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 $p$.

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.