Carlitz’s formalism and Euler’s $Gamma$-function

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It was always my fondest hope that the arithmetic of function fields in finite characteristic would finally become sophisticated enough so that it could be developed somewhat in tandem with classical arithmetic. In the recent past, this hope appears to becoming real. In particular, I would like to draw your attention to the new preprint by Federico Pellarin arXiv:1210.2490 “On the generalized Carlitz-module” where the formalism due to Carlitz is applied to deduce new functional equations for Euler’s $Gamma$-function (as well as applications to other classical special functions).

Let ${mathbf F}_q$ be the finite field with $q$ elements and let $theta$ an indeterminate; as usual we put $A:={mathbf F}_q[theta]$. When Carlitz was originally developing his module in the 1930’s he chose the mapping generated by $theta mapsto theta -tau$ where $tau$ is the $q$-th power mapping. I changed this to $theta + tau$ simply because this made computing Frobenius elements a bit easier.

As the theory of the Carlitz module, and general Drinfeld modules, began to develop, it was realized that one could abstract from the original settings of function fields over finite fields and this was developed e.g., by Mumford (1977) and Hellgouarch (1992 and 1997). In the paper by Federico this formalism is applied to complex analytic functions in a very concrete fashion.

More specifically, Let $F$ be the field of (complex valued) meromorphic functions in $s$ which are periodic of period $1$. Obviously, the function $s$ is not in $F$ and so one forms the polynomial ring (of functions) ${mathbf A}:=F[s]$; notice that the huge field $F$ is now playing the role of the finite field $mathbf F_q$….

The question is what plays the role of the $q$-th power mapping; here Federico makes the crucial choice $tau (f):=f(s+1)$ where $f$ is a meromorphic function (so that obviously $F$ is the fixed field of this action). One then has the (generalized) Carlitz module associated to $(F[s], tau)$ simply by following Carlitz’s idea of composing actions and his original formalism mentioned above! In particular, one is now able to talk about the torsion elements associated to an element $ain F[s]$. 

So what does this have to do with the $Gamma$-function? Well, virtually everything! Indeed, of course, everyone knows the (almost!) definitional functional equation
Federico now simply rewrites this as
In other words, the $Gamma$-function IS an $s$-torsion element for this Carlitz module. In fact, one sees that Carlitz-torsion is intimately connected with the $Gamma$-function in one form or another. (Indeed, I would not be surprised if there is yet another characterization of the $Gamma$-function which involves Carlitz-Pellarin torsion.)

But much more is true. In a previous blog I have discussed the applications of the Anderson-Thakur function $omega(t)$ to $L$-series. The function $omega(t)$ can abstractly be obtained as the Akheizer-Baker function associated to certain data. In the $F[s]$-case, Federico shows that the Akheizer-Baker function is precisely $Gamma(s-t)$ and this leads to other functional relations in line with the analogy to $omega(t)$.

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