Dinesh Thakur's remarkable recursion formla

In this blog entry, I would like to highlight a remarkable formula due to Dinesh Thakur in the arithmetic of function fields over finite fields. This formula appears in page 5 of his preprint "Power sums with applications to multizeta values and zeta zeros" which can be downloaded at

http://math.arizona.edu/~thakur/power.pdf

Before presenting Dinesh's formula, I will present a little history. Early on in the theory of characteristic $p$ zeta functions, I used a simple lemma to obtain strong enough estimates to establish that such functions, and their interpolations at finite primes, are indeed "entire" (which, in this case, means a family of entire power series $\zeta(x,y)$ in $x^{-1}$ where the parameter $y$ lies in the $p$-adic integers). In the middle of the 90's, I discovered some old formulas of Carlitz gave much better (exponential) estimates for some special values of $y$. At that point, Daqing Wan and Yuichiro Taguchi were visiting me to discuss applications of Dwork theory to general $L$-series of Drinfeld modules. So I asked Daqing if he could use their theory to obtain such exponential estimates. The next day he came and showed me his elementary calculations for the Newton polygons for $\zeta(x,y)$ where he worked in the simplest possible case of ${\bf F}_p[t]$. It was quite a shock when he stated that these calculations showed that the zeroes of $\zeta(x,y)$ were simple and in the field ${\bf F}_p((1/t))$ (indeed there was at most $1$ zero, with multiplicity, of a given absolute value); in other words, all the zeroes lie "on the line" given by ${\bf F}_p((1/t))$ itself. Clearly this was a form of the Riemann hypothesis for these functions and Wan's results marked the first indication that these characteristic $p$ functions possess a profound theory of their zeroes.

In the characteristic $p$ theory, the theory for ${\bf F}_p[t]$ and general ${\bf F}_q[t]$ ($q=p^m$, $m$ arbitrary) should be the same; so one wanted to know whether the Newton polygons associated to ${\bf F}_q[t]$ also had the same simple form as given in the $q=p$ case. This was finally proved by Jeff Sheats based on some ideas of Bjorn Poonen; see Dinesh's paper for more history and the exact references. In any case, the general ${\bf F}_q[t]$ case is much harder than the special case when $q=p$!

We still do not know exactly how to phrase an "Rh" in general because the trivial zeroes can have a very large impact on other zeroes due to the nonArchimedean topology of the spaces these functions are defined on. (Indeed, this was what made the calculations of Dinesh and Javier Diaz-Vargas on "nonclassical" trivial zeroes so important --- here, again, by nonclassical we mean trivial zeroes whose true order of
vanishing is higher than one would expect from classical theory). Moreover, even in the ${\bf F}_q[T]$ case one does not understand what sort of information is contained in the results of Wan and Sheats. However, Thakur's results may be giving as the first very serious clues.

What Dinesh does is to establish a fundamental recursion formula for the $\infty$-adic valuations of certain fundamental sums arising in the function field theory (see page 5 of his preprint). From this recursion, the "Rh" follows readily.

Here then is the recursion formula, which, you will see, is quite elementary to state. We follow the notation of the paper: Let $A={\bf F}_q[t]$ and let $d$ be a nonnegative integer and $k$ an arbitrary integer. Let $A_+(d)$ be the set of monic elements in $A$ of degree $d$. Define:

$$ S_d(k):=\sum_{a\in A_+(d)} 1/a^k$$

which is an element of ${\bf F}_q(T)$. Let $s_d(k)$ be the valuation of $S_d(k)$ at the place $\infty$ of ${\bf F}_q(t)$.

Dinesh's "main recursion formula" then states that:

$$s_d(k)=s_{d-1}(s_1(k)) + s_1(k)\,.$$

This then leads iteratively to the second recursion formula

$$s_d(k)=s_1^{(d)}(k)+\ldots +s_1^{(2)}(k) + s_1(k)\,.$$

where $s_1^{(i)}$ means the $i$-composition of the $s_1$ map with itself.

The main recursion formula is highly remarkable in that one computes a sum over the monics of degree $1$ and then finds its valuation at $\infty$ and *then* uses this integer as the exponent to raise the monics of degree $d-1$. This feedback loop is absolutely new in terms of anything that I have ever seen.

One can ask whether there are any classical analogs of the above recursion formulas. It may be that when things are much better known, the second recursion formula will be viewed as the $A$-analog of the basic formula

$$N_n(m)=q^{nm}+q^{(n-1)m}+\cdots + q^m+1$$

which gives the number of points over ${\bf F}_{q^m}$ of projective $n$-space. An analog of Dinesh's first recursion formula is now easy to construct.