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 $yin Z_p$ 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 $sigmain S_{(q)}$. Let $yin Z_p$ 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 $yin Z_p$. 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

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

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 $Xsubset Z_p$ 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.