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 $p$ zeta functions associated to ${\bf F}_q[t]$ 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 $-i$ 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 $p$.

Now, in order to show that $f(y) \neq 0$, for a given $y$ in $Z_p$, it is necessary and sufficient to simply show that there is ONE $k \in I$ such that ${y \choose k}$ is nonzero in ${\bf F}_p$. When $q=p$, this is readily accomplished.

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

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

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

Proof:

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

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

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 $p$ 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.

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
http://www.rse.org.uk/events/reports/2007-2008/presidential_address.pdf
and can read a report on a similar talk at
http://www.dailystar.com.lb/article.asp?edition_id=1&categ_id=2&article_id=97190
My best,
David

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!

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
http://www.youtube.com/user/JournalNumberTheory
or
http://www.youtube.com/user/AlainConnes
Alain's video is a terrific example of what is possible with these video abstracts!
My best,
David

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 $i$
$$a_if(1+x)(1+x)^{-i}=\frac{D^i}{i!}f(1+x)\,.$$
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 $u$.

IRONY

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.

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[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.]