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)! \.

${\displaystyle {\displaystyle }}$

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.

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 ${\displaystyle p}$ 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 ${\displaystyle \zeta (x,y)}$ in ${\displaystyle x^{-1}}$ where the parameter ${\displaystyle y}$ lies in the ${\displaystyle p}$-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 ${\displaystyle y}$. At that point, Daqing Wan and Yuichiro Taguchi were visiting me to discuss applications of Dwork theory to general ${\displaystyle L}$-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 ${\displaystyle \zeta (x,y)}$ where he worked in the simplest possible case of ${\displaystyle {\backslash bfF}_p\left[t\right]}$. It was quite a shock when he stated that these calculations showed that the zeroes of ${\displaystyle \zeta (x,y)}$ were simple and in the field ${\displaystyle {\backslash bfF}_p\left(\right(1/t\left)\right)}$ (indeed there was at most ${\displaystyle 1}$ zero, with multiplicity, of a given absolute value); in other words, all the zeroes lie "on the line" given by ${\displaystyle {\backslash bfF}_p\left(\right(1/t\left)\right)}$ itself. Clearly this was a form of the Riemann hypothesis for these functions and Wan's results marked the first indication that these characteristic ${\displaystyle p}$ functions possess a profound theory of their zeroes.

In the characteristic ${\displaystyle p}$ theory, the theory for ${\displaystyle {\backslash bfF}_p\left[t\right]}$ and general ${\displaystyle {\backslash bfF}_q\left[t\right]}$ (${\displaystyle q=p^m}$, ${\displaystyle m}$ arbitrary) should be the same; so one wanted to know whether the Newton polygons associated to ${\displaystyle {\backslash bfF}_q\left[t\right]}$ also had the same simple form as given in the ${\displaystyle q=p}$ 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 ${\displaystyle {\backslash bfF}_q\left[t\right]}$ case is much harder than the special case when ${\displaystyle q=p}$!

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 ${\displaystyle {\backslash bfF}_q\left[T\right]}$ 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 ${\displaystyle \infty }$-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 ${\displaystyle A={\backslash bfF}_q\left[t\right]}$ and let ${\displaystyle d}$ be a nonnegative integer and ${\displaystyle k}$ an arbitrary integer. Let ${\displaystyle A_{+}\left(d\right)}$ be the set of monic elements in ${\displaystyle A}$ of degree ${\displaystyle d}$. Define:

}${\displaystyle }$

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

${\displaystyle {\displaystyle }}$

which is an element of ${\displaystyle {\backslash bfF}_q\left(T\right)}$. Let ${\displaystyle s_d\left(k\right)}$ be the valuation of ${\displaystyle S_d\left(k\right)}$ at the place ${\displaystyle \infty }$ of ${\displaystyle {\backslash bfF}_q\left(t\right)}$.

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

}${\displaystyle }$

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

${\displaystyle {\displaystyle }}$

This then leads iteratively to the second recursion formula

}${\displaystyle }$

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

${\displaystyle {\displaystyle }}$

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

The main recursion formula is highly remarkable in that one computes a sum over the monics of degree ${\displaystyle 1}$ and then finds its valuation at ${\displaystyle \infty }$ and *then* uses this integer as the exponent to raise the monics of degree ${\displaystyle d-1}$. 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 ${\displaystyle A}$-analog of the basic formula

}${\displaystyle }$

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

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