Operator/scalar fusion in finite characteristic and remarkable formulae

Let $E$ be a curve of genus $1$ over the rational field $\mathbf Q$. One of the glories of mathematics is the discovery that (upon choosing a fixed rational point "$\mathbf O$") $E$ comes equipped with an addition which makes its points over any number field (or $\mathbf R$ or $\mathbf C$) a very natural abelian group. (In the vernacular of algebraic geometry, one calls $E$ an "abelian variety" of dimension $1$ or an "abelian curve".)

Built into this setup is a natural tension between the two different avatars of the integers $\mathbf Z$ which now arise. On the one hand, an integer $n$ is an element of the scalars $\mathbf Q$ over which our curve $E$ lies; on the other hand, $n$ is also an operator on the group formed  by the elliptic curve (and, in fact, it is well known that this operator is actually a morphism on the elliptic curve).

One would, somehow, like to form a ring that encompasses both of these avatars. An obvious way to do this would be to form ${\mathbf Z}\otimes {\mathbf Z}$ but, alas, this fails as this tensor product is simply $\mathbf Z$. I have always thought, perhaps naively, that one of the motivations in studying ${\mathbf F}_1$ was the hope that progress could be made here....

In any case, in finite characteristic we are blessed with more flexibility. Let $q$ be a power of a prime $p$ and let ${\mathbf F}_q$ by the field with $q$-elements with $A:={\mathbf F}_q[\theta]$ the polynomial ring in the indeterminate $\theta$. In the 1970's, soon after he defined elliptic modules (a.k.a., Drinfeld modules) Drinfeld was influenced by the work of Krichever to define an associated vector bundle called a "shtuka". In order to do so, Drinfeld worked with the $2$-dimensional algebra $A\otimes_{\mathbf F_q} A$ which precisely combined the roles of operator and scalar. Soon after that, Greg Anderson used this algebra to develop his higher dimensional analog of Drinfeld modules (called "$t$-modules"); in particular, Anderson's theory allowed one to create a good category of "motives" out of Drinfeld modules which is, itself, equipped with a good notion of a tensor product.

One can associate to Drinfeld modules analogs of classical special functions such as $L$-series, gamma functions; etc. Classical theory leads to the expectation that these gamma functions should somehow be related to the $L$-series much as gamma functions are "Euler-factors at infinity" in classical algebraic number theory. But so far that has not been the case and the connection, if one exists, remains unknown.

The basic Drinfeld module is the rank $1$ module $C$ discovered by L. Carlitz in the 1930's (in a triumph of old school algebra!); it is a function field analog of the algebraic group ${\mathbf G}_m$ and its exponential is a function field analog of the classical exponential function.  Let $\tau(z):=z^{q}$ be the $q$-th power mapping with $\tau^i$ defined by composition; the Carlitz module is then the $\mathbf F_q$-algebra map defined by $C_\theta:=\theta \tau^0+\tau$. Using Anderson's notion of a tensor product, Greg and Dinesh Thakur rapidly defined, and studied, the $n$-tensor power $C^{\otimes n}$ of the Carlitz module in "Tensor powers of the Carlitz module and zeta values," Ann. of Math. 132 (1990), 159–191.  In particular, they defined the following marvelous function
$$\omega (t):=\theta_1 \prod_{i=0}^\infty \left(1-\frac{t}{\theta^{q^i}}\right)^{-1}\,,$$
where $\theta_1$ is a fixed $(q-1)$-st root of $-\theta$. Notice that $\omega(t)$ is obviously the reciprocal of an entire function and, in that, it reminds one of Euler's gamma function.

However, much more profound is the result of Anderson/Thakur (loc. cit.) that $\lim_{t\mapsto\theta}(t-\theta)\omega(t)$ is the period $\tilde{\xi}$ of the Carlitz module. Here one can't help but be reminded of the famous equality $\Gamma(1/2)=\sqrt \pi$; so one is led to view $\omega(t)$ as yet another function field manifestation of the notion of a gamma function. Indeed, in a tour de force, "Determination  of the algebraic relations among special $\Gamma$-values in positive characteristic," (Ann. of Math. (2) (2004), 237-313), Anderson, Dale Brownawell, and Matt Papanikolas used $\omega(t)$ to establish virtually all the transcendence results one would want of the geometric gamma function.

So it was apparent, to me anyway, that this magical $\omega(t)$ should also make itself known in the theory of characteristic $p$ $L$-series. However, I simply did not see how this could happen. This impasse was recently broken by some fantastic results of Federico Pellarin ("Values of Certain $L$-series in positive characteristic," Ann. of Math. to appear, http://arxiv.org/abs/1107.4511) and these results precisely provide the operator/scalar fusion mentioned in the title of this blog!

So I would like to finish by describing some of Federico's results, and also those of my student Rudy Perkins in this regard. They both are obtaining all sorts of beautiful formulae of the sort one might find in the famous book by Whittaker and Watson which is very exciting and certainly bodes very well for the future of the subject. But before doing so, we do need one more result of Anderson/Thakur.

As in my previous blog put $K:={\mathbf F}_q((1/\theta))$ with the canonical absolute value. Put ${\mathbf T}:=\{\sum_{i=0}^\infty a_it^i\}$ where $\{a_i\}\subset K$ and $a_i\to 0$ as $i \to \infty$; so $\mathbf T$ is simply the Tate algebra of functions with coefficients in $K$ converging on the closed unit disc.

The algebra $\mathbf T$ comes equipped with two natural operators: First of all, the usual hyperdifferential operators act on $\mathbf T$ via differentation with respect to $t$ in the standard fashion.  Now let $f(t)=\sum a_it^i\in \mathbf T$; we then set $\tau (f):=\sum a_i^qt^i$ and call it the
"partial Frobenius operator" (in an obvious sense). Note that, in this setting, $\tau$ is actually $\mathbf F_q[t]$-linear. Note also, because we are in characteristic $p$ these operators commute.

Anderson and Thakur look at the following partial Frobenius equation on $\mathbf T$: $\tau \phi=(t-\theta)\phi$ (N.B.: $t-\theta$ is the "shtuka function" associated to the Carlitz module). The solutions to this equation clearly form an $\mathbf F_q[t]$-module and the remarkable result of A/T is that this module is free of rank $1$ and generated by $\omega(t)$.

One can rewrite the fundamental equation $\tau \omega=(t-\theta)\omega$ as
$$(\theta \tau^0+\tau)\omega=t\cdot \omega\,;$$
in other words, if we use the partial Frobenius operators to extend the Carlitz module to $\mathbf T$ then $\omega$ trivializes this action. So if $f(\theta)\in A$ one sees immediately that  $C_f\omega=f(t)\omega$.

Abstracting a bit, if $t$ is a scalar, then one defines the "quasi-character" $\chi_t(f):=f(t)$ simply by evaluation. It is Federico's crucial insight that this quasi-character is exactly the necessary device to fuse both the scalars and operators in the theory of characteristic $p$ $L$-series by defining the associated $L$-series $L(\chi_t,s)$ (in the standard fashion). These functions have all the right analytic properties  in the $s$-variable and also have excellent analytic properties in the $t$-variable!

(The reader might have imagined, as I did at first, that the poles $\{\theta^{q^i}\}$ of $\omega(t)$ are too specialized to be associated to something canonical. However, we now see that these poles correspond to the quasi-characters $f(\theta)\mapsto f(\theta^{q^i})=f(\theta)^{q^i}$ and so are completely canonical...)

The introduction of the variable $t$ is, actually, a realization of the notion of "families" of $L$-series. Indeed, if $t$ belongs to the algebraic closure of $\mathbf F_q$, then $\chi_t$ is a character modulo $p(\theta)$, where $p(\theta)$ is the minimal polynomial of $t$.

Theorem: (Pellarin) We have $(t-\theta)\omega(t)L(\chi_t,1)= -\tilde{\xi}\,.$

And so $\omega(t)$ makes its appearance in $L$-series! (One is  also reminded a bit of Euler's famous formula $e^{\pi i}=-1$.) Now let $n$ be a positive integer $\equiv 1$ mod $(q-1)$.

Theorem: (Pellarin) There exists a rational function $\lambda_n\in {\mathbf F}_q(t,\theta)$ such that
$$(t-\theta)\omega(t)L(\chi,n)=\lambda_n \tilde{\xi}^n\,.$$
In "Explicit formulae for $L$-values in finite characteristic" (just uploaded to the arXiv as http://arxiv.org/abs/1207.1753), my student Rudy Perkins gives a simple closed form expression for these $\lambda_n$ as well as all sorts of connections with other interesting objects (such as the Wagner expansion of $\mathbf F_q$-linear functions, recursive formulae for Bernoulli-Carlitz elements, etc.).

So the introduction of $\chi_t$ has opened the door to all sorts of remarkable results. Still, the algebraic closure of $K$ is such a vast thing (with infinitely many extensions of bounded degree etc.), that there may be other surprises we do not yet know. Moreover, we do know that the algebras of measures can be interpreted as hyperdifferential operators on $\mathbf T$. Where are they in the game Federico started?