Tuesday, March 18, 2014

Review of a paper by Gebhard Boeckle and the group S_(q)

So this post is a bit of an experiment. My friends at Math Reviews recently sent me a really interesting Math. Z. paper by Gebhard Boeckle. I spent some time reviewing it and found it contained very interesting results and calculations that pointed, yet again, to some possible underlying action of the group $S_{(q)}$ that I have discussed in other posts here. If you combine it with the new results of Rudy Perkins in http://arxiv.org/abs/1402.4000, the situation becomes even more intriguing....

The review at Math Reviews has number MR3127039.  Here is the link: http://www.ams.org/mathscinet-getitem?mr=3127039

With the concurrence of MR, and with my sincere gratitude, I am posting the review below; if you find it too small, you can simply increase the size of the font (by something like 'command +'). (I should also say that I converted the original pdf to jpegs which could then be uploaded to Blogger...)






Tuesday, February 4, 2014

zeta zeroes AND gamma poles

The arithmetic of function fields over finite fields has always been a ``looking-glass'' window into the standard arithmetic of number fields, varieties, motives etc.; sort of ``life based on silicon'' as opposed to the classical ``carbon-based'' complex-valued constructions. It has constantly amazed me, and frankly given me great pleasure, to see the way that analogies always seem to work out in one form or another. Often these analogies are not at all obvious and I want to report here on the existence of a certain analogy that I find particularly satisfying and greatly encouraging.

One of my great desires in working in this area was, and of course is, to have a fully analytic theory of $L$-series in characteristic $p$ based on Drinfeld modules, $t$-modules, etc., (as opposed to the fundamentally algebraic nature of the complex valued functions traditionally defined for function fields). For a long time we have known the correct definitions of Euler factors at the good primes and, with the work of Gardeyn, we also know the correct definitions at the bad places (at least in the case of Drinfeld modules). We further know that these $L$-series have excellent analyticity properties with associated ``trivial zeroes''; moreover, in the simplest case of $A:=F_q[\theta]$, the infinite prime, and the associated zeta function, we know that the zeroes are actually are simple and ``lie on the line'' $F_q((1/\theta))$. Until now these trivial zeroes arose by using the (polynomial) Euler factors at infinity coming from classical theory, or cohomology of crystals, etc., and some auxiliary arguments.

More recently, beginning with the work of Taelman and Lafforgue, there has been really exciting progress in establishing the ``correct'' analogs of the class group and class number formulae in this context. Indeed, this an area of great current excitement and active research. See for instance: http://hal.archives-ouvertes.fr/hal-00940567 .

Given these very strong indications, it is not unreasonable to expect that many, if not all, of the remaining properties from the complex (``carbon-based'') $L$-series should ultimately show up in some form or other in the finite characteristic theory. So, in this post I will briefly describe an observation about trivial zeroes due to Rudy Perkins, and based on the wonderful preprint http://arxiv.org/abs/1301.3608v2 of Bruno Angles and Federico Pellarin, which shows, yet again, the remarkable similarities between the classical theory of $L$-series and their finite characteristic cousins.

As every arithmetician knows, in order to truly appreciate the analytic properties of classical $L$-series (of number fields) one must adjoin to them a finite number of Euler factors at the infinite primes. These Euler factors are, of course, created out of Euler's fabulous gamma function $\Gamma (s)$. And everybody knows that $\Gamma (s)$ is nowhere zero with simple poles at the nonpositive integers. Via the functional equation of a given $L$-series, these poles translate into the fundamental ``trivial zeroes'' of the $L$-series (which often times may also be deduced in a more elementary fashion) as well as determining the exact order of these zeroes.

In the finite characteristic case, using the Carlitz exponential and factorial, I was able to define a number of continuous (and even rigid analytic) $\Gamma$-analogs which capture many of the properties of Euler's $\Gamma(s)$ (due to the fundamental work of Greg Anderson, Dinesh Thakur, Dale Brownawell, Matt Papanikolas,...). However, there was no obvious connection with $L$-series or their trivial zeroes (which were originally obtained using the polynomial Euler factors associated to the infinite primes as mentioned above). Of course, as can be imagined, this was truly a disappointment.

On the other hand, beginning with their fundamental work on tensor powers of the Carlitz module, Greg Anderson and Dinesh Thakur introduced another $\Gamma$-analog  denoted $\omega (t)$. This is a nowhere zero function with simple poles at {$\theta$qj} for $j\geq 0$. Subsequently, this function proved to be instrumental in studying the properties of the previously mentioned gamma functions. When I first saw it, I noticed how natural it seemed as a deformation of the Carlitz period (a $\Gamma$-type property after all!).  However, I also found the collection of poles of $\omega(t)$ too specialized to somehow be related to $L$-series; in this I was simply wrong (for which I am grateful!).

The reason I was wrong is due to the fundamental work of Federico Pellarin over the past few years. Federico introduced the natural (but seemingly highly non-classical) set {$\chi$t} of quasi-characters of $A$ given simply by the maps $f(\theta)\mapsto f(t)$ where $t$ is some constant. He then naturally defines the $L$-series $L(\chi$t,s) and, most importantly, establishes a wonderful formula relating the special values of these $L$-series to the Carlitz period and $\omega(t)$. Federico also made the elementary but totally key observation that $L(\chi$u, s)=$\zeta$(s-qj), where $u=\theta$qj.

Thus, the poles of $\omega(t)$ are completely canonical, and actually represent the qj-th power morphisms on $A$. I can't help but wonder if there is a some sort of similar interpretation of the poles of Euler's $\Gamma(s)$.

Still, what about $\zeta(s-i)$ for any positive $i$?????

Obviously [C R]=2; but  R and C are the only local fields with the property that their algebraic closure comprises a finite dimensional extension. For function fields over finite fields, the
algebraic closure of the associated local fields are vast objects with a huge amount of  ``room to move.'' Put more directly, one can simply add quasi-characters at will and consider $L$-series of the form $L(\chi$t1,...,$\chi$te, s) for arbitrary e. This gives a staggering and bewildering (at least with the current state of the art) amount of flexibility, but it really does work and one can indeed clearly specialize (in many ways) to $\zeta(s-i)$. In other words, it is mandatory to adjoin an arbitrary number of $\Gamma$-factors to a fixed $L$-series.

And this brings us back to the preprint http://arxiv.org/abs/1301.3608v2 of Bruno and Federico. Here a beautiful integrality result, Theorem 4, is obtained for  $L(\chi$t...$\chi$te, $\alpha$)  and
$\Pi \omega($ti) where $\alpha$ is a positive integer and $\alpha\equiv e$ mod (q-1).

Finally to close the circle of arguments, Rudy Perkins has just shown me a quick and elegant argument, by specializing the {ti}, how this Theorem 4 implies the existence of trivial zeroes in great generality (and certainly those of $\zeta(s)$ at the negative integers divisible by $q-1$). Briefly here is what Rudy does: Given an integer s divisible by q-1, one uses s+1 and 1 in this theorem; on the left one then has an expression involving the $L$-function (viewed as a function of the {ti}) and on the right one has a multi-variable polynomial (which is the "integrality" part of the result). Upon taking the limit of the last variable at $\theta$, one obtains that the value in question times $\Pi$ $\omega$(ti) is still a polynomial. But the value in question can easily be seen to also be a polynomial and it must have zeros all over the place in order to cancel the $Gamma$-poles. So many zeroes, in fact, that it identically vanishes!

The order of these trivial zeroes is another matter. While one can compute these orders using elementary arguments in certain cases, a more deeper approach now seems truly to be indicated...


(Added2-6-2014: Lenny Taelman has produced some highly valuable notes of his Beijing lectures and these are now in a form, while still preliminary, that can be shared: please see
http://www.math.leidenuniv.nl/~lenny/beijing.pdf
Also along these very same lines, please see the preprint by Jiangxue Fang
http://arxiv.org/abs/1401.1293v1  )

(Added 2-18-2014: Perkins' paper "An exact degree for multivariate special polynomials" is on the arXiv at http://arxiv.org/abs/1402.4000 .)

Sunday, September 8, 2013

Trimester program on Non-commutative Geometry and its Applications

From September-December 2014 there will be a trimester program on Non-commutative Geometry and its Applications at the Hausdorff Research Institute for Mathematics.

There will be four workshops during the trimester:
  • September 15-19, Non-commutative geometry's interactions with mathematics.
  • September 22-26, Quantum physics and non-commutative geometry.
  • November 24-28, Number theory and non-commutative geometry.
  • December 15-18, Future directions for non-commutative geometry.
There will be a series of lecture courses aimed at postgraduate students and postdoctoral level researchers.
  • September 1-12, Introductory series.
  • October 6 - November 21, Special topics series.
There will also be a weekly seminar series on current research topics and a working seminar within that part of the program aimed at junior researchers. 

Places are available for junior researchers, who can apply here
(deadline: January 31, 2014).

Thursday, September 5, 2013

Analytic continuation in the blogosphere....

Hi. For those interested, I have started another blog at http://dmgoss.wordpress.com/ to cover items that are probably not appropriate (too technical, specialized, etc.) for this wonderful blog.... Best, David

Friday, August 23, 2013

Website Noncommutative Geometry and Particle Physics

A new website on noncommutative geometry has been created, connected to the workshop Noncommutative Geometry and Particle Physics organized at the Lorentz Centre in Leiden in October 2013. As this type of workshop only allows for a limited number of participants, this website will form the virtual portal for a wider audience.


It will contain updates during the workshop, documents with background material, discussions that take place during the workshop, a glossary, and much more.

In the future, we plan to extend this website as a repository for noncommutative geometry and its applications to particle physics.

Tuesday, July 9, 2013

A-expansions

As I have written about before, the integers Z play a dual role in arithmetic. On the one hand, they are obviously scalars in terms of the fields of definitions of varieties etc.; yet, on the other hand, they are also operators, as in the associated Z-action on multiplicative groups (or the groups of rational points of abelian varieties etc.). This is absolutely so basic that we do not notice it in day-to-day mathematics.

Yet these dual notions are there and are highlighted by the curious cases of similar phenomena in the arithmetic of function fields. This is what I want to discuss here. So, as usual in the characteristic $p$ game, let $A$:=Fq$[\theta]$, $K$:=Fq$(\theta)$ and $K$$\infty$:=Fq$((1/\theta))$. Recall also the Carlitz module $C$ given by $C$$\theta$(z):=$\theta z+z$q; one always views $C$ as the analog of the multiplicative group Gm (indeed its division values generate abelian extensions etc.).

Of course every algebra lies over Z and thus one can always study the corresponding "Z-invariants'' such as class groups or class numbers etc. But the analogy between Z and $A$ really calls out for "$A$-invariants'' also. When I mentioned the possibility of such $A$-objects way back in 1980 at a conference, the participants looked at me like I had lost my mind. Be that as it may, within the past few years such $A$-invariants have indeed been produced in the seminal work of Lenny Taelman; so we now have "class $A$-modules'' and "$A$ class numbers'' (really generators of the Fitting ideals of these finite class $A$-modules). In fact, these notions fit beautifully into the special values of $L$-series in direct analogy with algebraic number theory.

In a similar way, the very notion of  "analytic function'' is clearly Z-based; i.e., based on the notion of power series $\sum$ ai xi; built directly into the definition of power series is the standard Z-action arising from multiplication (i.e., the mapping $(i,x)$ $\mapsto x$i).

So the idea of this blog is that there should analogously be "$A$-expansions'' where we now sum over the monic elements of $A$ (and not elements of  Z) in the theory and, remarkably, such things do exist. We are, by no means, close to a full theory of such expansions but rather we have a number of highly intriguing results.

Here is a very cool example of what I am talking about (essentially due to Greg Anderson in his famous "log-algebraicity" paper: Journal Number Theory 60, 165-209 (1996)): We begin by recalling, from Calculus 1, the basic expansion
$\log (1+x)=x$-x2/2+x3/3+...
Now let $C$ be the Carlitz module and $\log$C(z) it's logarithm. For i $\geq$ 1, we put $[i]:=\theta$qi-$\theta\in A$ and also Li:=[i][i-1]...[1]. One can easily see that Li is the least common multiple of the monic elements of $A$ of degree $i$. One then has the Z-expansion
$\log$C(z)=$\sum$i zqi/Li.
For the $A$-expansion we have Greg's formula
$\log$C (z)=$\hat\sum$a Ca(z)/a,
where $a$ runs over the monics of $A$ and $\hat \sum$ means that we compute the sum as the limit of  {Sd(z)} where Sd(z) is the above sum truncated over the (finite number of) monics of degree $\leq$ d (so, alas, we have not fully removed here after all!). Note also that without such a truncation, the convergence of the sum is extremely tricky and rare! The analogies between the $A$-expansion of $\log$C(z) and the usual expansion for $\log (1+x)$ are very clear....

(Greg calls the power series x-x2/2+x3/3+... "log-algebraic" since it is clearly the log of an algebraic
function. Once one views power series this way, many examples spring to mind; indeed Dwork's famous result on points over finite fields can be viewed in this optic. Greg's log-alg ideas are currently having an extremely large impact on research; for more, see Rudy's blog
https://rudyperkins.wordpress.com/   .)

For a monic $a$, the additive polynomial $C$a(z) has derivative identically equal to $a$. As such one can find a formal composition inverse denoted $C$a-1 (z) as an Fq-linear power series. To obtain an $A$-expansion for the Carlitz exponential we then have the beautiful, unpublished, formula of Federico Pellarin:
$\exp$C(z)=$\hat\sum$a$C$a-1 (az),
where one must now "renormalize'' the sum in two steps: First of all, we truncate the sum over the monics of degree $d$, as before, and then we also truncate the resulting expression (which is an additive power series) to only include the terms of degree $\leq$ qd. Again, as before, without these operations there is no hope of convergence.

Next let's move on to $L$-series in finite characteristic.  Again we find that there is a mix between $A$-expansions and Z-expansions. For purposes of illustration we only treat the simplest case; thus given a monic $a$ in $A$ of degree $d$, we set
$\langle a\rangle$:= $a/\theta$d.
Notice that $\langle a \rangle$ is a $1$-unit in K$\infty$, and, as such, the expression $\langle a \rangle$y makes sense for $y \in$ Zpvia the Binomial Theorem (and with the usual exponential properties). We put S$\infty$:=K$\infty$*$\times$ Zp with its obvious abelian group structure, and for s=(x,y)$\in$ S$\infty$, as:=xd$\langle a\rangle$y . One then has the zeta function of $A$ defined by the $A$-expansion
$\zeta$A(s):=$\sum$a a-s.
For x not in Fq[[$1/\theta$]], this expansion converges without further manipulation. For the rest of S$\infty$ we rewrite $\zeta$A(s)=$\hat \sum$a a-s where, to guarantee convergence, we again truncate by the degree $d$ and take the limit....

There is yet a third place where $A$-expansions are now playing a very interesting role  and which is presumably somehow related to the above cases. Let $f(\tau)$ be a classical elliptic modular form on the upper half plane associated to SL2(Z). As everybody knows, the form $f$ has an expansion $f=\sum$anqn where q:=e$2\pi i \tau. Now let $g(z)$ be a modular form on the Drinfeld upper half-plane. In particular, $g(z)$ is, by definition, invariant under transformations of the form $z\mapsto z+h$ for $h\in A$; as such I showed long ago that $g(z)$ has a Z-expansion $\sum$cn un where $u(z):=\exp$C$(\pi z)$-1. Noting that
$\sum$ anqn=$\sum$ ane$n 2\pi i \tau$
leads one to suspect that $g(z)$ might also have an expression of the form $\sum$adaua where $a$ runs over the monics and ua:=u(az). In fact, this is almost the correct idea: Let $G(X)$ be a fixed function (so far only polynomials have been considered). Then we call an expansion of the form c0+$\sum$acaG(ua) an "$A$-expansion''. While it turns out that not all forms in finite characteristic have such expansions (at least for the class of functions G considered up till now), it has recently become very clear that a great many important ones do!

For instance, all Eisenstein series have such expansions. More importantly, the two basic cusp forms $\Delta$ and $h$ also have them: $\Delta$=$\sum$a aq(q-1) uaq-1 and $h$ (which is a $q-1$-st root of $\Delta$) has the expansion $h=\sum$a aq ua. These expansions are due to B. L'opez, Arch. Math. 95 (2010), 143–150. Very recently, in Journal of Number Theory 133 (2013) 2247–2266, A. Petrov has shown how to construct families of cusp forms
$\sum$a at Gn(u a)
for certain positive integers $t$ and polynomials {Gn(X)}. Moreover he proves that these forms are, in fact, all Hecke eigenforms with easily computed eigenvalues. In arXiv:1306.4344 I showed how these forms give rise to non-trivial interpolations at the finite primes $\mathfrak v$ of $A$ in the sense of Serre's construction of p-adic modular forms (something I have long wanted to do). This also fits perfectly in to the theory of such forms created by C. Vincent in her 2012 Wisconsin thesis.

I would like to finish by explaining how Petrov's sums, just above, have elliptic modular analogs. Put G(X):=X/(1-X) and qn:=qn. Then, indeed, the normalized Eisenstein series of weight 2k has the Lambert expansion
1+2/$\zeta(1-2k)$ $\sum$n n2k-1 G(qn) .

It is my pleasure to thank Rudy Perkins and Federico Pellarin for their invaluable input.

Friday, April 19, 2013

Wronski, Vandermonde, and Moore!

This post is based on a recent letter by Matt Papanikolas outlining some results he has discovered whilst writing a (highly anticipated!) monograph on $L$-values in finite characteristic. In staring at Matt's letter, I realized that he allowed one to relate the big 3 matrices (Wronski, Vandermonde and Moore) in one simple formula which I will present below and then pose a related question.

I apologize if this is challenging to read; I am having a devil of a time getting my tex to compile correctly.

Matt's perspicacious insight was based on a paper by Felipe Voloch which was published in the Journal of Number Theory  in 1998. (Full disclosure: I had previously played with Voloch's paper and totally missed these ideas...)

Let me begin with some standard notation. As is now becoming standard, following Greg Anderson, let us put $A:= F$_q$[\theta]$. For each nonnegative integer $i$, let's put $A(i):= \{a\in A$ $\mid $ $ \deg (a)\leq  i\}$. Moreover, for such $i$, following Carlitz, we set $[i]:=\theta$^{q^i}-$\theta$. These are basic building blocks of $A$; for instance, one easily sees that $[i]$ is the product of all monic primes of degree dividing $i$. Various types of factorials are products of these elements....

Now recall that the Wronskian matrix is made up out of the derivatives of a function. Obviously in finite characteristic one can only differentiate a little bit before ending up with identically zero functions. As such we modify the Wronskian slightly here and use the "divided derivatives" $\partial$_i (i.e., those elements actually giving the coefficients of Taylor series etc.) in order to end up with nontrivial matrices.

Let $x$_0,..., $x$_i be i indeterminates. We define the Vandermonde matrix $V(i)(x$_0,..,$x$_i) as usual by having the $i$-th row consist of $(1$,$a$ ,..., $a$^i$)$, where $a=x$_{i-1}, and so on.

Finally of course the Moore matrix is made up out of the $q$^i powers of elements in a very similar
fashion to both the Wronski and Vandermonde matrices.

Let $g\in F[[\theta]]$. Notice that $g$^{q^i} is simply $g(\theta$^{q^i}) as the elements of $F$ are
fixed.  Thus, as Felipe observed, a bit of calculus (e.g., Taylor expansions) immediately gives $g$^{q^i}=$\sum$_j $\partial$_j(g) [i]^j.

Let $a\in A(i)$. Let M(i)=(a, a^q,..., a^{q^i})^t be the column vector consisting of a and its q^j-th powers and let W(i):=(a, $\partial a$, ..., $\partial$_ia)^t be the column vector consisting of the divided derivatives of a. Matt realized that Felipe's result implies
$$M(i)=V(i)([0],[1],...,[i])W(i).$$
Now simply choose a bunch more elements in A(i) and sling the above equations together. We immediately deduce an equality of matrices of the form M=VW where M is a Moore matrix, V is the Vandermonde and W the Wronskian. Thus taking determinants allows us, in this case, to calculate the Wronskian determinant in terms of the Moore and Vandermonde determinants.

It is very wild that one can relate the q-th power mapping, which is a field embedding, in this fashion with the divided derivatives which satisfy the Leibniz identity and I wonder if there is much more here.

In any case, the Moore determinant is intimately related to having elements linearly independent over F_q.  In classical theory, the Wronskian of analytic functions determines whether they are linearly independent (a result evidently due to Peano); see arXiv:1301.6598 for a modern approach. In characteristic p, the p-th power mapping clearly wreaks some havoc as usual (think of the Wronskian of x^p and x^{p^2}). But I wonder if there might not be some clear statement about linear dependence involving both the Wronskian and the p-th power mapping...?

(Updated 4-22-13: If you choose as your elements $a\in A(i)$ the set $\{1,\theta,...,\theta$^i}$, you
obtain a Wronskian with determinant 1 thus giving an equality between the determinant of Matt's Vandermonde matrix and the determinant of the Moore matrix. As in Carlitz's original paper back in the 1930's, this Moore determinant is then easily computed to be a Carlitz factorial... It should also be possible to compute the determinant of Matt's matrix directly using the well-known formula for the Vandermonde determinant.)

(Updated 4-23-13: I thank Sangtae Jeong for pointing out the work of F.K. Schmidt, in 1939, and the work of Felipe Voloch and Arnaldo Garcia, in 1987, that goes very far with Wronskians in finite characteristic. See "Wronskians and linear independence in fields of prime characteristic", Manuscripta Math., 59, 1987, 457-469.)

(:1301.6598arXiv:1301.6598