## Saturday, December 26, 2009

### 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!

## Wednesday, October 7, 2009

### I.M. Gelfand, 1913-2009

## Sunday, March 29, 2009

### A DAY OF JAMI MEETING AT JHU

**Morita Equivalence Revisited**

**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**

**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**---------------------------------

**Essential dimension**

## Monday, January 19, 2009

### A very simple example

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

2. The orders of the trivial zeroes should be an invariant of the action of the group $S_{(q)}$ of homeomorphisms of $Z_p$ which permute the $q$-adic digits of a $p$-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 $d$ 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 $q=p$ case in Warren's paper "Dirichlet Series in function fields" (J. Number Th. 128 (2008) 1893-1899). The $L$-functions that occur in the theory of Drinfeld modules and the like are functions of two

variables $(x,y)$. If one fixes $x$, the functions in $y\in Z_p$ that one obtains are uniform limits of finite sums of exponentials $u^y$ where $u$ is a $1$-unit. In his paper Warren studies such functions and shows that if $f(y)$ 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 $p$ so that the basic lemma of Lucas holds for them.

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

$${y \choose k}= {\sigma(y) \choose \sigma (k)} \,.$$

2. Let $i,j$ be two nonnegative integers. Then

$${i +j \choose j}= {\sigma (i) +\sigma (j) \choose \sigma (j)}\,.$$

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

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

$$ f(y):= \sum_{g\in A^+(1)} (\pi g)^y \,;$$

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

$$ f(y)=\sum_{\alpha \in \Fq}(1+\alpha \pi)^y .

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

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

where $I$ is the set of positive integers divisible by $q-1$.

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

such that the reduction of ${y \choose k}$ is nonzero. QED

There are other important results that arise from the first part of the

Lemma. Indeed, upon replacing $k$ with $\sigma^{-1}(t)$, we obtain

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

This immediately gives the action of $S_{(q)}$ on the Mahler expansion of

a continuous function from $Z_p$ to characteristic $p$. One also obviously has

$$\sum_k {\sigma y \choose k} x^k=

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

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

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

which is a sort of change of variable formula.

As the action of $S_{(q)}$ is continuous on $Z_p$ there is a dual action

on measures; if the measures are characteristic $p$ valued, then this action is

easy to compute from (*) above.

However, there is ALSO a highly mysterious action of $S_{(q)}$ on the

*convolution algebra* of characteristic $p$ valued measures on

the maximal compact subrings in the completions of $F_q(T)$ at its

places of degree $1$ (e.g, the place at $\infty$ or associated to $(t)$, if the place has higher degree one replaces $S_{(q)}$ with the appropriate subgroup).

Indeed, given a Banach basis for the space of $Fq$-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 $\sigma \in S_{(q)}$ and define

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

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.

## Sunday, January 18, 2009

### Dinesh Thakur's remarkable recursion formla

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 $p$ 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 $\zeta(x,y)$ in $x^{-1}$ where the parameter $y$ lies in the $p$-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 $y$. At that point, Daqing Wan and Yuichiro Taguchi were visiting me to discuss applications of Dwork theory to general $L$-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 $\zeta(x,y)$ where he worked in the simplest possible case of ${\bf F}_p[t]$. It was quite a shock when he stated that these calculations showed that the zeroes of $\zeta(x,y)$ were simple and in the field ${\bf F}_p((1/t))$ (indeed there was at most $1$ zero, with multiplicity, of a given absolute value); in other words, all the zeroes lie "on the line" given by ${\bf F}_p((1/t))$ itself. Clearly this was a form of the Riemann hypothesis for these functions and Wan's results marked the first indication that these characteristic $p$ functions possess a profound theory of their zeroes.

In the characteristic $p$ theory, the theory for ${\bf F}_p[t]$ and general ${\bf F}_q[t]$ ($q=p^m$, $m$ arbitrary) should be the same; so one wanted to know whether the Newton polygons associated to ${\bf F}_q[t]$ also had the same simple form as given in the $q=p$ 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 ${\bf F}_q[t]$ case is much harder than the special case when $q=p$!

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 ${\bf F}_q[T]$ 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 $\infty$-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 $A={\bf F}_q[t]$ and let $d$ be a nonnegative integer and $k$ an arbitrary integer. Let $A_+(d)$ be the set of monic elements in $A$ of degree $d$. Define:

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

which is an element of ${\bf F}_q(T)$. Let $s_d(k)$ be the valuation of $S_d(k)$ at the place $\infty$ of ${\bf F}_q(t)$.

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

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

This then leads iteratively to the second recursion formula

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

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

The main recursion formula is highly remarkable in that one computes a sum over the monics of degree $1$ and then finds its valuation at $\infty$ and *then* uses this integer as the exponent to raise the monics of degree $d-1$. 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 $A$-analog of the basic formula

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

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