# gamma poles and zeta zeroes

In this post, I am going to be brief and give you a link to the noncommutative geometry blog where I have posted the details. Briefly, the arithmetic of function fields over the finite field ${\mathbb F}_q$ is a decades long dream I had of recreating classical arithmetic in the setting of Drinfeld modules and the like. This is very much like the dream of some biologists to create artificial life or even, more remotely, life based on silicon. Quite recently there has been a deepening of our understanding the relationship between the appropriate $\Gamma$-function analog, denoted $\omega(t)$ and $L$-series. In particular, we now know how the poles of $\omega(t)$ actually give rise to the “trivial zeta zeroes” in sharp analogy with classical theory. So, if you are interested, please check out http://noncommutativegeometry.blogspot.com/2014/02/zeta-zeroes-and-gamma-poles.html

# Measures and their discontents, er, automorphisms…

So one of the reasons I started to blog in the first place was that it is a great way to reach out to the community and, in particular, to younger mathematicians. It has always amazed and delighted me the way young mathematicians have approached function field arithmetic with wide open eyes and seen things that I have totally missed. Frankly, while I feel a bit mortified that I missed something or other, this is more than made up with by the joy in seeing these new results…

One of the points of function field arithmetic is that it allows one to revisit classical ideas “through the looking glass” (in a Lewis Carroll sort of way!).  One instance of this lies in the application of integration in characteristic $p$. In fact, the contrast between classical integration (with Haar Measures, Cauchy’s Theorem or whatever) and integration in finite characteristic could not seem greater. Classical integration is a tool of almost unlimited flexibility; one only has to think of the connection with anti-differentiation, the change of variables formula, the Fourier transform, and on and on. Non-Archimedean integration is not so obviously well equipped.

Yet there is a large paradox in all of this. The functional equation of the Riemann zeta function is associated to the group of order $2$; for the Fourier transform we end up with the cyclic group of order 4. But for the basic case of function field arithmetic, $A:={\mathbb F}_q[\theta]$, $q=p^{m_0}$, $p$ prime, we appear to end up with a group with the cardinality of the continuum! More precisely, let $\rho$ be a permutation of the set $\{0,1,\ldots\}$ and let $y$ be a $p$-adic integer written $q$-adically as $\sum_{j=0}^\infty c_j q^j\,.$ One then sets

$\rho_\ast (y):=\sum_j c_j q^{\rho j}\,.$

One can see without too much difficulty that $y\mapsto \rho_\ast y$ is continuous and that it stabilizes both the non-positive and non-negative integers. Moreover, it is clearly almost additive; i.e., additive as long as we have a sum not involving a carry-over of $q$-adic digits. In this fashion we obtain a group of homeomorphisms of $\mathbb Z_p$ which is denoted $S_{(q)}$. And one computes that this group has the cardinality of the continuum.

I discovered this group by staring at some early, and prescient, calculations done by Dinesh Thakur on those peculiar cases where a “trivial zero” has a higher order than one would expect. But then I went on to discover the following connection with measure theory. So let’s put $A_\infty:={\mathbb F}_q[[1/\theta]]$. We fix a parameter $\mathfrak t\in A_\infty$ of the form $1/\theta+${ higher terms in $1/\theta$}. (So, for instance, $\mathfrak t=1/\theta$!); therefore $A_\infty=\mathbb F_q[[\mathfrak t]]$.

Following Carlitz, one defines $e_d(z):=z\prod_{\alpha\in {\mathbb F}_q[{\mathfrak t}](d)}(z+\alpha)$ where ${\mathbb F}_q[\mathfrak t](d)$ is the set of polynomials of degree $. This is seen to be $\mathbb F_q$-linear. By using these and the $q$-adic expansion of a positive integer $n$ one forms the Carlitz polynomials $\{G_n(z)\}$. These form a Banach basis of the space of continuous functions $A_\infty \to A_\infty$ (think of Mahler’s Theorem for $\mathbb Z_p$). The dual of this Banach space is the space $\mathfrak M$ of measures (and one can form integrals via Riemann sums!). In the usual way, the space of measures is an algebra under convolution.

So now we come to the first  connection of $S_{(q)}$ with the algebra $\mathfrak M$. Because of the additivity of the polynomials $e_d(z)$, the polynomials $\{G_n(z)\}$ satisfy the Binomial Theorem. Thus, as Greg Anderson observed, the algebra $\mathfrak M$ is isomorphic  the algebra $\mathfrak D$ of formal divided derivatives.  Let $\rho$ be as above and let $\mathfrak d\in \mathfrak D$. We define $\rho_\ast (d)$ by extending the map $\displaystyle \frac{D^i}{i!} \mapsto \frac{D^{\rho_\ast (i)}}{\rho_\ast (i)!}$ linearly. Combining the almost additivity of $\rho_\ast$ with Lucas’ Theorem, one sees that one obtains an algebra automorphism of $\mathfrak D$ and thus $\mathfrak M$. This automorphism is extrinsic in that we needed the identification in order to see it.

I recently realized that there is also an intrinsic action of this crazy group $S_{(q)}$ on $\mathfrak M$ as follows. Let $x=\sum c_i\mathfrak t^i\in A_\infty$. Define $\rho_\ast (x):=\sum c_i \mathfrak t^{\rho_ast i}$Now, we obtain a fully linear continuous action on $A_\infty$ (which I originally wrote down as it appears to be related to $\zeta$-zeroes…). I just simply realized that the linearity of this last action implies that the dual action on measures preserves convolution!

So: are these two actions the same?  This would be quite remarkable if true. If not, it is hard to believe that these two natural actions are not somehow naturally related!

# My first post here on WordPress

Hi. Although I happily blog on the noncommutative geometry blog, there are many times I would like to post things that are too specialized and/or inappropriate for that blog. As such I have decided to open up a second front here on WordPress. I hope it will be a forum for sharing information and ideas on function field arithmetic (or anything that analytically continues from there!). My best, David