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 $L(\chi$t1 ...$\chi$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 .)
