A very simple example

In my previous contributions to this blog , I have mentioned how the calculations of Dinesh Thakur and Javier Diaz-Vargas suggested that the nonclassical trivial zeroes of characteristic $p$ zeta functions associated to ${\bf F}_q[t]$ should have the following two properties (where nonclassical means that the actual order is higher than what one would expect from classical theory):

1. If a nonclassical trivial zero occurs at $-i$ then the sum of the $p$-adic digits of $i$ must be bounded.

2. The orders of the trivial zeroes should be an invariant of the action of the group $S_{(q)}$ of homeomorphisms of $Z_p$ which permute the $q$-adic digits of a $p$-adic integer.

In my last entry, I discussed Dinesh's remarkable result on valuations of certain basic sums in this game; one key point is that the valuations for arbitrary $d$ iteratively reduced to valuations just involving sums of monics of degree one. Here I want to again use monics of degree one to give a very simple example with properties very similar to 1 and 2 above. We will then draw some conclusions for the relevant theory of nonArchimedean measures.

The example presented here was first mentioned by Warren Sinnott, in the $q=p$ case in Warren's paper "Dirichlet Series in function fields" (J. Number Th. 128 (2008) 1893-1899). The $L$-functions that occur in the theory of Drinfeld modules and the like are functions of two
variables $(x,y)$. If one fixes $x$, the functions in $y\in Z_p$ that one obtains are uniform limits of finite sums of exponentials $u^y$ where $u$ is a $1$-unit. In his paper Warren studies such functions and shows that if $f(y)$ is a nonzero such function, its zero set *cannot* contain an open set (unlike arbitrary continuous functions such as step-functions).

In what follows ALL binomial coefficients are considered modulo $p$ so that the basic lemma of Lucas holds for them.

Lemma: 1. Let $\sigma\in S_{(q)}$. Let $y\in Z_p$ and $k$ a nonnegative integer. Then

$${y \choose k}= {\sigma(y) \choose \sigma (k)} \,.$$

2. Let $i,j$ be two nonnegative integers. Then

$${i +j \choose j}= {\sigma (i) +\sigma (j) \choose \sigma (j)}\,.$$

Proof: 1 is simply $q$-Lucas. For 2 note that if there is carry over of digits in the addition for $i+j$ then there is also in the sum for $\sigma (i)+\sigma (j)$, and vice versa; in this case, both sides are $0$. If there is no carry over the result follows from $q$-Lucas again. QED

As before, let $q=p^m$ and let $y\in Z_p$. Let $A=Fq[t]$ and let $\pi=1/t$; so $\pi$ is a positive uniformizer at the place $\infty$ of ${\bf F}_q(t)$. Define

$$ f(y):= \sum_{g\in A^+(1)} (\pi g)^y \,;$$

where $A^+(1)$ is just the set of monic polynomials of degree $1$. The sum can clearly be rewritten as

$$ f(y)=\sum_{\alpha \in \Fq}(1+\alpha \pi)^y .

Upon expanding out via the binomial theorem, and summing over $\alpha$, we find

$$ f(y)= -\sum_{k \in I} {y \choose k} \pi^k$$

where $I$ is the set of positive integers divisible by $q-1$.

Let $X\subset Z_p$ be the zeroes of $f(y)$; it is obviously closed. When $q=p$, Warren (in his paper and in personal communication) showed that $X$ consists pricisely of those non-negative integers $i$ such that the sum of the $p$-adic digits of $i$ is less than $p$.

Now, in order to show that $f(y) \neq 0$, for a given $y$ in $Z_p$, it is necessary and sufficient to simply show that there is ONE $k \in I$ such that ${y \choose k}$ is nonzero in ${\bf F}_p$. When $q=p$, this is readily accomplished.

However, when $q$ is general it gets much more subtle to make sure that the reduced binomial coefficient is non-zero.

Proposition: The set $X$ is stable under $S_{(q)}$. Moreover, there is an explicit constant $C$ (which depends on $q$) such that the elements of $X$ have their sum of $q$-adic coefficients less than $C$.

(As Warren has remarked, the Proposition then reduces the problem of finding the zero set to checking *finitely many* orbits!)

Proof:

Let $\sigma \in S_{(q)}$. The first part follows immediately from the first part of the Lemma and the fact that $I$ is stable under $S_{(q)}$.

To see the second part, let $C: = (q-2)(1+2+\cdots+ q-1)=(q-2)(q-1)q/2$. Let $y$ be any $p$-adic integer with the property that its sum of $q$-adic digits is greater than $C$. Then there must be at least one $e$ with $e$ between $0$ and $q-1$ such that $e$ occurs at least $q-1$ times in the expansion of $y$. It is then easy to find $k$ such that the reduction of ${y \choose k}$ is nonzero. QED

There are other important results that arise from the first part of the Lemma. Indeed, upon replacing $k$ with $\sigma^{-1}(t)$, we obtain

$${y \choose \sigma^{-1}(t)}= {\sigma(y) \choose t\,.$$ (*)

This immediately gives the action of $S_{(q)}$ on the Mahler expansion of a continuous function from $Z_p$ to characteristic $p$. One also obviously has

$$\sum_k {\sigma y \choose k} x^k=
\sum_k {\sigma(y) \choose \sigma (k)}x^{\sigma(k)\,.$$

But, by the first part of the Lemma, this then equals

$$\sum {y \choose k}x^{\sigma k}\,,$$

which is a sort of change of variable formula.

As the action of $S_{(q)}$ is continuous on $Z_p$ there is a dual action on measures; if the measures are characteristic $p$ valued, then this action is easy to compute from (*) above.

However, there is ALSO a highly mysterious action of $S_{(q)}$ on the *convolution algebra* of characteristic $p$ valued measures on the maximal compact subrings in the completions of $F_q(T)$ at its places of degree $1$ (e.g, the place at $\infty$ or associated to $(t)$, if the place has higher degree one replaces $S_{(q)}$ with the appropriate subgroup). Indeed, given a Banach basis for the space of $Fq$-linear continuous functions from that local ring to itself, the "digit expansion principle"gives a basis for ALL continuous functions of the ring to itself (see, e.g., Keith Conrad, "The Digit Principle", J. Number Theory 84(2000) 230-257). In the 1980's Greg Anderson and I realized that this gives an isomorphism of the associated convolution algebra of measures with the ring of formal *divided power series* over the local ring.

But let $\sigma \in S_{(q)}$ and define

$$\sigma (z^i/i!):= z^{\sigma (i)}/\sigma(i)! \.$$

The content of the second part of the Lemma is precisely that this definition gives rise to an algebra automorphism of the ring of formal divided power series.