Bjorn Poonen; see Dinesh's paper for more history and the exact references. In any case, the general ${\displaystyle {\backslash bfF}_q\left[t\right]}$ case is much harder than the special case when ${\displaystyle q=p}$!

We still do not know exactly how to phrase an "Rh" in general because the trivial zeroes can have a very large impact on other zeroes due to the nonArchimedean topology of the spaces these functions are defined on. (Indeed, this was what made the calculations of Dinesh and Javier Diaz-Vargas on "nonclassical" trivial zeroes so important --- here, again, by nonclassical we mean trivial zeroes whose true order of
vanishing is higher than one would expect from classical theory). Moreover, even in the ${\displaystyle {\backslash bfF}_q\left[T\right]}$ case one does not understand what sort of information is contained in the results of Wan and Sheats. However, Thakur's results may be giving as the first very serious clues.

What Dinesh does is to establish a fundamental recursion formula for the ${\displaystyle \infty }$-adic valuations of certain fundamental sums arising in the function field theory (see page 5 of his preprint). From this recursion, the "Rh" follows readily.

Here then is the recursion formula, which, you will see, is quite elementary to state. We follow the notation of the paper: Let ${\displaystyle A={\backslash bfF}_q\left[t\right]}$ and let ${\displaystyle d}$ be a nonnegative integer and ${\displaystyle k}$ an arbitrary integer. Let ${\displaystyle A_{+}\left(d\right)}$ be the set of monic elements in ${\displaystyle A}$ of degree ${\displaystyle d}$. Define:

}${\displaystyle }$

S_d(k):=\sum_{a\in A_+(d)} 1/a^k

${\displaystyle {\displaystyle }}$

which is an element of ${\displaystyle {\backslash bfF}_q\left(T\right)}$. Let ${\displaystyle s_d\left(k\right)}$ be the valuation of ${\displaystyle S_d\left(k\right)}$ at the place ${\displaystyle \infty }$ of ${\displaystyle {\backslash bfF}_q\left(t\right)}$.

Dinesh's "main recursion formula" then states that:

}${\displaystyle }$

s_d(k)=s_{d-1}(s_1(k)) + s_1(k)\,.

${\displaystyle {\displaystyle }}$

This then leads iteratively to the second recursion formula

}${\displaystyle }$

s_d(k)=s_1^{(d)}(k)+\ldots +s_1^{(2)}(k) + s_1(k)\,.

${\displaystyle {\displaystyle }}$

where ${\displaystyle s_1^{\left(i\right)}}$ means the ${\displaystyle i}$-composition of the ${\displaystyle s_1}$ map with itself.

The main recursion formula is highly remarkable in that one computes a sum over the monics of degree ${\displaystyle 1}$ and then finds its valuation at ${\displaystyle \infty }$ and *then* uses this integer as the exponent to raise the monics of degree ${\displaystyle d-1}$. This feedback loop is absolutely new in terms of anything that I have ever seen.

One can ask whether there are any classical analogs of the above recursion formulas. It may be that when things are much better known, the second recursion formula will be viewed as the ${\displaystyle A}$-analog of the basic formula

}${\displaystyle }$

N_n(m)=q^{nm}+q^{(n-1)m}+\cdots + q^m+1${\displaystyle \$}$

which gives the number of points over ${\displaystyle {\backslash bfF}_{q^m}}$ of projective ${\displaystyle n}$-space. An analog of Dinesh's first recursion formula is now easy to construct.