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 ${\displaystyle p}$ zeta functions associated to ${\displaystyle {\backslash bfF}_q\left[t\right]}$ 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 ${\displaystyle -i}$ then the sum of the ${\displaystyle p}$-adic digits of ${\displaystyle i}$ must be bounded.

2. The orders of the trivial zeroes should be an invariant of the action of the group ${\displaystyle S_{\left(q\right)}}$ of homeomorphisms of ${\displaystyle Z_p}$ which permute the ${\displaystyle q}$-adic digits of a ${\displaystyle p}$-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 ${\displaystyle d}$ 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 ${\displaystyle q=p}$ case in Warren's paper "Dirichlet Series in function fields" (J. Number Th. 128 (2008) 1893-1899). The ${\displaystyle L}$-functions that occur in the theory of Drinfeld modules and the like are functions of two
variables ${\displaystyle (x,y)}$. If one fixes ${\displaystyle x}$, the functions in ${\displaystyle y\in Z_p}$ that one obtains are uniform limits of finite sums of exponentials ${\displaystyle u^y}$ where ${\displaystyle u}$ is a ${\displaystyle 1}$-unit. In his paper Warren studies such functions and shows that if ${\displaystyle f\left(y\right)}$ 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 ${\displaystyle p}$ so that the basic lemma of Lucas holds for them.

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

${\displaystyle {\displaystyle \left(\genfrac{}{}{0ex}{}{y}{k}\right)=\sigma \left(y\left(\genfrac{}{}{0ex}{}{)}{\sigma }\right)\right(k).}}$

2. Let ${\displaystyle i,j}$ be two nonnegative integers. Then

${\displaystyle {\displaystyle i+\left(\genfrac{}{}{0ex}{}{j}{j}\right)=\sigma \left(i\right)+\sigma \left(j\left(\genfrac{}{}{0ex}{}{)}{\sigma }\right)\right(j).}}$

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

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

${\displaystyle {\displaystyle f\left(y\right):=\sum _{g\in A^{+}\left(1\right)}(\pi g{)}^y;}}$

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

${\displaystyle {\displaystyle f\left(y\right)=\sum _{\alpha \in \backslash Fq}(1+\alpha \pi {)}^y.}}$

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

}${\displaystyle }$

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

${\displaystyle {\displaystyle }}$

where ${\displaystyle I}$ is the set of positive integers divisible by ${\displaystyle q-1}$.

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

Now, in order to show that ${\displaystyle f\left(y\right)\ne 0}$, for a given ${\displaystyle y}$ in ${\displaystyle Z_p}$, it is necessary and sufficient to simply show that there is ONE ${\displaystyle k\in I}$ such that ${\displaystyle \left(\genfrac{}{}{0ex}{}{y}{k}\right)}$ is nonzero in ${\displaystyle {\backslash bfF}_p}$. When ${\displaystyle q=p}$, this is readily accomplished.

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

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

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

Proof:

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

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

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

}${\displaystyle }$

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

${\displaystyle {\displaystyle (*)}}$

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

}${\displaystyle }$

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

${\displaystyle {\displaystyle }}$

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

}${\displaystyle }$

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

${\displaystyle {\displaystyle }}$

which is a sort of change of variable formula.

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

However, there is ALSO a highly mysterious action of ${\displaystyle S_{\left(q\right)}}$ on the *convolution algebra* of characteristic ${\displaystyle p}$ valued measures on the maximal compact subrings in the completions of ${\displaystyle F_q\left(T\right)}$ at its places of degree ${\displaystyle 1}$ (e.g, the place at ${\displaystyle \infty }$ or associated to ${\displaystyle \left(t\right)}$, if the place has higher degree one replaces ${\displaystyle S_{\left(q\right)}}$ with the appropriate subgroup). Indeed, given a Banach basis for the space of ${\displaystyle Fq}$-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 ${\displaystyle \sigma \in S_{\left(q\right)}}$ and define

}${\displaystyle }$

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

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.