# Universal families of Eulerian multiple zeta values in positive
characteristic

Kwun Chung  Department of Mathematics
University of California, San Diego (UCSD)
9500 Gilman Drive 0112
La Jolla, CA 92093-0112
United States of America (USA<EMAIL_ADDRESS>,  Tuan Ngo Dac  Univ Lyon,
CNRS, Université Claude Bernard Lyon 1, Institut Camille Jordan, UMR 5208,
F-69622 Villeurbanne, France<EMAIL_ADDRESS>and  Federico Pellarin
Institut Camille Jordan, UMR 5208, Site de Saint-Etienne, 23 rue du Dr. P.
Michelon, F-42023 Saint-Etienne, France<EMAIL_ADDRESS>
###### Abstract.

We study positive characteristic multiple zeta values associated to general
curves over $\mathbb{F}_{q}$ together with an $\mathbb{F}_{q}$-rational point
$\infty$ as introduced by Thakur. For the case of the projective line these
values were defined as analogues of classical multiple zeta values. In the
present paper we first establish a general non-commutative factorization of
exponential series associated to certain lattices of rank one. Next we
introduce universal families of multiple zeta values of Thakur and show that
they are Eulerian in full generality. In particular, we prove a conjecture of
Lara Rodríguez and Thakur in [28]. One of the main ingredients of the proofs
is the notion of $L$-series in Tate algebras introduced by the third author
[31] in 2012.

## Introduction

###  0.1. Classical multiple zeta values

Multiple zeta values are positive real numbers that have been involved in
certain investigations by Euler in the late eighteenth century. They
generalize the values of Riemann’s zeta function at integers $\geq 2$ and
occur naturally in the computation of their products. Surprisingly, these
particular real numbers are ubiquitous in several mathematical and physical
theories. For instance, they are connected with periods of mixed Tate motives,
and with values of Feynman integrals in perturbative quantum field theory. It
is also known that they are related to a certain universal vector bundle with
connection on the projective line with the points $0,1,\infty$ removed. These,
and other properties, made multiple zeta values the center of intensive
studies, especially in the last three decades. We refer the reader to the
survey of Zagier [43] and the book of Burgos Gil and Fresán [15] for a
detailed introduction to these topics, as well as for further references.

Let $\mathbb{N}$ be the set of positive integers. By definition, the classical
multiple zeta values are the values of the convergent series

$\zeta(n_{1},\dots,n_{r})=\sum_{k_{1}>\dots>k_{r}>0}\frac{1}{k_{1}^{n_{1}}\dots
k_{r}^{n_{r}}},\quad\text{where }n_{i}\in\mathbb{N},\quad n_{1}\geq 2.$

Here $r$ is called the depth and $w=n_{1}+\dots+n_{r}$ the weight of the
presentation $\zeta(n_{1},\dots,n_{r})$. For notational convenience, we set
$\zeta(\emptyset)=1$. For $r=1$ and $n\geq 2$ we recover the special values
$\zeta(n)$ of the Riemann zeta function.

In 1735, Euler proved that

(0.1)
$\frac{\zeta(2n)}{(2\pi)^{2n}}=(-1)^{n+1}\frac{1}{2}\frac{B_{2n}}{(2n)!}\quad\text{
for all }n\geq 1,$

where for $k\geq 0$, $B_{k}$ denotes the $k$-th Bernoulli number. In view of
this, it is natural to classify tuples $(n_{1},\dots,n_{r})$ with $n_{i}\geq
1,n_{1}\geq 2$ such that the quotient
$\zeta(n_{1},\dots,n_{r})/\pi^{n_{1}+\dots+n_{r}}$ is rational. In this case,
following [27, 18], we say that the multiple zeta value
$\zeta(n_{1},\dots,n_{r})$ is Eulerian.

For example $\zeta(\underbrace{2,\dots,2}_{n})$ with $n\geq 1$ is Eulerian.
This follows from the identity

(0.2) $\sum_{n\geq
0}(-1)^{n}\zeta(\underbrace{2,\dots,2}_{n})x^{2n+1}=\frac{\sin(\pi
x)}{\pi}=\sum_{n\geq 0}(-1)^{n}\frac{\pi^{2n}}{(2n+1)!}x^{2n+1},$

an elegant proof of which is given by Zagier in [44]. Further, one can show
that $\zeta(\underbrace{3,1,\dots,3,1}_{n})$ is Eulerian (conjectured in [43]
and proved in [13]). We refer the reader to [27, Remark after Conjecture 4.3]
and [39, §7.5] for more details and more complete references on Eulerian
multiple zeta values. Brown formulated a sufficient (and conjecturally
necessary) condition for Eulerian multiple zeta values in terms of motivic
multiple zeta (see [14, Theorem 3.3]). This condition is not completely
effective (see [18, §1] for more details).

Finally, we mention that Siegel and Klingen were able to extend Euler’s
formulas (0.1) for Dedekind zeta values $\zeta_{F}(.)$ attached to a number
field $F$ and showed that for a totally real number field $F$ and all even
$n\geq 2$, $\zeta_{F}(n)$ is an algebraic multiple of $\pi^{n[F:\mathbb{Q}]}$
(see [33] for further details).

###  0.2. Multiple zeta values in positive characteristic

There is a deep analogy between the arithmetic of number fields and that of
global function fields of positive characteristic. Through this analogy one
can track similarities in the comparison of the arithmetic over the ring
$\mathbb{Z}$ on one side, and the arithmetic over the ring $A$ of rational
functions over a given smooth curve $X$ over a finite field, which are regular
away from a distinguished point (see below for a rigorous description of the
settings). Carlitz, in the years 1930s, inaugurated this study in the case of
$X$ the projective line (the genus zero case). As a natural consequence, a
theory of positive characteristic multiple zeta values associated to $A$
emerged, pioneered by Thakur. At this level of generality ($X$ a curve over a
finite field), the description of the algebraic relations connecting multiple
zeta values offers new challenges that do not seem to be immediately visible
in the classical setting. It is our aim to shed new light and perspectives on
these aspects.

Throughout this paper, let $\mathbb{F}_{q}$ be the finite field with $q$
elements and characteristic $p>0$. Let $K$ be the function field of a
geometrically connected smooth projective curve $X$ over $\mathbb{F}_{q}$ and
let $\infty$ be a place of $K$ of degree $d_{\infty}=1$. Let $g$ be the genus
of $X$ and let

(0.3) $A=H^{0}(X\setminus\\{\infty\\},\mathcal{O}_{X})$

be the ring of elements of $K$ which are regular outside $\infty$ ($A$ is
called the base ring). The $\infty$-adic completion $K_{\infty}$ of $K$ is
equipped with the normalized additive $\infty$-adic valuation
$v_{\infty}:K_{\infty}\rightarrow\mathbb{Z}\cup\\{+\infty\\}$. Let us set
$\deg(\cdot):=-v_{\infty}(\cdot)$. The completion $\mathbb{C}_{\infty}$ of a
fixed algebraic closure $\overline{K}_{\infty}$ of $K_{\infty}$ comes equipped
with a unique valuation extending $v_{\infty}$, which will be again denoted by
$v_{\infty}$. We fix a uniformizer $\pi$ of $K_{\infty}$ so that we can
identify $K_{\infty}$ and the field of formal Laurent series
$\mathbb{F}_{q}((\pi))$. This choice also allows to introduce a sign function
$\operatorname{sgn}:K_{\infty}^{\times}\to\mathbb{F}_{q}^{\times}$. This is
the unique group homomorphism defined by $\operatorname{sgn}(\pi)=1$. We
define $A_{+}$ to be the set of monic elements of $A$, i.e., the set of $a\in
A$ such that $\operatorname{sgn}(a)=1$. For all $i\in\mathbb{N}$, we also set
$A_{+,i}=\\{a\in A_{+}:\deg(a)=i\\}$.

### The genus 0 case

We consider the genus $0$ case where $X=\mathbb{P}^{1}$ and $\infty$ is an
$\mathbb{F}_{q}$-rational point on it. Then $A=\mathbb{F}_{q}[\theta]$ the
polynomial ring in $\theta$, a rational function over $\mathbb{P}^{1}$ which
has a simple pole at $\infty$ and which is regular everywhere else on
$\mathbb{P}^{1}$. We denote by $K=\mathbb{F}_{q}(\theta)$ the fraction field
of $A$ and by $K_{\infty}=\mathbb{F}_{q}((1/\theta))$ the completion of $K$ at
the place associated with $\infty$.

Recall that $\mathbb{N}$ denotes the set of positive integers. In [16] Carlitz
introduced, for $n\in\mathbb{N}$, the elements $\zeta_{A}(n)$ given by the
convergent series

$\zeta_{A}(n):=\sum_{a\in A_{+}}\frac{1}{a^{n}}\in K_{\infty}$

which, through the previously mentioned analogy, correspond to classical
special zeta values in the function field setting and are called the Carlitz
zeta values. For any tuple of positive integers
$\mathfrak{s}=(s_{1},\ldots,s_{r})\in\mathbb{N}^{r}$, Thakur [36] considered
the elements $\zeta_{A}(\mathfrak{s})$ (or $\zeta_{A}(s_{1},\ldots,s_{r})$) of
$K_{\infty}$ defined by the convergent series

(0.4) $\zeta_{A}(\mathfrak{s}):=\sum\frac{1}{a_{1}^{s_{1}}\ldots
a_{r}^{s_{r}}}\in K_{\infty}$

where the sum runs through the set of tuples $(a_{1},\ldots,a_{r})\in
A_{+}^{r}$ with $\deg a_{1}>\ldots>\deg a_{r}$. These are Thakur’s multiple
zeta values, characteristic $p$ analogues of the classical multiple zeta
values. We call $r$ the depth of $\zeta_{A}(\mathfrak{s})$ and
$w(\mathfrak{s})=s_{1}+\dots+s_{r}$ the weight of $\zeta_{A}(\mathfrak{s})$.
We extend the definition of $\zeta_{A}(\mathfrak{s})$ to the empty tuple by
defining the associated zeta value $\zeta_{A}(\emptyset)$ to be 1. We note
that Carlitz zeta values $\zeta_{A}(n)$ are exactly depth one multiple zeta
values. It can be proved that for any $\mathfrak{s}$ as above,
$\zeta_{A}(\mathfrak{s})\in K_{\infty}^{\times}$ (see [38]). A much stronger
property holds: Thakur’s multiple zeta values $\zeta_{A}(\mathfrak{s})$ and
$1$ span an $\mathbb{F}_{p}$-vector space which is also an
$\mathbb{F}_{p}$-algebra, graded by the weights (see [17, 39]).

In 1935 Carlitz [16] introduced analogues $BC_{n}\in K^{\times}$ ($n\geq 1$)
of the Bernoulli numbers of even order and proved an analogue of Euler’s
result (0.1) (see also [25, §9.2]):

$\frac{\zeta_{A}(n)}{\widetilde{\pi}^{n}}=\frac{BC_{n}}{\Gamma_{n}}\quad\text{
for all }n\geq 1,\ n\equiv 0\pmod{q-1}.$

Here $\Gamma_{n}\in A$ is the $n$-th Carlitz factorial, and
$\widetilde{\pi}\in\mathbb{C}_{\infty}^{\times}$ denotes the Carlitz period
which is defined up to multiplication by an element of
$\mathbb{F}_{q}^{\times}$ (see [25, 36]).

Thakur called $\zeta_{A}(\mathfrak{s})$ Eulerian (see [36, Definition 5.10.8])
if $\zeta_{A}(\mathfrak{s})/\widetilde{\pi}^{w(\mathfrak{s})}$ belongs to $K$.
Since $\zeta(ps_{1},\dots,ps_{r})=\zeta(s_{1},\dots,s_{r})^{p}$, we restrict
ourselves to primitive tuples $\mathfrak{s}$, characterized by the property
that not all the coefficients $s_{i}$ are divisible by $p$.

Chen [20], Lara Rodríguez and Thakur [27] proved the following remarkable
theorem:

######  Theorem 0.1 (Chen, Lara Rodríguez and Thakur).

Assume that $A=\mathbb{F}_{q}[\theta]$. Then for all $n\geq 1$ and $k\geq 0$,
the multiple zeta value

$\zeta_{A}\left(q^{n}-1,(q-1)q^{n},\ldots,(q-1)q^{n+k-1}\right)$

is Eulerian.

Contrary to the classical setting, for $q>2$, it is conjectured that these
series exhaust all the (primitive) Eulerian tuples when the depth exceeds $2$
(see [18, §6.2] and [39, §7.3]). The depth is formally defined in §1.2.

### The higher genus case

The analogy mentioned in the beginning of §0.2 focuses on the comparison
between the arithmetic theory over $\mathbb{Z}$ and that over an
$\mathbb{F}_{q}$-algebra $A$ as in (0.3). For this reason, Carlitz’s theory
only gives an incomplete view. We now move to include the general situation of
the higher genus cases and present our main results.

The definition (0.4) of $\zeta_{A}(\mathfrak{s})$ extends without modification
to the case of general $A$ with $\infty\in X(\mathbb{F}_{q})$ such that
$d_{\infty}=\deg(\infty)=1$ as in (0.3) (see also §1.2 for precise
definitions). Following Lara Rodríguez and Thakur in [28], we say that
$\zeta_{A}(\mathfrak{s})$ is Eulerian if
$\zeta_{A}(\mathfrak{s})/\widetilde{\pi}_{A}^{w(\mathfrak{s})}$ is algebraic
over $K$, where
$\widetilde{\pi}_{A}:=\widetilde{\pi}_{\phi}\in\mathbb{C}_{\infty}$ is the
period attached to the standard sign-normalized Drinfeld module of rank one
associated to $A$ (see §3.1).

It turns out that for all $n\geq 1$ and $n\equiv 0\pmod{q-1}$, $\zeta_{A}(n)$
is Eulerian (see [25, §8.18] and note the proximity with the Klingen-Siegel
theorem mentioned above). However, only five other cases of Eulerian multiple
zeta values were known by the recent work of Lara Rodríguez and Thakur [28]:
they are all of the form $\zeta_{A}\left(q-1,q(q-1)\right)$ or
$\zeta_{A}\left(q^{2}-1,(q-1)q^{2}\right)$ for one of the four base rings $A$
of class number one. The proofs are based on explicit log-algebraicity
formulas developed in [34, 37] (see for examples [1, 2, 5, 7, 30] for further
developments about log-algebraicity identities). Based on extensive use of
computer aided numerical experiments, Lara Rodríguez and Thakur conjectured
that the appropriate generalization of the statement of Theorem 0.1 remains
valid for these four base rings (see [28, Conjecture 3.1]).

###  0.3. Main results

In this paper we work with the general case where $A$ is an arbitrary base
ring as given in §0.2. We first establish a non-commutative analogue of
Zagier’s formula (0.2) in our settings, with $A$ as in (0.3) and
$d_{\infty}=1$.

######  Theorem 0.2 (Theorem 3.4).

Let $A$ be an arbitrary base ring and let $\phi$ be the corresponding standard
sign-normalized Drinfeld module of rank one. Then

$\sum_{k\geq
0}(-1)^{k}\zeta_{A}\left(q-1,\ldots,(q-1)q^{k-1}\right)\tau^{k}=\widetilde{\pi}_{\phi}^{-1}\exp_{\phi}\widetilde{\pi}_{\phi}\in
K_{\infty}[[\tau]].$

In particular, for all $k\in\mathbb{N}$,
$\zeta_{A}(q-1,q(q-1),\dots,q^{k-1}(q-1))$ is Eulerian.

Here $\widetilde{\pi}_{\phi}\in\mathbb{C}_{\infty}$ is any fundamental period
attached to $\phi$ and $\exp_{\phi}\in K_{\infty}[[\tau]]$ is the exponential
function associated to $\phi$ (see §3.1). As $\widetilde{\pi}_{\phi}$ is
defined up to multiplication by an element of $\mathbb{F}_{q}^{\times}$, the
formula does not depend on its choice.

We then extend Theorem 0.1 with explicit formulas for a base ring $A$ as in
the hypotheses of Theorem 0.2, and prove the aforementioned conjecture of Lara
Rodríguez and Thakur in full generality.

######  Theorem 0.3 (Theorem 3.6).

Let $A$ be an arbitrary base ring and let $n\geq 1$ be an integer. We denote
by $H\subset K_{\infty}$ the Hilbert class field of $A$. Then for all $k\geq
0$, there exists an explicitly computable constant $\alpha_{n,k}\in H$ (see
(3.8)) such that

$\zeta_{A}\left(q^{n}-1,(q-1)q^{n},\ldots,(q-1)q^{n+k-1}\right)=\alpha_{n,k}\zeta_{A}\left(q^{n+k}-1\right).$

In particular,
$\zeta_{A}\left(q^{n}-1,(q-1)q^{n},\ldots,(q-1)q^{n+k-1}\right)$ is Eulerian.

As pointed out by Lara Rodríguez and Thakur in [28], such identities also
hold, in the genus zero case, at ‘finite levels’ (that is, at the level of
partial sums). However, different phenomena occur in the case of $X$ of genus
$g>0$. In this case, identities of multiple zeta values (‘infinite level’
identities) are regulated in a different way and Theorems 0.2 and 0.3 do not
seem to mirror identities at the finite level. It is in fact the analysis of
the interesting phenomenology described in [28] that brought us to the path
that gave rise to the present paper. The reader will also notice similarities
in the processes behind the proofs of, on one side, [27, Theorem 3.2] and, on
the other side, the proof of [24, Lemma 6.11].

We now briefly describe the ingredients of the proofs. Our approach is
completely different from that of the previous works. To prove Theorem 0.2 we
provide a non-commutative factorization of the so-called Carlitz-Hayes
polynomials. Then, passing to the limit, we obtain a non-commutative
factorization of the function
$\exp_{A}:=\widetilde{\pi}_{\phi}^{-1}\exp_{\phi}\widetilde{\pi}_{\phi}$ from
which Theorem 0.2 follows.

The proof of Theorem 0.3 is based on the key notion of $L$-series in Tate
algebras introduced by the third author in [31]. This notion has turned out to
be very fruitful in function field arithmetic, see for example [5, 6, 7, 9,
10, 11, 12, 24, 26]. In our setting, these functions can be identified with
rigid analytic functions over the analytification
$(X\setminus\\{\infty\\})^{an}_{\mathbb{C}_{\infty}}$ of the affine curve
$\operatorname{Spec}(A\otimes_{\mathbb{F}_{q}}\mathbb{C}_{\infty})$ over
$\mathbb{C}_{\infty}$. Note that other classical functions of the theory, such
as the exponential functions $\exp_{\phi}$ and $\exp_{A}$, are rather rigid
analytic functions over the rigid affine line
$\mathbb{A}^{1,an}_{\mathbb{C}_{\infty}}$.

We prove a key and unexpected new general identity in Theorem 2.4 among
$L$-series in Tate algebras at finite levels. As a consequence, we obtain a
harmonic product formula in Proposition 3.5 which allows us to conclude.

Finally, as an application, for class number one base rings, we show in
Theorem 4.2 that our explicit expressions match with those conjectured by Lara
Rodríguez and Thakur [28, Conjecture 3.3].

### Acknowledgements

The authors are deeply thankful to Dinesh Thakur for a careful reading of a
preliminary version of the present manuscript, and for useful corrections,
suggestions and discussions.

##  1\. Background

###  1.1. Notation

We keep the notation of the Introduction. Recall that $\mathbb{N}$ is the set
of positive integers. We denote by $\mathbb{Z}^{\geq 0}$ the set of non-
negative integers. Let $K$ be the function field of a geometrically connected
smooth projective curve $X$ over $\mathbb{F}_{q}$ and $\infty$ be a place of
$K$ of degree $d_{\infty}=1$. Let $g$ be the genus of $X$ and $A$ be the base
ring of elements of $K$ which are regular outside $\infty$. Let $v_{\infty}$
be the additive valuation at $\infty$ and let us set
$\deg(\cdot):=-v_{\infty}(\cdot)$. We denote by $K_{\infty}$ the completion of
$K$ at $\infty$ and by $\mathbb{C}_{\infty}$ the completion of an algebraic
closure $\overline{K}_{\infty}$. Let $\overline{K}$ be the algebraic closure
of $K$ in $\overline{K}_{\infty}$.

We fix a uniformizer $\pi$ of $K_{\infty}$ so that we can identify
$K_{\infty}$ and the field of formal Laurent series $\mathbb{F}_{q}((\pi))$.
This choice also allows to associate to it a sign function
$\operatorname{sgn}:K_{\infty}^{\times}\to\mathbb{F}_{q}^{\times}$ which is
the unique group homomorphism such that $\epsilon(\pi)=1$ (see [25, §7.2]). We
define $A_{+}$ to be the set of monic elements of $A$, i.e., the set of $a\in
A$ such that $\operatorname{sgn}(a)=1$. For all $i\in\mathbb{Z}^{\geq 0}$, we
also set $A_{+,i}=\\{a\in A_{+}:\deg(a)=i\\}$, and

$\displaystyle A(\leq i)=\\{a\in A:\deg(a)\leq i\\},\quad A(<i)=\\{a\in
A:\deg(a)<i\\}.$

Further, for any field extension $L$ of $\mathbb{F}_{q}$, we have a uniquely
determined sign function

$\widetilde{\operatorname{sgn}}:(L\otimes_{\mathbb{F}_{q}}K_{\infty})^{\times}\to
L^{\times}$

extending $\operatorname{sgn}$. We also set

$X_{L}:=X\times_{\mathbb{F}_{q}}L.$

###  1.2. Multiple zeta values of Thakur, and in Tate algebras

We follow Thakur in [39, §3] and Lara Rodríguez-Thakur in [28, §2]. To the
datum $(K,\infty,\operatorname{sgn})$ we associate certain multiple zeta
values

$\zeta_{A}\big{(}s_{1},\ldots,s_{r}\big{)}\in K_{\infty}$

in the following way. For $d\geq 0$ and $n\in\mathbb{N}$ we consider the power
sum of degree $d$ and order $n$ given by

$S_{d}(n)=\sum_{a\in A_{+,d}}\frac{1}{a^{n}}\in K.$

By convention we define empty sums to be 0 and empty products to be 1. Thus if
$A_{+,d}=\emptyset$, then $S_{d}(n)=0$. Note that some such sums can vanish.
Along with this, we also set

$S_{<d}(n):=\sum_{i=0}^{d-1}S_{i}(n)\in K$

and iteratively for $r>1$ and $s_{1},\ldots,s_{r}\in\mathbb{N}$,

$\displaystyle S_{d}(s_{1},\ldots,s_{r})$
$\displaystyle:=S_{d}(s_{1})S_{<d}(s_{2},\ldots,s_{r}),$ $\displaystyle
S_{<d}(s_{1},\ldots,s_{r})$
$\displaystyle:=\sum_{i=0}^{d-1}S_{i}(s_{1},\ldots,s_{r})\in K.$

It is easily seen that $S_{d}(n)$ tends to $0$ in $K_{\infty}$, as $d$ tends
to $\infty$, the sequence $S_{<d}(s_{1},\ldots,s_{r})$ converges in
$K_{\infty}$ for any choice of $s_{1},\ldots,s_{r}\in\mathbb{N}$. We write

(1.1)
$\zeta_{A}\big{(}s_{1},\ldots,s_{r}\big{)}:=\lim_{d\rightarrow\infty}S_{<d}(s_{1},\ldots,s_{r})\in
K_{\infty}.$

For notational convenience, we also set

$\zeta_{A}(\emptyset):=1\in K_{\infty}.$

For an $r$-tuple $\mathfrak{s}=(s_{1},\dots,s_{r})\in\mathbb{N}^{r}$ we shall
write $\zeta_{A}(\mathfrak{s})$ instead of using the long expression (1.1).
The weight of $\mathfrak{s}$ is $w(\mathfrak{s})=s_{1}+\dots+s_{r}$ and its
depth is $r$. We shall also say that the presentation
$\zeta_{A}(\mathfrak{s})$ is of depth $r$ and weight $w(\mathfrak{s})$.

These elements of $K_{\infty}$ are the $\infty$-adic multiple zeta values of
the papers [28, 39, 40]. At this level of generality, not all the basic
properties of the multiple zeta values of §The genus 0 case are known to be
true. For instance, it is unclear if all the values defined in (1.1) are non-
zero. Moreover, while it is fairly well known that the
$\mathbb{F}_{p}$-subvector space they span in $K_{\infty}$ also is an
$\mathbb{F}_{p}$-algebra (with the product rules being identical to those in
the genus zero case), it seems that we miss a proof that the algebra they
generate is graded by the weights.

####  1.2.1. A family of rigid analytic functions

We recall that $\mathbb{C}_{\infty}$ is the completion of an algebraic closure
$\overline{K}_{\infty}$ of $K_{\infty}$. We present here the basic multiple
zeta values in Tate algebras that we study in the paper. We consider the
$\mathbb{F}_{q}$-algebra homomorphism

$\chi:A\rightarrow A\otimes_{\mathbb{F}_{q}}\mathbb{C}_{\infty}$

determined by $a\mapsto a\otimes 1$. There is a natural bijection
$x\mapsto\chi_{x}$ between closed points $x\in
X(\mathbb{C}_{\infty})\setminus\\{\infty\\}$ (where for $F$ a field extension
of $\mathbb{F}_{q}$, $X(F)$ denotes the set of $F$-rational points of $X$) and
$\mathbb{F}_{q}$-algebra homomorphisms
$A\xrightarrow{\chi_{x}}\mathbb{C}_{\infty}$. We denote by $\Xi$ the unique
point of $X(\mathbb{C}_{\infty})\setminus\\{\infty\\}$ such that $\chi_{\Xi}$
is the natural inclusion $A\subset\mathbb{C}_{\infty}$. For any given $a\in A$
the assignment $x\mapsto\chi_{x}(a)\in\mathbb{C}_{\infty}$ defines a rational
map

$X(\mathbb{C}_{\infty})\setminus\\{\infty\\}\xrightarrow{\chi(a)}\mathbb{C}_{\infty}$

which is regular everywhere away from $\infty$, with a pole of order $\deg(a)$
at $\infty$. If $x\in X(\mathbb{C}_{\infty})\setminus\\{\infty\\}$ we denote
by $\chi_{x}(a)$, or $\chi(a)_{x}$, or $\chi(a)|_{x}$, the image of $x$ by
$\chi(a)$ and we call it the evaluation of $\chi(a)$ at $x$.

If $F$ is a subfield of $\mathbb{C}_{\infty}$, we consider the Tate algebra

$\mathbb{T}(F):=\widehat{A\otimes_{\mathbb{F}_{q}}F}_{\|\cdot\|}$

with the completion $\widehat{(\cdot)}$ relative to $\|\cdot\|$ the
multiplicative Gauss valuation extending a fixed multiplicative valuation
$|\cdot|$ over $1\otimes F$, trivial over $A\otimes 1$. Let
$\mathcal{B}=(\beta_{i})_{i\in\mathcal{I}}$ be a basis of the
$\mathbb{F}_{q}$-vector space $A$. Note that $\mathcal{I}$ is countable set.
Every element $f$ of $\mathbb{T}(F)$ can be expanded in a unique way as a
series

$f=\sum_{i\in\mathcal{I}}\beta_{i}f_{i},\quad f_{i}\in\widehat{F},$

with $\widehat{F}$ the completion of $F$, where $f_{i}\rightarrow 0$ for the
Fréchet filter over $\mathcal{I}$. Then $\|f\|=\sup_{i\in\mathcal{I}}|f_{i}|$.
An entire function $f$ over $X(\mathbb{C}_{\infty})\setminus\\{\infty\\}$ is
by definition one such formal series in $\mathbb{T}(\mathbb{C}_{\infty})$ such
that additionally, $|f_{i}|\rho^{\deg(\beta_{i})}\rightarrow 0$ for all
$\rho>1$. Equivalently, the sequence $(f_{i}\chi_{x}(\beta_{i}))_{i}$ tends to
zero for all $x\in X(\mathbb{C}_{\infty})\setminus\\{\infty\\}$. We denote by
$f(x)$ or $f|_{x}$ the sum of the convergent series
$\sum_{i}f_{i}\chi_{x}(\beta_{i})\in\mathbb{C}_{\infty}$. This is the
evaluation of the entire function $f$ at $x\in
X(\mathbb{C}_{\infty})\setminus\\{\infty\\}$. It is not difficult to see that
this evaluation does not depend on the choice of the basis $\mathcal{B}$, and
that the evaluations taken with respect to different bases agree. Also, let
$X^{an}_{\mathbb{C}_{\infty}}$ be the analytification of $X$ over
$\mathbb{C}_{\infty}$. It is easy to verify that if
$f=\sum_{i\in\mathcal{I}}\beta_{i}f_{i}$ is an entire function over
$X(\mathbb{C}_{\infty})\setminus\\{\infty\\}$, then the everywhere convergent
series defines a rigid analytic function
$X^{an}_{\mathbb{C}_{\infty}}\setminus\\{\infty\\}\rightarrow\mathbb{C}_{\infty}$.
In the case $X=\mathbb{P}^{1}$ with $\infty$ an $\mathbb{F}_{q}$-rational
point, this coincides with the usual notion of an entire function
$f:\mathbb{C}_{\infty}\rightarrow\mathbb{C}_{\infty}$ (see [25, Chapter 3]).
We denote by $\mathbb{E}$ the $\mathbb{C}_{\infty}$-algebra of entire
functions
$X(\mathbb{C}_{\infty})\setminus\\{\infty\\}\rightarrow\mathbb{C}_{\infty}$.

The unique continuous and open $A\otimes 1$-linear automorphism $\tau$ of
$\mathbb{T}(\mathbb{C}_{\infty})$ extending $c\mapsto c^{q}$ over
$1\otimes\mathbb{C}_{\infty}$ induces an automorphism of $\mathbb{E}$.
Explicitly, if

$f=\sum_{i\in\mathcal{I}}\beta_{i}f_{i}$

is the series expansion of the entire function $f$, relative to the basis
$\mathcal{B}$, then

$f^{(k)}:=\tau^{k}(f)=\sum_{i\in\mathcal{I}}\beta_{i}f_{i}^{q^{k}},\quad
k\in\mathbb{Z},$

is another entire function over $X(\mathbb{C}_{\infty})\setminus\\{\infty\\}$,
well defined independently of the choice of the basis $\mathcal{B}$. If
$x\mapsto x^{(1)}$ denotes the Frobenius map
$X(\mathbb{C}_{\infty})\rightarrow X(\mathbb{C}_{\infty})$ then
$f^{(1)}(x^{(1)})=f(x)^{q}$ for all $x\in
X(\mathbb{C}_{\infty})\setminus\\{\infty\\}$.

We consider the rational functions
$X(\mathbb{C}_{\infty})\setminus\\{\infty\\}\rightarrow\mathbb{C}_{\infty}$

(1.2) $\mathcal{S}_{d}:=\sum_{a\in A_{+,d}}\frac{\chi(a)}{a}\in
A\otimes_{\mathbb{F}_{q}}K.$

Note that they are also entire in the above terminology.

In the present paper, we study the functions $\tau^{n}(Z_{k})$ where, for all
$k\geq 0$, we define $Z_{k}$ by the formal series

$Z_{k}:=\sum_{d\geq
0}\mathcal{S}_{d}S_{<d}\Big{(}q-1,(q-1)q,\ldots,(q-1)q^{k-1}\Big{)}$

which converges in $\mathbb{T}(K)$. They are all entire functions, as shown by
the following lemma:

######  Lemma 1.1.

For all $k\geq 0$, the series $Z_{k}\in\mathbb{T}(K)$ defines an entire
function
$X(\mathbb{C}_{\infty})\setminus\\{\infty\\}\rightarrow\mathbb{C}_{\infty}$.

###### Proof.

It suffices to show that for all $x\in
X(\mathbb{C}_{\infty})\setminus\\{\infty\\}$, the sequence of evaluations

$\big{(}\mathcal{S}_{i}|_{x}\big{)}_{i\geq 0}$

of $\mathcal{S}_{i}$ at $x$ tends to zero uniformly for the standard
admissible covering of $X^{an}_{\mathbb{C}_{\infty}}\setminus\\{\infty\\}$.

The proof we give is a simple adaptation of the proof of [8, Lemma 7]. We set,
for $r\geq 0$, $y_{r}=q^{r+1}-1$ which $p$-adically tends to
$-1\in\mathbb{Z}_{p}$ as $r\to\infty$. Note that for all $r\geq 0$, assuming
that $A_{+,i}\neq\emptyset$,

$\sum_{a\in A_{+,i}}\chi(a)a^{y_{r}}=\sum_{b\in
A(<i)}L(\eta_{i}+b)L_{0}(\eta_{i}+b)^{q-1}\cdots L_{r}(\eta_{i}+b)^{q-1}\in
A\otimes_{\mathbb{F}_{q}}A,$

where $\eta_{i}$ is any element of $A_{+,i}$, $L$ is the
$\mathbb{F}_{q}$-linear form $a\mapsto a\otimes 1$ and $L_{j}$ is the
$\mathbb{F}_{q}$-linear form $a\mapsto 1\otimes a^{q^{j}}$ for $j=0,\ldots,r$.
By [25, Lemma 8.8.1], $\sum_{a\in A_{+,i}}\chi(a)a^{y_{r}}$ vanishes if
$1+(r+1)(q-1)<\dim_{\mathbb{F}_{q}}(A(<i))$. We can therefore choose

$r=\dim_{\mathbb{F}_{q}}(A(<i))-\begin{cases}2\quad\text{if }q>2,\\\
3\quad\text{otherwise.}\end{cases}$

assuming at once $i$ large enough so that $r$ is non-negative. In particular
we have, for this choice of $r$, the identity $\sum_{a\in
A_{+,i}}\chi(a)a^{y_{r}}=0$.

We consider $a\in A_{+,i}$. Since $\epsilon(a)=1$, there exists a unique
element $\langle a\rangle\in
K_{\infty}^{\times}\cap\operatorname{Ker}(\epsilon)$, depending on $\pi$, such
that $a=\langle a\rangle\pi^{-i}$. Note that $|\langle
a\rangle-1|\leq|\pi|<1$. We have

$\langle a\rangle^{-1}-\langle a\rangle^{y_{r}}=\sum_{j\geq
0}\Big{(}\binom{-1}{j}-\binom{y_{r}}{j}\Big{)}(\langle
a\rangle-1)^{j}\in\mathbb{F}_{q}[[\pi]].$

Since $\binom{-1}{j}-\binom{y_{r}}{j}=0$ for $j=0,\ldots,q^{r+1}-1$, the above
difference has multiplicative valuation strictly smaller than
$|\pi|^{q^{r+1}}$. This implies that $\mathcal{S}_{i}\in
A\otimes_{\mathbb{F}_{q}}K_{\infty}$ has the property that there is a finite
expansion $\mathcal{S}_{i}=\sum_{a\in A_{+,i}}a\otimes\alpha_{a}$ with, for
$a\in A_{+,i}$, $\alpha_{a}\in K_{\infty}$ such that
$|\alpha_{a}|<|\pi|^{i+q^{r+1}}$. From this we deduce that the sequence of
entire functions $\mathcal{S}_{i}$ tends to zero uniformly for the standard
admissible covering of $X^{an}_{\mathbb{C}_{\infty}}\setminus\\{\infty\\}$.
The covering is given by the filtered union

$\bigcup_{\begin{smallmatrix}\rho\in|\mathbb{C}_{\infty}^{\times}|\\\
\rho>1\end{smallmatrix}}\operatorname{Spm}\Big{(}\widehat{A\otimes\mathbb{C}_{\infty}}_{\|\cdot\|_{\rho}}\Big{)},$

where $\operatorname{Spm}$ denotes the maximal spectrum, and where we take the
completions with respect to $\|\cdot\|_{\rho}$ the multiplicative valuation
which restricts to $|\cdot|$ over $1\otimes\mathbb{C}_{\infty}$, such that
$\|a\otimes 1\|_{\rho}=\rho^{\deg(a)}$. By the above discussion, for all
$x\in\operatorname{Spm}\big{(}\widehat{A\otimes\mathbb{C}_{\infty}}_{\|\cdot\|_{\rho}}\big{)}$
we have, by the maximum principle, and for large values of $i$:

$|\mathcal{S}_{i}(x)|\leq\|\mathcal{S}_{i}\|_{\rho}\leq\rho^{i}|\pi|^{i+q^{r+1}}$

which tends to zero for any fixed $\rho>1$ as $i\rightarrow\infty$.∎

######  Remark 1.2.

The entire function $Z_{k}$ can be viewed as an example of multiple zeta value
in the Tate algebra $\mathbb{T}(K)$. These functions have been investigated in
[24] in the case of $X=\mathbb{P}^{1}$ and $\infty$ an
$\mathbb{F}_{q}$-rational point. We will not pursue the general theory of such
functions because in the present paper we only need to work with the entire
functions

$\tau^{n}(Z_{k}),\quad n,k\geq 0.$

However, we mention here that, in agreement with the notations of ibid., we
could have written:

$Z_{k}=\zeta_{A}\begin{pmatrix}\\{1\\}&\emptyset&\cdots&\emptyset\\\
1&q-1&\cdots&(q-1)q^{k-1}\end{pmatrix}\in\mathbb{T}(K),$

to stress the analogy. Note that, following this notation, we have
$Z_{0}=\zeta_{A}\binom{\\{1\\}}{1}$. We also set $Z_{-1}:=0$. Also, Lemma 1.1
can be easily generalized to match with the formalism of [24] and the reader
can see that all the functions that naturally arise in this picture define
entire functions over products of the curve
$X(\mathbb{C}_{\infty})\setminus\\{\infty\\}$ in an appropriate sense.

##  2\. A universal formula

###  2.1. Auxiliary results

We recall several results from [34, §0 and §1]. For all $i\in\mathbb{Z}^{\geq
0}$ and $n\in\mathbb{N}$, we set, with appropriate conventions for empty sums
and products,

$\displaystyle D_{i}$ $\displaystyle:=\prod_{a\in A_{+,i}}a,$ $\displaystyle
e_{i}(z)$ $\displaystyle:=\prod_{a\in A(<i)}(z-a),$
$\displaystyle\frac{e_{i}(z)}{D_{i}}$ $\displaystyle:=\sum_{k\geq
0}A_{ik}z^{q^{k}}$ $\displaystyle S_{i}(n)$ $\displaystyle:=\sum_{a\in
A_{+,i}}\frac{1}{a^{n}},$ $\displaystyle S_{i}$
$\displaystyle:=S_{i}(1)=\sum_{a\in A_{+,i}}\frac{1}{a}.$

If $A_{+,i}\neq\emptyset$, then by [34, Eq. (13)],

$e_{i+1}(z)=e_{i}(z)^{q}-D_{i}^{q-1}e_{i}(z).$

Otherwise, if $A_{+,i}=\emptyset$, we get $e_{i+1}(z)=e_{i}(z)$. In
particular, for all $i\in\mathbb{Z}^{\geq 0}$, $D_{i+1}A_{(i+1)0}$ and
$D_{i}^{q}A_{i0}$ agree up to a sign so that we can write
$D_{i+1}A_{(i+1)0}=\pm D_{i}^{q}A_{i0}$ and

$A_{i0}=\pm\frac{(D_{0}D_{1}\dots D_{i-1})^{q-1}}{D_{i}}\neq 0.$

Now, if $A_{+,i}\neq\emptyset$, by [34, Eq. (18)], we have

$\frac{A_{i0}}{1-\sum_{k=0}^{i}A_{ik}z^{q^{k}}}=-\sum_{a\in
A_{+,i}}\frac{1}{z-a}=\sum_{n\geq 0}S_{i}(n+1)z^{n}.$

Consequently,

(2.1) $S_{i}=S_{i}(1)=A_{i0}\neq 0,$

and for all $1\leq n\leq q-1$,

(2.2) $S_{i}(n)=S_{i}^{n}.$

For all $d\in\mathbb{Z}^{\geq 0}$, we recall that we have set (see (1.2))

$\displaystyle\mathcal{S}_{d}$ $\displaystyle=\sum_{a\in
A_{+,d}}\frac{\chi(a)}{a}.$

We additionally set

$\displaystyle\mathcal{S}_{<d}:=\sum_{i=0}^{d-1}\mathcal{S}_{i},\quad
S_{<d}:=\sum_{i=0}^{d-1}S_{i}.$

We put $\mathcal{S}_{\leq d}=\mathcal{S}_{<d+1}$ and $S_{\leq d}=S_{<d+1}$.

If $D$ is a divisor over $X_{\overline{K}}$, then we denote by
$\mathcal{L}(D)$ the $\overline{K}$-subvector space of the fraction field of
$X_{\overline{K}}$ consisting of functions $f$ such that $(f)+D$ is effective:
$(f)\geq-D$. For all $m\in\mathbb{Z}^{\geq 0}$, we define $j_{m}$ to be the
smallest non-negative integer such that
$\dim_{\overline{K}}\mathcal{L}(j_{m}\infty)=m+1$. Note that
$\mathcal{L}(j_{m}(\infty))=A(\leq
j_{m})\otimes_{\mathbb{F}_{q}}\overline{K}$, where $A(\leq\ell)$ is the
$\mathbb{F}_{q}$-vector space spanned by the elements $a\in A$ such that
$\deg(a)\leq\ell$. We have $j_{0}=0<j_{1}<j_{2}<...$ and for $m$ sufficiently
large, $j_{m}=m+g$ by the Riemann-Roch theorem.

######  Lemma 2.1.

For all $m\geq 1$, the function $\mathcal{S}_{j_{m}}$ vanishes at
$\Xi,\dots,\Xi^{(m-1)}$.

###### Proof.

Let us choose an element $\eta_{m}\in A_{+,j_{m}}$. Then
$A_{+,j_{m}}=\eta_{m}+A(\leq j_{m-1})$. Note that by the definition of the
sequence $\\{j_{m}\\}_{m\geq 0}$, $\dim_{\mathbb{F}_{q}}A(\leq j_{m-1})=m$.
Since

$\mathcal{S}_{j_{m}}(\Xi^{(j)})=\sum_{a\in A_{+,j_{m}}}a^{q^{j}-1}=\sum_{b\in
A(\leq j_{m-1})}L_{0}(\eta_{m}+b)^{q-1}\cdots L_{j-1}(\eta_{m}+b)^{q-1}$

where $L_{i}$ is the $\mathbb{F}_{q}$-linear form $a\mapsto a^{q^{i}}$ for
$i=0,\ldots,j-1$, $\mathcal{S}_{j_{m}}(\Xi^{(j)})$ vanishes for
$j=0,\ldots,m-1$ in virtue of [25, Lemma 8.8.1]. ∎

######  Lemma 2.2.

The set $\\{\mathcal{S}_{j_{0}},\dots,\mathcal{S}_{j_{m}}\\}$ forms a basis of
$\mathcal{L}(j_{m}\infty)$.

###### Proof.

We observe that for all $k\in\mathbb{Z}^{\geq 0}$, $\mathcal{S}_{j_{k}}$ has a
pole of order $j_{k}$ at $\infty$. The integers $j_{0},\dots,j_{m}$ being
distinct, $\mathcal{S}_{j_{0}},\dots,\mathcal{S}_{j_{m}}$ are linearly
independent over $\overline{K}$. Since
$\dim_{\overline{K}}\mathcal{L}(j_{m}\infty)=m+1$, the Lemma follows. ∎

######  Lemma 2.3.

The function $\mathcal{S}_{j_{m}}$ does not vanish at $\Xi^{(m)}$.

###### Proof.

We suppose by contradiction that $\mathcal{S}_{j_{m}}$ vanishes at
$\Xi^{(m)}$. Combining with Lemma 2.1 implies that for all $k\geq m$,
$\mathcal{S}_{j_{m}},\dots,\mathcal{S}_{j_{k}}$ belong to
$\mathcal{L}(j_{k}\infty-\Xi-\dots-\Xi^{(m)})$. We choose $k$ sufficiently
large so that by the Riemann-Roch theorem,

  * •

$\dim_{\overline{K}}\mathcal{L}(j_{k}\infty)=j_{k}+1-g$,

  * •

$\dim_{\overline{K}}\mathcal{L}(j_{k}\infty-\Xi-\dots-\Xi^{(m)})=j_{k}-(m+1)+1-g$.

The first equation implies $j_{k}+1-g=k+1$. Putting this into the second
equation yields

$\dim_{\overline{K}}\mathcal{L}(j_{k}\infty-\Xi-\dots-\Xi^{(m)})=j_{k}-(m+1)+1-g=k-m.$

By Lemma 2.2, we know that $\mathcal{S}_{j_{m}},\dots,\mathcal{S}_{j_{k}}$ are
linearly independent over $\overline{K}$. Thus we get a contradiction since

$\displaystyle k-m$
$\displaystyle=\dim_{\overline{K}}\mathcal{L}(j_{k}\infty-\Xi-\dots-\Xi^{(m)})$
$\displaystyle\geq\dim_{\overline{K}}\text{Vect}(\mathcal{S}_{j_{m}},\dots,\mathcal{S}_{j_{k}})$
$\displaystyle=k-m+1.$

We conclude that $\mathcal{S}_{j_{m}}$ does not vanish at $\Xi^{(m)}$. ∎

###  2.2. A key identity

We now prove the main result of this section.

######  Theorem 2.4.

For all $i\in\mathbb{Z}^{\geq 0}$, we have

$\mathcal{S}_{i}^{(1)}=\mathcal{S}_{\leq i}\cdot S_{i}(q-1).$

###### Proof.

Note that if $i\neq j_{m}$ for all $m$, then $A_{+,i}=\emptyset$. Thus
$\mathcal{S}_{i}=S_{i}(q-1)=0$ and we are done. Therefore, it suffices to
prove that for all $m\in\mathbb{Z}^{\geq 0}$,

(2.3) $\mathcal{S}_{j_{m}}^{(1)}=\mathcal{S}_{\leq j_{m}}\cdot
S_{j_{m}}(q-1).$

We observe

$\displaystyle\mathcal{S}_{j_{m}}^{(1)}\Big{|}_{\Xi}=S_{j_{m}}(q-1)=S_{j_{m}}^{q-1}.$

Here the second equality holds by (2.2). By (2.1), it follows that
$\mathcal{S}_{j_{m}}^{(1)}\Big{|}_{\Xi}\neq 0$. Thus (2.3) is equivalent to

(2.4)
$\frac{\mathcal{S}_{j_{m}}^{(1)}}{\mathcal{S}_{j_{m}}^{(1)}\Big{|}_{\Xi}}=\mathcal{S}_{\leq
j_{m}}.$

We now prove (2.4) by induction on $m$. For $m=0$, (2.4) holds since
$\mathcal{S}_{\leq j_{m}}=\mathcal{S}_{j_{m}}^{(1)}=S_{j_{m}}(q-1)=1$. Suppose
that for all $0\leq k\leq m-1$,

$\frac{\mathcal{S}_{j_{k}}^{(1)}}{\mathcal{S}_{j_{k}}^{(1)}\Big{|}_{\Xi}}=\mathcal{S}_{\leq
j_{k}}.$

We show that

$\frac{\mathcal{S}_{j_{m}}^{(1)}}{\mathcal{S}_{j_{m}}^{(1)}\Big{|}_{\Xi}}=\mathcal{S}_{\leq
j_{m}}.$

In fact,

$F:=\frac{\mathcal{S}_{j_{m}}^{(1)}}{\mathcal{S}_{j_{m}}^{(1)}\Big{|}_{\Xi}}\in\mathcal{L}(j_{m}\infty).$

By Lemma 2.2, the set $\mathcal{S}_{j_{0}},\dots,\mathcal{S}_{j_{m}}$ forms a
basis of the $\overline{K}$-vector space $\mathcal{L}(j_{m}\infty)$.
Therefore, there exist $a_{0},\dots,a_{m}\in\overline{K}$ such that

(2.5) $F=a_{0}\mathcal{S}_{j_{0}}+\dots+a_{m}\mathcal{S}_{j_{m}}.$

We show by induction on $0\leq k\leq m-1$ that $a_{k}=1$. For $k=0$, we
evaluate (2.5) at $\Xi$. The left-hand side gives $F|_{\Xi}=1$. By Lemma 2.1,
the right-hand side returns

$a_{0}\mathcal{S}_{j_{0}}\Big{|}_{\Xi}+\dots+a_{m}\mathcal{S}_{j_{m}}\Big{|}_{\Xi}=a_{0}.$

We then obtain $a_{0}=1$.

Suppose that for all $j$ such that $0\leq j<k$, we have $a_{j}=1$. We have to
prove that $a_{k}=1$. We evaluate (2.5) at $\Xi^{(k)}$. Note that $1\leq k\leq
m-1$. By Lemma 2.1,

$F|_{\Xi^{(k)}}=0.$

We get

$\displaystyle(a_{0}\mathcal{S}_{j_{0}}+\dots+a_{m}\mathcal{S}_{j_{m}})\Big{|}_{\Xi^{(k)}}$
$\displaystyle=a_{0}\mathcal{S}_{j_{0}}\Big{|}_{\Xi^{(k)}}+\dots+a_{m}\mathcal{S}_{j_{m}}\Big{|}_{\Xi^{(k)}}$
$\displaystyle=a_{0}\mathcal{S}_{j_{0}}\Big{|}_{\Xi^{(k)}}+\dots+a_{k}\mathcal{S}_{j_{k}}\Big{|}_{\Xi^{(k)}}$
$\displaystyle=\mathcal{S}_{j_{0}}\Big{|}_{\Xi^{(k)}}+\dots+\mathcal{S}_{j_{k}}\Big{|}_{\Xi^{(k)}}+(a_{k}-1)\mathcal{S}_{j_{k}}\Big{|}_{\Xi^{(k)}}$
$\displaystyle=(\mathcal{S}_{j_{0}}+\dots+\mathcal{S}_{j_{k}})\Big{|}_{\Xi^{(k)}}+(a_{k}-1)\mathcal{S}_{j_{k}}\Big{|}_{\Xi^{(k)}}$
$\displaystyle=\mathcal{S}_{\leq
j_{k}}\Big{|}_{\Xi^{(k)}}+(a_{k}-1)\mathcal{S}_{j_{k}}\Big{|}_{\Xi^{(k)}}.$

Here the second equality follows from Lemma 2.1 again. The third equality
holds as $a_{0}=\dots=a_{k-1}=1$.

Since $k\leq m-1$, by the induction hypothesis and Lemma 2.1,

$\mathcal{S}_{\leq
j_{k}}\Big{|}_{\Xi^{(k)}}=\frac{\mathcal{S}_{j_{k}}^{(1)}\Big{|}_{\Xi^{(k)}}}{{\mathcal{S}_{j_{k}}^{(1)}\Big{|}_{\Xi}}}=0.$

We then get

$\displaystyle(a_{0}\mathcal{S}_{j_{0}}+\dots+a_{m}\mathcal{S}_{j_{m}})\Big{|}_{\Xi^{(k)}}=(a_{k}-1)\mathcal{S}_{j_{k}}\Big{|}_{\Xi^{(k)}}.$

Putting all together yields

$0=F|_{\Xi^{(k)}}=a_{0}\mathcal{S}_{j_{0}}\Big{|}_{\Xi^{(k)}}+\dots+a_{m}\mathcal{S}_{j_{m}}\Big{|}_{\Xi^{(k)}}=(a_{k}-1)\mathcal{S}_{j_{k}}\Big{|}_{\Xi^{(k)}}.$

By Lemma 2.3, $\mathcal{S}_{j_{k}}\Big{|}_{\Xi^{(k)}}\neq 0$. We deduce that
$a_{k}=1$ as desired.

To finish the proof, we show that $a_{m}=1$. We look at the sign of both sides
of (2.5). The left-hand side gives

$\displaystyle\widetilde{\operatorname{sgn}}(F)=\frac{\widetilde{\operatorname{sgn}}(\mathcal{S}_{j_{m}}^{(1)})}{\mathcal{S}_{j_{m}}^{(1)}\Big{|}_{\Xi}}=\frac{S_{j_{m}}^{q}}{S_{j_{m}}(q-1)}=S_{j_{m}}.$

Here the last equality follows from Lemma 2.2. The right-hand side equals
$a_{m}S_{j_{m}}$. Thus $S_{j_{m}}=a_{m}S_{j_{m}}$. As $S_{j_{m}}\neq 0$, we
deduce $a_{m}=1$ and the proof is complete. ∎

As a direct consequence, since
$Z_{0}=\lim_{n\rightarrow\infty}\mathcal{S}_{\leq n}\in\mathbb{T}(K)$, we
deduce that the entire function $Z_{0}$ vanishes at the points
$\Xi^{(1)},\Xi^{(2)},\ldots$ and $Z_{0}\big{|}_{\Xi}=1$. The genus one case of
Theorem 2.4 can also be deduced from the explicit arguments of Green and
Papanikolas in [26].

##  3\. Universal families of Eulerian multiple zeta values

###  3.1. Sign-normalized rank one Drinfeld modules

If $R$ is an $\mathbb{F}_{q}$-algebra we denote by $R[\tau]$ the skew
polynomial ring in $\tau$ (denoted by $R\\{\tau\\}$ in [25, Chapter 1]), and
by $R[[\tau]]$ the ‘ring of Frobenius power series’ (denoted by
$R\\{\\{\tau\\}\\}$ in [25, §4.6].) Let
$\phi:A\rightarrow\overline{K}_{\infty}[\tau]$ be a rank one Drinfeld module
such that for all $a\in A\setminus\\{0\\}$,

$\phi_{a}=a+\cdots+\operatorname{sgn}(a)\tau^{\deg a}.$

Such a Drinfeld module $\phi$ is said to be sign-normalized. In [26] a sign-
normalized rank one Drinfeld module is called a Drinfeld-Hayes module. By [25,
Theorem 7.2.15], there always exist sign-normalized rank one Drinfeld modules.
Further, if we denote by $H$ the Hilbert class field of $A$ and $O_{H}$ the
integral closure of $A$ in $H$, then for all $a\in A$, $\phi_{a}\in
O_{H}[\tau]$.

There exist unique elements $\exp_{\phi},\log_{\phi}\in H[[\tau]]$ such that

$\exp_{\phi},\log_{\phi}\in 1+H[[\tau]]\tau,$ $\forall a\in
A,\quad\exp_{\phi}a=\phi_{a}\exp_{\phi},$
$\exp_{\phi}\log_{\phi}=\log_{\phi}\exp_{\phi}=1.$

We write

$\displaystyle\exp_{\phi}$ $\displaystyle=\sum_{i\geq
0}\epsilon_{i}\tau^{i},\quad\epsilon_{i}\in H,$ $\displaystyle\log_{\phi}$
$\displaystyle=\sum_{i\geq 0}\lambda_{i}\tau^{i},\quad\lambda_{i}\in H.$

The formal series $\exp_{\phi},$ and $\log_{\phi}$ are respectively called the
exponential series and the logarithm series associated to $\phi$.
Additionally, $\exp_{\phi}$ defines, in a unique way, an
$\mathbb{F}_{q}$-linear entire function
$\mathbb{C}_{\infty}\rightarrow\mathbb{C}_{\infty}$ (unlike the functions
$Z_{k}$ which are entire
$X(\mathbb{C}_{\infty})\setminus\\{\infty\\}\rightarrow\mathbb{C}_{\infty}$).
In what follows, we identify the operator $\exp_{\phi}$ with this entire
function $\mathbb{C}_{\infty}\rightarrow\mathbb{C}_{\infty}$ hence tolerating
an abuse of notation. The kernel of $\exp_{\phi}$ is a projective $A$-module
of rank one embedded discretely in $\mathbb{C}_{\infty}$. By [35, Theorem
5.8], the coefficients $\lambda_{i}$ are non-zero for all $i\geq 0$. Theorem
3.2 of [35] describes non-vanishing criteria for the coefficients
$\epsilon_{i}$’s at a superior level of generality, i.e. with $\infty$ a
closed point of $X$ not necessarily $\mathbb{F}_{q}$-rational. Note that the
condition $d_{\infty}=1$ ensures that they are all non-zero.

Drinfeld proved that there is a bijection between the set of sign-normalized
rank one Drinfeld module $\phi$ and a set of certain functions $f$ called
shtuka functions in the rational function field
$F:=\operatorname{Frac}(A\otimes_{\mathbb{F}_{q}}\overline{K})$ of
$X_{\overline{K}}$ (see [22, 23, 35]). It follows that one can associate a
shtuka function $f_{\phi}\in F^{\times}$ to any sign-normalized Drinfeld
module of rank one $\phi$. The divisor $(f_{\phi})$ associated to $f_{\phi}$
can be expressed in the following way:

$(f_{\phi})=V_{\phi}^{(1)}-V_{\phi}+\Xi-\infty,$

for some effective divisor $V_{\phi}$ of $X(\overline{K})$ of degree $g$. We
call it the Drinfeld divisor attached to $\phi$. Note that the points $\Xi$
and $\infty$ do not belong to the support of $V_{\phi}$ (see [35, Corollary
0.3.3]).

There exists a unique sign-normalized Drinfeld module of rank one $\phi$,
called the standard sign-normalized Drinfeld module, such that the kernel of
$\exp_{\phi}$ is a free $A$-module of rank one. From now on, we only work with
this standard Drinfeld module $\phi$. Then,

$\operatorname{Ker}(\exp_{\phi})=A\widetilde{\pi}_{\phi}$

for some element $\widetilde{\pi}_{\phi}\in\mathbb{C}_{\infty}^{\times}$,
uniquely defined up to multiplication by a factor in
$\mathbb{F}_{q}^{\times}$. We also denote by $f$ and $V$, respectively, the
associated shtuka function and the associated Drinfeld divisor.

We now come back to the multiple zeta values of Thakur, in $K_{\infty}$. We
choose an $r$-tuple $\mathfrak{s}=(s_{1},\ldots,s_{r})\in\mathbb{N}^{r}$, with
$r>0$. We recall from the introduction the following definition (see the first
paragraph concerning the higher genus cases).

######  Definition 3.1.

The multiple zeta value $\zeta_{A}(\mathfrak{s})$ is Eulerian if
$\zeta_{A}(\mathfrak{s})/\widetilde{\pi}_{\phi}^{w(\mathfrak{s})}\in\mathbb{C}_{\infty}$
is algebraic over $K$.

######  Remark 3.2.

We consider the genus $0$ case, i.e., $X=\mathbb{P}^{1}$ with $\infty$ an
$\mathbb{F}_{q}$-rational point. Then as already noticed,
$A=\mathbb{F}_{q}[\theta]$ and $K=\mathbb{F}_{q}(\theta)$ for some rational
function $\theta$ over $\mathbb{P}^{1}$. In this case we have $\phi=C$ the
Carlitz module. It is uniquely defined by the identity
$C_{\theta}=\theta+\tau$ in $K[\tau]$. We write
$\widetilde{\pi}=\widetilde{\pi}_{C}$ and we have already mentioned that if
$q-1\mid n$ with $n\geq 1$, $\zeta_{A}(n)$ is Eulerian, see [25, 36] for more
details. By using a positive characteristic variant of Baker’s theory and a
seminal result of Anderson and Thakur [4], Yu proved the following result for
$n>0$: $\zeta_{A}(n)/\widetilde{\pi}^{n}\in\overline{K}$ if and only if
$\zeta_{A}(n)/\widetilde{\pi}^{n}\in K$, and of course the latter property
occurs if $q-1\mid n$ (see [41, Corollary 2.6]). This result was generalized
for multiple zeta values associated to $\mathbb{F}_{q}[\theta]$, of arbitrary
depth, by Chang [17]. Chang and Yu explicitly computed in [19, 42] the
transcendence degree of the subfield of $\mathbb{C}_{\infty}$ generated by $K$
and the zeta values $\zeta_{A}(n)$, $n\geq 1$. The methods of [17, 19] do not
use Baker’s theory but rest on the transcendence methods of Anderson,
Brownawell and Papanikolas [3], and Papanikolas [29].

Recall that $\exp_{\phi}=\sum_{i\geq 0}\epsilon_{i}\tau^{i}$ with
$\epsilon_{i}\in H$. Let us consider

$\exp_{A}(z):=z\prod_{a\in A\setminus\\{0\\}}\left(1-\frac{z}{a}\right),\quad
z\in\mathbb{C}_{\infty}.$

This product converges for all $z\in\mathbb{C}_{\infty}$ and defines an entire
$\mathbb{F}_{q}$-linear function
$\mathbb{C}_{\infty}\rightarrow\mathbb{C}_{\infty}$ with kernel
$A\subset\mathbb{C}_{\infty}$ that we identify with an operator

$\exp_{A}=\sum_{k\geq 0}e_{i}\tau^{i}\in K_{\infty}[[\tau]].$

The link between the operators $\exp_{A}$ and $\exp_{\phi}$ is given by the
following identity in $\mathbb{C}_{\infty}[[\tau]]$:

(3.1) $\exp_{A}=\widetilde{\pi}_{\phi}^{-1}\exp_{\phi}\widetilde{\pi}_{\phi}.$

We also have the identity of meromorphic functions over $\mathbb{C}_{\infty}$
and of formal series of $K_{\infty}[[z]]$:

$\frac{z}{\exp_{A}(z)}=1+\sum_{\begin{smallmatrix}n\geq 1\\\ (q-1)\mid
n\end{smallmatrix}}\zeta_{A}(n)z^{n}\in K_{\infty}[[z]].$

From it we get

(3.2) $\zeta_{A}(q^{k}-1)=\lambda_{k}\widetilde{\pi}_{\phi}^{q^{k}-1},\quad
k\geq 0.$

Note that this implies $\lambda_{k}\neq 0$ for all $k\geq 0$, which is Theorem
5.8 of [35] (and essentially the same proof). Further, we deduce that for all
$n\geq 1$ with $n\equiv 0\pmod{q-1}$, $\zeta_{A}(n)$ is Eulerian (see [25,
Lemma 8.18.1]).

###  3.2. Carlitz-Hayes polynomials

We know that for all $n\geq 1$ there exists a unique polynomial called the
$n$-th Carlitz-Hayes polynomial

$\widetilde{\mathcal{E}}_{n}\in K[\tau]$

such that, for all $z\in\mathbb{C}_{\infty}$,

$\widetilde{\mathcal{E}}_{n}(z)=z\prod_{a\in
A(<n)\setminus\\{0\\}}\left(1-\frac{z}{a}\right).$

Here $\widetilde{\mathcal{E}}_{n}(z)$ denotes the evaluation of
$\widetilde{\mathcal{E}}_{n}$ at $z$. Let $\kappa_{n,m}$ be the coefficient of
$\tau^{m}$ in $\mathcal{E}_{n}$ seen as a linear combination of
$1,\tau,\tau^{2},\ldots$

######  Proposition 3.3.

With the above notation, we have

$\kappa_{n,m}=(-1)^{m}S_{<n}(q-1,q(q-1),\dots,q^{m-1}(q-1))$

###### Proof.

Since the zeros of $\widetilde{\mathcal{E}}_{n}$ are exactly the elements of
the $\mathbb{F}_{q}$-vector space $A(<n)$ with multiplicity $1$, we can write

$\widetilde{\mathcal{E}}_{n}=(1-\alpha_{n-1}\tau)\widetilde{\mathcal{E}}_{n-1}$

for some $\alpha_{n-1}\in K$. Thus we obtain a non-commutative factorization

$\widetilde{\mathcal{E}}_{n}=(1-\alpha_{n-1}\tau)\cdots(1-\alpha_{1}\tau)(1-\alpha_{0}\tau)$

with $\alpha_{0}=1$. It follows that

(3.3) $\kappa_{n,1}=-(\alpha_{0}+\alpha_{1}+\cdots+\alpha_{n-1}).$

Now, by considering the zeros, degree and the constant term, we see that, for
any $a\in\mathbb{C}_{\infty}^{\times}$,

$\prod_{c\in\mathbb{F}_{q}^{\times}}\left(1-\frac{z}{ca}\right)=1-\frac{z^{q-1}}{a^{q-1}}.$

Thus

$\widetilde{\mathcal{E}}_{n}(z)=z\prod_{\begin{subarray}{c}a\in
A(<n)\setminus\\{0\\},\\\ a\text{
monic}\end{subarray}}\left(1-\frac{z^{q-1}}{a^{q-1}}\right).$

We deduce

(3.4) $\kappa_{n,1}=-\sum_{\begin{subarray}{c}a\in A(<n)\setminus\\{0\\},\\\
a\text{ monic}\end{subarray}}\frac{1}{a^{q-1}}.$

Combining (3.3) and (3.4) yields, inductively,

$\alpha_{n-1}=\sum_{a\in A_{+,n-1}}\frac{1}{a^{q-1}}=S_{n-1}(q-1).$

We then get an explicit factorization

$\widetilde{\mathcal{E}}_{n}=\Big{(}1-S_{n-1}(q-1)\tau\Big{)}\dots\Big{(}1-S_{1}(q-1)\tau\Big{)}.$

The Proposition follows. ∎

Note that the multiple zeta value
$\zeta_{A}\left(q-1,\ldots,(q-1)q^{k-1}\right)$ has depth $k$. If $k=0$, we
have set
$\zeta_{A}\left(q-1,\ldots,(q-1)q^{k-1}\right)=\zeta_{A}(\emptyset)=1$.

######  Theorem 3.4.

We have the following identity

(3.5) $\exp_{A}=\sum_{k\geq
0}(-1)^{k}\zeta_{A}\left(q-1,\ldots,(q-1)q^{k-1}\right)\tau^{k}\in
K_{\infty}[[\tau]].$

In particular, for all $k\in\mathbb{N}$,

(3.6)
$\zeta_{A}(q-1,q(q-1),\dots,q^{k-1}(q-1))=(-1)^{k}\epsilon_{k}\widetilde{\pi}_{\phi}^{q^{k}-1}.$

###### Proof.

We observe that $\widetilde{\mathcal{E}}_{n}(z)$ converges uniformly on every
compact subset of $\mathbb{C}_{\infty}$ to the function $\exp_{A}\in
K_{\infty}[[\tau]]$. By the previous discussion, (3.5) follows immediately
and, viewing (3.1), it suffices to justify (3.6). ∎

As a direct consequence of Theorem 3.4 we deduce that the multiple zeta values
$\zeta_{A}(q-1,q(q-1),\dots,q^{k-1}(q-1))$ are all Eulerian in the sense of
Definition 3.1.

Note that the special case of $A=\mathbb{F}_{q}[\theta]$ is considered in [32,
Proposition 4.4.9]. Unlike (3.2), (3.6) does not seem to deliver non-vanishing
properties of the coefficients $\epsilon_{k}$. We can however consider the
reverse implication and deduce that the multiple zeta values
$\zeta_{A}(q-1,q(q-1),\dots,q^{k-1}(q-1))$ are non-zero by using [35, Theorem
3.2].

###  3.3. Harmonic products

We are ready to prove our main result, by variants of the harmonic products
discussed in [24].

######  Proposition 3.5.

For all $k\in\mathbb{N}$ we have the following identity of entire functions
over $\operatorname{Spec}(A)^{an}_{\mathbb{C}_{\infty}}$:

$\displaystyle
Z_{0}\cdot\zeta_{A}(q-1,\dots,q^{k-1}(q-1))=Z_{k}+Z_{k-1}^{(1)}.$

###### Proof.

By Lemma 2.4,

$\displaystyle Z_{0}\cdot\zeta_{A}(q-1,\dots,q^{k}(q-1))-Z_{k}$
$\displaystyle=\sum_{i\geq 0}\mathcal{S}_{\leq i}\cdot
S_{i}(q-1,q(q-1),\dots,q^{k}(q-1))$ $\displaystyle=\sum_{i\geq
0}\mathcal{S}_{\leq i}\cdot S_{i}(q-1)\cdot S_{<i}(q(q-1),\dots,q^{k}(q-1))$
$\displaystyle=\sum_{i\geq 0}\mathcal{S}_{i}^{(1)}\cdot
S_{<i}(q(q-1),\dots,q^{k}(q-1))$ $\displaystyle=Z_{k-1}^{(1)}.$

∎

As a consequence, we obtain

######  Theorem 3.6.

Let $n\geq 1$ be an integer. Then for all $k\geq 0$, there exists
$\alpha_{n,k}\in H$ such that

(3.7)
$\zeta_{A}\left(q^{n}-1,(q-1)q^{n},\ldots,(q-1)q^{k-1+n}\right)=\alpha_{n,k}\zeta_{A}\left(q^{n+k}-1\right).$

Further,

(3.8)
$\alpha_{n,k}=(-1)^{k}\lambda_{n+k}^{-1}\sum_{i=0}^{k}\lambda_{n+i}\epsilon_{k-i}^{q^{n+i}}.$

In particular, the multiple zeta value
$\zeta_{A}(q^{n}-1,q^{n}(q-1),\dots,q^{n+k-1}(q-1))$ is Eulerian.

Note that the depth of the multiple zeta value in the left-hand side of (3.7)
equals $k+1$. The coefficient $\alpha_{n,k}$ is well defined because
$\lambda_{n+k}$ never vanishes, as previously noticed. If $k=0$ the left-hand
side is just $\zeta_{A}(q^{n}-1)$ so that $\alpha_{n,k}=1$ in this case.

###### Proof of Theorem 3.6.

Let us write

$\mathcal{F}:=\sum_{k\geq 0}(-1)^{k}Z_{k}\tau^{k}\in\mathbb{T}(K)[[\tau]].$

The coefficient of $1=\tau^{0}$ is therefore $Z_{0}$. We also set

$\mathcal{G}:=\sum_{k\geq
0}(-1)^{k}\zeta_{A}\left(q-1,\ldots,(q-1)q^{k-1}\right)\tau^{k}\in
K_{\infty}[[\tau]].$

We recall that by (3.5),

$\mathcal{G}=\exp_{A}\in K_{\infty}[[\tau]].$

Hence Proposition 3.5 implies the identity in $\mathbb{T}(K)[[\tau]]$:

$\mathcal{F}-\tau\mathcal{F}=Z_{0}\exp_{A}.$

This allows to write

$\mathcal{F}=\sum_{i\geq 0}\tau^{i}(Z_{0})\tau^{i}\exp_{A},$

a series expansion which is convergent for the $\tau$-adic topology.
Explicitly, equating the coefficients of $\tau^{k}$, we find

(3.9)
$Z_{k}=(-1)^{k}\sum_{i=0}^{k}\epsilon_{k-i}^{q^{i}}\tau^{i}(Z_{0})\widetilde{\pi}_{\phi}^{q^{k}-q^{i}},\quad
k\geq 0.$

Applying $\tau^{n}$ we get

$\tau^{n}(Z_{k})=(-1)^{k}\sum_{i=0}^{k}\epsilon_{k-i}^{q^{i+n}}\tau^{i+n}(Z_{0})\widetilde{\pi}_{\phi}^{(q^{k}-q^{i})q^{n}}.$

Evaluating at $\Xi$ the above identity of entire functions and using (3.1) and
(3.2) together with the identities

$\tau^{i}(Z_{0})_{\Xi}=\zeta_{A}(q^{i}-1),\quad i\geq 0,$

we obtain

$\zeta_{A}\left(q^{n}-1,(q-1)q^{n},\ldots,(q-1)q^{k-1+n}\right)=\\\
=(-1)^{k}\lambda_{n+k}^{-1}\left(\sum_{i=0}^{k}\lambda_{n+i}\epsilon_{k-i}^{q^{n+i}}\right)\zeta_{A}(q^{k+n}-1),$

and (3.8) holds. This completes the proof of the Theorem. ∎

######  Remark 3.7.

Applying $\tau^{n}$ for $n>0$ to both sides of the identity of Proposition
3.5, evaluating at $\Xi$, and assuming that $X=\mathbb{P}^{1}$, yields
directly [20, Theorem 5.1]. If $k=1$, (3.9) yields

$Z_{k}=-\epsilon_{1}Z_{0}\widetilde{\pi}^{q-1}-Z_{0}^{(1)}.$

Since $\epsilon_{1}=-\lambda_{1}$ the above is equivalent, by (3.2), to the
harmonic product formula

(3.10) $Z_{0}\zeta_{A}(q-1)=Z_{0}^{(1)}+Z_{1}.$

A particular case of this formula, when $\phi=C$ is Carlitz’s module, so that
$A=\mathbb{F}_{q}[\theta]$, appears in [24, §7.2.1].

######  Remark 3.8.

Applying $\tau^{n}$ with $n>0$ to both sides of (3.10) and evaluating at $\Xi$
yields the formula

$\zeta_{A}(q^{n}-1)\zeta_{A}(q-1)^{q^{n}}=\zeta_{A}(q^{n+1}-1)+\zeta_{A}\left(q^{n}-1,(q-1)q^{n}\right)$

which is a generalization of [37, Theorem 4]. The case $n=0$ makes sense too,
but returns a tautological identity. Theorem 3.6 in the case $k=1$ rewrites as

(3.11)
$\zeta_{A}\left(q^{n}-1,(q-1)q^{n}\right)=-\left(\frac{\lambda_{n}\epsilon_{1}^{q^{n}}}{\lambda_{n+1}}+1\right)\zeta_{A}(q^{n+1}-1)$

and agrees with the above formula again in virtue of the fact that
$\epsilon_{1}=-\lambda_{1}$.

##  4\. Applications

In this section, we consider Drinfeld modules on elliptic curves and ramifying
hyperelliptic curves as studied in [21, 26]. We then derive a simple formula
for the ratio $\alpha_{n,1}$. As a consequence, we prove a conjecture of Lara
Rodríguez and Thakur (see Theorem 4.2).

###  4.1. Elliptic curves: the genus 1 case

We work with Drinfeld modules on elliptic curves (i.e., $g=1$ and
$d_{\infty}=1$) which were studied in detail in [26]. We let $E$ be the
elliptic curve defined over $\mathbb{F}_{q}$ given by

$E:\eta^{2}+a_{1}\theta\eta+a_{3}\eta=\theta^{3}+a_{2}\theta^{2}+a_{4}\theta+a_{6},\quad
a_{i}\in\mathbb{F}_{q}.$

We denote by $A=\mathbb{F}_{q}[\theta,\eta]$ the ring of functions on $E$ and
$K$ its fraction field. We recall the $\mathbb{F}_{q}$-algebra homomorphism
$\chi:A\to A\otimes\mathbb{C}_{\infty}$ given by $a\mapsto a\otimes 1$. We put
$y=\chi(\eta)$ and $t=\chi(\theta)$.

The shtuka function attached to the standard sign-normalized Drinfeld module
of rank one is given by

$f=\frac{\nu}{\delta}=\frac{y-\eta-m(t-\theta)}{t-\alpha}$

where $m=\frac{\eta-\beta^{q}}{\theta-\alpha^{q}}$ and $V=(\alpha,\beta)\in
E(H)$ is the Drinfeld divisor. Recall that for all $i\geq 0$,

$\epsilon_{i}=\frac{1}{ff^{(1)}\dots
f^{(i-1)}}\Big{|}_{\Xi^{(i)}},\quad\lambda_{i}=\frac{\delta^{(i+1)}}{\delta^{(1)}f^{(1)}\dots
f^{(i)}}\Big{|}_{\Xi}$

The first equality is proved by Thakur [35, Proposition 0.3.6] as mentioned
before. The second equality is shown by Anderson [35, Proposition 0.3.8],
Green and Papanikolas [26, Theorem 3.4 and Corollary 3.5].

By (3.11),

$\displaystyle\alpha_{n,1}$
$\displaystyle=-\left(\frac{\lambda_{n}\epsilon_{1}^{q^{n}}}{\lambda_{n+1}}+1\right)$
$\displaystyle=-\left(\frac{\delta^{(n+1)}f^{(n+1)}}{\delta^{(n+2)}}\Big{|}_{\Xi}\frac{1}{(f\big{|}_{\Xi^{(1)}})^{q^{n}}}+1\right)$
$\displaystyle=-\left(\frac{\nu^{(n+1)}}{\delta^{(n+2)}}\Big{|}_{\Xi}\frac{1}{(f\big{|}_{\Xi^{(1)}})^{q^{n}}}+1\right).$

###  4.2. Ramifying hyperelliptic curves

In [21], the first author extended the previous results to the case of
ramifying hyperelliptic curves. More precisely, letting $g\geq 1$ be an
integer, we consider a ramifying hyperelliptic curve $X$ defined over
$\mathbb{F}_{q}$ is given by

$X:\eta^{2}+F_{2}(\theta)\eta=F_{1}(\theta)$

with $F_{i}\in\mathbb{F}_{q}[\theta]$, $F_{1}$ monic of degree $2g+1$ and
$F_{2}$ of degree at most $g$. Thus the genus of $X$ equals $g$. We denote by
$A=\mathbb{F}_{q}[\theta,\eta]$ the ring of functions on $X$ and $K$ its
fraction field. We recall the $\mathbb{F}_{q}$-algebra homomorphism $\chi:A\to
A\otimes\mathbb{C}_{\infty}$ given by $a\mapsto a\otimes 1$. We put
$y=\chi(\eta)$ and $t=\chi(\theta)$.

By [21, Proposition 3.1], the shtuka function attached to the standard sign-
normalized Drinfeld module of rank one can be expressed as

$f=\frac{\nu}{\delta}$

where $\nu=y-\eta-Q$ and $\delta,Q\in H[t]$ of degree $g$ in $t$. Further, we
have
$\widetilde{\operatorname{sgn}}(\nu)=\widetilde{\operatorname{sgn}}(\delta)=1$.
Then by [35, Proposition 0.3.6] and [21, Proposition 3.3], we still get

$\epsilon_{i}=\frac{1}{ff^{(1)}\dots
f^{(i-1)}}\Big{|}_{\Xi^{(i)}},\quad\lambda_{i}=\frac{\delta^{(i+1)}}{\delta^{(1)}f^{(1)}\dots
f^{(i)}}\Big{|}_{\Xi}$

Thus we obtain a similar expression for $\alpha_{n,1}$:

$\displaystyle\alpha_{n,1}=-\left(\frac{\nu^{(n+1)}}{\delta^{(n+2)}}\Big{|}_{\Xi}\frac{1}{(f\big{|}_{\Xi^{(1)}})^{q^{n}}}+1\right).$

###  4.3. Class number one base rings

Following Thakur [35] (see also [28]), another important class of base rings
consists of class number one rings, i.e., we require that $A$ is a principal
ideal domain. There are exactly four of them listed as Examples (i)-(iv) in
[28]:

  1. (i)

$A=\mathbb{F}_{2}[\theta,\eta]/(\eta^{2}+\eta+\theta^{3}+\theta+1)$;

  2. (ii)

$A=\mathbb{F}_{3}[\theta,\eta]/(\eta^{2}-(\theta^{3}-\theta-1))$;

  3. (iii)

$A=\mathbb{F}_{4}[\theta,\eta]/(\eta^{2}+\eta+\theta^{3}+w)$, where
$w^{2}+w+1=0$, i.e., $w$ is a generator of $\mathbb{F}_{4}^{\times}$;

  4. (iv)

$A=\mathbb{F}_{2}[\theta,\eta]/(\eta^{2}+\eta+\theta^{5}+\theta^{3}+1)$.

They considered the following ratios (in [28], these ratios are denoted by
$R_{n}$)

$\alpha_{n,1}:=\frac{\zeta_{A}(q^{n}-1,q^{n+1}-q^{n})}{\zeta_{A}(q^{n+1}-1)},$

and conjectured that they are in $K$ in the above four cases. In each of the
four cases, we will set $X$ and $Y$ to be some $q^{n}$-th power of $\theta$
and $\eta$ respectively.

######  Conjecture 4.1 ([28], Conjecture 3.3).

The following formulas hold in Examples (i)-(iv) above:

  1. (i)

Set $X:=\theta^{2^{n}}$ and $Y:=\eta^{2^{n}}$. Then

(4.1)
$\begin{split}\alpha_{n,1}&=\frac{\theta^{2^{n+1}}+\theta^{2}}{\eta^{2^{n+1}}+\eta+\theta^{2^{n+1}+1}+\theta}\\\
&=\frac{X^{2}+\theta^{2}}{Y^{2}+\eta+\theta X^{2}+\theta}.\end{split}$

  2. (ii)

Set $X:=\theta^{3^{n}}$ and $Y:=\eta^{3^{n}}$. Then

(4.2)
$\begin{split}\alpha_{n,1}&=\frac{(\theta^{3^{n}}-\theta)(\eta^{3^{n}}-\eta)^{2}+(\theta^{3^{n}}-\theta)(-\theta^{3^{n}}-\theta^{3}-\theta+1)}{\theta^{2\cdot
3^{n+1}}+\theta^{3^{n+1}+1}+\theta^{3^{n+1}}+\eta^{3\cdot
3^{n}+1}+\theta^{2}-\theta+1}\\\
&=\frac{(X-\theta)(Y-\eta)^{2}-X^{2}+(-\theta^{3}+1)X+\theta^{2}-\theta}{\eta
Y^{3}+X^{6}+(\theta+1)X^{3}+\theta^{2}-\theta+1}.\end{split}$

  3. (iii)

Set $X:=\theta^{4^{n}}$ and $Y:=\eta^{4^{n}}$. Then

(4.3)
$\begin{split}\alpha_{n,1}&=\frac{(\theta^{4^{n}}+\theta)(\eta^{4^{n+1}}+\eta^{4})+(\theta^{4^{n+1}}+\theta^{4})(\theta^{4^{n}+2}+\theta^{3}+1)+(\theta^{4^{n}}+\theta)}{\theta^{2\cdot
4^{n+1}+2}+(\theta^{4^{n+1}}+\theta)(\eta^{4^{n+1}}+\eta)+\theta^{4^{n+1}}}\\\
&=\frac{(X+\theta)(Y^{4}+\eta^{4})+(X^{4}+\theta^{4})(\theta^{2}X+\theta^{3}+1)+X+\theta}{\theta^{2}X^{8}+(X^{4}+\theta)(Y^{4}+\eta)+X^{4}}\\\
&=\frac{(X+\theta)(Y^{4}+\eta^{4})+\theta^{2}X^{5}+(\theta^{3}+1)X^{4}+(\theta^{6}+1)X+\theta^{7}+\theta^{4}+\theta}{(X^{4}+\theta)(Y^{4}+\eta)+\theta^{2}X^{8}+X^{4}}.\end{split}$

  4. (iv)

Set $X:=\theta^{2^{n-1}}$ and $Y:=\eta^{2^{n-1}}$. Then

(4.4) $\begin{split}\alpha_{n,1}&=\frac{1}{X^{24}+\theta
X^{16}+(\theta+1)X^{8}+\theta^{2}+\theta}\bigg{(}X^{22}+(1+\theta)\\\
&\phantom{==}\cdot(X^{20}+X^{18}+X^{16})+(\theta^{2}+\theta+1)(X^{12}+X^{10})+X^{9}+\theta
X^{8}\\\
&\phantom{==}+X^{5}+(Y+\eta)(X^{4}+X^{2})+\theta^{2}+\theta\bigg{)}.\end{split}$

######  Theorem 4.2.

Conjecture 4.1 holds.

###### Proof.

In the case of elliptic curve and ramifying hyperelliptic curves, we have
proved an expression in terms of the shtuka function $f$ for the standard
sign-normalized rank one Drinfeld module $\phi$:

$\alpha_{n,1}=-\left(\left.\frac{\nu^{(n+1)}}{\delta^{(n+2)}}\right|_{\Xi}\frac{1}{(f|_{\Xi^{(1)}})^{q^{n}}}+1\right).$

Here $\nu,\delta$ are respectively the numerator and denominator of the shtuka
function $f$, as given in §4.1 and §4.2. Upon evaluation and reusing the
notations from Conjecture 4.1, we have

  1. (i)

$f=\frac{y-\eta-\theta(t-\theta)}{t-\theta-1},\qquad f|_{\Xi^{(1)}}=1,$ (4.5)
$\begin{split}\alpha_{n,1}&=\frac{\eta+\eta^{2^{n+1}}+\theta^{2^{n+1}+1}+\theta+1}{\theta^{2^{n+2}}+\theta+1}\\\
&=\frac{Y^{2}+\eta+\theta X^{2}+\theta+1}{X^{4}+\theta+1}.\end{split}$

  2. (ii)

$f=\frac{y-\eta-\eta(t-\theta)}{t-\theta-1},\qquad
f|_{\Xi^{(1)}}=\frac{1}{\eta},$ (4.6)
$\begin{split}\alpha_{n,1}&=\frac{(\eta-\eta^{3^{n+1}}-\eta^{3^{n+1}}(\theta-\theta^{3^{n+1}}))\eta^{3^{n}}-\theta^{3^{n+2}}+\theta-1}{\theta^{3^{n+2}}-\theta+1}\\\
&=\frac{Y^{4}(X^{3}-\theta-1)+\eta
Y-X^{9}+\theta-1}{X^{9}-\theta+1}.\end{split}$

  3. (iii)

$f=\frac{y-\eta-\theta^{2}(t-\theta)}{t-\theta},\qquad
f|_{\Xi^{(1)}}=\frac{1}{\theta^{4}+\theta},$ (4.7)
$\begin{split}\alpha_{n,1}&=\frac{(\eta^{4^{n+1}}+\eta+\theta^{2\cdot
4^{n+1}}(\theta+\theta^{4^{n+1}}))(\theta^{4^{n+1}}+\theta^{4^{n}})+\theta^{4^{n+2}}+\theta}{\theta^{4^{n+2}}+\theta}\\\
&=\frac{(Y^{4}+\eta)(X^{4}+X)+X^{13}+\theta X^{12}+\theta
X^{9}+\theta}{X^{16}+\theta}.\end{split}$

  4. (iv)

$f=\frac{y+\eta-(t+\theta)(\theta^{4}+\theta^{3}+\theta^{2}(t+1))}{\theta^{3}+t\theta^{2}+(1+t)\theta+t^{2}+t},\qquad
f|_{\Xi^{(1)}}=\frac{1}{\theta^{2}+\theta},$ (4.8)
$\begin{split}\alpha_{n,1}&=\frac{1}{\theta^{3\cdot 2^{n+2}}+\theta^{2\cdot
2^{n+2}+1}+(1+\theta)\theta^{2^{n+2}}+\theta^{2}+\theta}\bigg{(}\big{(}\eta+\eta^{2^{n+1}}\\\
&\phantom{==}+(\theta+\theta^{2^{n+1}})(\theta^{4\cdot 2^{n+1}}+\theta^{3\cdot
2^{n+1}}+\theta^{2\cdot
2^{n+1}}(\theta+1))\big{)}\cdot(\theta^{2^{n+1}}+\theta^{2^{n}})\\\
&\phantom{==}+(\theta^{3\cdot 2^{n+2}}+\theta^{2\cdot
2^{n+2}+1}+(1+\theta)\theta^{2^{n+2}}+\theta^{2}+\theta)\bigg{)}\\\
&=\frac{1}{X^{24}+\theta
X^{16}+(\theta+1)X^{8}+\theta^{2}+\theta}\bigg{(}\big{(}\eta+Y^{4}\\\
&\phantom{==}+(\theta+X^{4})(X^{16}+X^{12}+(\theta+1)X^{8})\big{)}\cdot(X^{4}+X^{2})\\\
&\phantom{==}+(X^{24}+\theta
X^{16}+(\theta+1)X^{8}+\theta^{2}+\theta)\bigg{)}\\\ &=\frac{1}{X^{24}+\theta
X^{16}+(\theta+1)X^{8}+\theta^{2}+\theta}\bigg{(}(\eta+Y^{4})(X^{4}+X^{2})\\\
&\phantom{==}+X^{22}+(\theta+1)(X^{20}+X^{18})+(\theta+1)X^{16}+X^{14}\\\
&\phantom{==}+(\theta^{2}+\theta)(X^{12}+X^{10})+(\theta+1)X^{8}+\theta^{2}+\theta\bigg{)}.\end{split}$

We now prove that our expressions match those in Conjecture 4.1.

Example (i)
$A=\mathbb{F}_{2}[\theta,\eta]/(\eta^{2}+\eta+\theta^{3}+\theta+1)$.

Set (4.1) and (4.5) to be equal and cross multiply. We want to show that the
following equality holds:

$Y^{4}+\eta^{2}+\theta^{2}X^{4}+\theta^{2}+Y^{2}+\eta+\theta
X^{2}+\theta=X^{6}+\theta^{2}X^{4}+(\theta+1)X^{2}+\theta^{3}+\theta^{2}.$

Using $\eta^{2}+\eta=\theta^{3}+\theta+1$, we can simplify the above equation
as

$Y^{4}+Y^{2}=X^{6}+X^{2}+1.$

This is true since

$Y^{2}+Y=\eta^{2\cdot
2^{n}}+\eta^{2^{n}}=(\eta^{2}+\eta)^{2^{n}}=(\theta^{3}+\theta+1)^{2^{n}}=X^{3}+X+1,$

and we obtain the desired equality by squaring this equation. This proves that
the expressions (4.5) and (4.1) are equal.

Example (ii) $A=\mathbb{F}_{3}[\theta,\eta]/(\eta^{2}-(\theta^{3}-\theta-1))$.

By squaring $\eta^{2}=\theta^{3}-\theta-1$, we have
$\eta^{4}=\theta^{6}+\theta^{4}+\theta^{3}+\theta^{2}-\theta+1$. Hence

$Y^{4}=(\eta^{4})^{3^{n}}=X^{6}+X^{4}+X^{3}+X^{2}-X+1.$

Thus we can rewrite the expression in (4.6) as

(4.9) $\alpha_{n,1}=\frac{1}{X^{9}-\theta+1}\big{(}\eta Y+X^{7}-\theta
X^{6}+X^{5}\\\ +(-\theta+1)X^{4}-\theta
X^{3}-(\theta+1)X^{2}+(\theta+1)X+1\big{)}.$

We also expand the terms involving $Y$ in (4.2). First, we have that

$Y^{2}=(\eta^{2})^{3^{n}}=(\theta^{3}-\theta-1)^{3^{n}}=X^{3}-X-1.$

Thus

$\begin{split}(Y-\eta)^{2}&=Y^{2}+\eta Y+\eta^{2}\\\ &=(X^{3}-X-1)+\eta
Y+(\theta^{3}-\theta-1)\\\ &=\eta Y+X^{3}-X+\theta^{3}-\theta+1.\end{split}$

At the same time,

$Y^{3}=Y^{2}\cdot Y=(X^{3}-X-1)Y.$

Putting these into (4.2), we have that the expression equals

$\begin{split}&\phantom{=1}\frac{(X-\theta)(\eta
Y+X^{3}-X+\theta^{3}-\theta+1)-X^{2}+(-\theta^{3}+1)X+\theta^{2}-\theta}{(X^{3}-X-1)\eta
Y+X^{6}+(\theta+1)X^{3}+\theta^{2}-\theta+1}\\\ &=\frac{(X-\theta)\eta
Y+X^{4}-\theta X^{3}+X^{2}-X-\theta^{2}+\theta}{(X^{3}-X-1)\eta
Y+X^{6}+(\theta+1)X^{3}+\theta^{2}-\theta+1}.\end{split}$

Set this and (4.9) to be equal and cross multiply, we want to show that the
following equality holds:

$\begin{split}&\phantom{=1}\big{(}X^{10}-\theta
X^{9}+(-\theta+1)X+\theta^{2}-\theta\big{)}\eta Y+X^{13}-\theta
X^{12}+X^{11}\\\
&\phantom{=1}-X^{10}+(-\theta^{2}+\theta)X^{9}+(-\theta+1)X^{4}+(\theta^{2}-\theta)X^{3}\\\
&\phantom{=1}+(-\theta+1)X^{2}+(\theta-1)X+\theta^{3}+\theta^{2}+\theta\\\
&=\eta^{2}(X^{3}-X-1)Y^{2}+\big{(}X^{10}-\theta
X^{9}+(-\theta+1)X+\theta^{2}-\theta\big{)}\eta Y\\\
&\phantom{=1}+X^{13}-\theta
X^{12}+X^{11}-X^{10}+(-\theta^{2}+\theta)X^{9}+(-\theta^{3}+\theta+1)X^{6}\\\
&\phantom{=1}-(\theta^{3}+1)X^{4}+(-\theta^{3}+\theta^{2}+1)X^{3}+(-\theta^{3}-1)X^{2}+(\theta^{3}+1)X\\\
&\phantom{=1}+\theta^{2}-\theta+1.\end{split}$

Cancelling common terms and rearranging, we want the following equality to
hold:

$\begin{split}\phantom{=1}\eta^{2}(X^{3}-X-1)Y^{2}&=(\theta^{3}-\theta-1)X^{6}+(\theta^{3}-\theta-1)X^{4}+(\theta^{3}-\theta-1)X^{3}\\\
&\phantom{=1}+(\theta^{3}-\theta-1)X^{2}-(\theta^{3}-\theta-1)X+(\theta^{3}-\theta-1).\end{split}$

This is indeed true since $\eta^{2}=\theta^{3}-\theta-1$, and

$(X^{3}-X-1)Y^{2}=(X^{3}-X-1)^{2}=X^{6}+X^{4}+X^{3}+X^{2}-X+1.$

This proves that the expressions (4.6) and (4.2) are equal.

Example (iii) $A=\mathbb{F}_{4}[\theta,\eta]/(\eta^{2}+\eta+\theta^{3}+w)$
where $w^{2}+w+1=0$.

Set the expressions (4.3) and (4.7) to be equal and cross multiply. We want to
show that the following equality holds:

$\begin{split}&\phantom{=1}(X^{17}+\theta X^{16}+\theta
X+\theta^{2})(Y^{4}+\eta^{4})+\theta^{2}X^{21}+(\theta^{3}+1)X^{20}+(\theta^{6}+1)X^{17}\\\
&\phantom{=1}+(\theta^{7}+\theta^{4}+\theta)X^{16}+\theta^{3}X^{5}+(\theta^{4}+\theta)X^{4}+(\theta^{7}+\theta)X+\theta^{8}+\theta^{5}+\theta^{2}\\\
&=(X^{8}+X^{5}+\theta X^{4}+\theta X)(Y^{4}+\eta)^{2}+\big{(}X^{17}+\theta
X^{16}+X^{8}+X^{5}\\\ &\phantom{=1}+\theta
X^{4}+\theta^{2}\big{)}(Y^{4}+\eta)+\theta^{2}X^{21}+\theta^{3}X^{20}+(\theta^{3}+1)X^{17}+\theta
X^{16}\\\ &\phantom{=1}+\theta X^{13}+\theta^{3}X^{8}+\theta
X^{4}.\end{split}$

Recall that $\eta^{2}=\eta+\theta^{3}+w$. Squaring both sides, we have that

$\eta^{4}=\eta^{2}+\theta^{6}+w^{2}=\eta+\theta^{6}+\theta^{3}+w^{2}+w=\eta+\theta^{6}+\theta^{3}+1.$

Using this and rearranging the long equation, we want to show that the
following equality holds:

$\begin{split}&\phantom{=1}(X^{8}+X^{5}+\theta X^{4}+\theta
X)(Y^{4}+\eta)^{2}+(X^{8}+X^{5}+\theta X^{4}+\theta X)(Y^{4}+\eta)\\\
&=(X^{17}+\theta X^{16}+\theta
X+\theta^{2})(\theta^{6}+\theta^{3}+1)+X^{20}+(\theta^{6}+\theta^{3})X^{17}+(\theta^{7}+\theta^{4})X^{16}\\\
&\phantom{=1}+\theta
X^{13}+\theta^{3}X^{8}+\theta^{3}X^{5}+\theta^{4}X^{4}+(\theta^{7}+\theta)X+\theta^{8}+\theta^{5}+\theta^{2}.\end{split}$

The right hand side can be simplified to

$\begin{split}&\phantom{=1}X^{20}+X^{17}+\theta X^{16}+\theta
X^{13}+\theta^{3}X^{8}+\theta^{3}X^{5}+\theta^{4}X^{4}+\theta^{4}X\\\
&=(X^{8}+X^{5}+\theta X^{4}+\theta X)(X^{12}+\theta^{3})\end{split}$

Hence, it suffices to show that

$(Y^{4}+\eta)^{2}+(Y^{4}+\eta)=X^{12}+\theta^{3}.$

Raising $\eta^{2}+\eta=\theta^{3}+w$ by $4\cdot 4^{n}$-th power, we have that

$Y^{8}+Y^{4}=X^{12}+w.$

In particular, $w^{4}=w$ as $w\in\mathbb{F}_{4}$. Thus,

$(Y^{4}+\eta)^{2}+(Y^{4}+\eta)=Y^{8}+Y^{2}+\eta^{2}+\eta=X^{12}+w+\theta^{3}+w=X^{12}+\theta^{3},$

completing the proof.

Example (iv)
$A=\mathbb{F}_{2}[\theta,\eta]/(\eta^{2}+\eta+\theta^{5}+\theta^{3}+1)$.

This one is the easiest among the four examples. The expression (4.8) has the
same denominator as Lara Rodríguez-Thakur’s expression (4.4). Hence it
suffices to show that their numerators are the same. By setting the numerators
to be equal, it boils down to showing that

$Y^{4}+Y=X^{10}+X^{6}+X^{5}+X^{3},$

Or equivalently,

$\eta^{2^{n+1}}+\eta^{2^{n-1}}=\theta^{10\cdot 2^{n-1}}+\theta^{6\cdot
2^{n-1}}+\theta^{5\cdot 2^{n-1}}+\theta^{3\cdot 2^{n-1}}$

for $n\geq 1$. This follows from the equality

$\eta^{4}+\eta=\theta^{10}+\theta^{6}+\theta^{5}+\theta^{3}$

by squaring $\eta^{2}=\eta+\theta^{5}+\theta^{3}+1$. The proof is complete. ∎

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