# Distributive properties of division points and discriminants of Drinfeld
modules

Ernst-Ulrich Gekeler  FR 6.1 Mathematik, Universität des Saarlandes, Postfach
15 11 50 D-66041 Saarbrücken<EMAIL_ADDRESS>
(Date: September 1, 2024)

###### Abstract.

We present a new notion of distribution and derived distribution of rank
$r\in\mathds{N}$ for a global function field $K$ with a distinguished place
$\infty$. It allows to describe the relations between division points,
isogenies, and discriminants both for a fixed Drinfeld module of rank $r$ for
the above data, or for the corresponding modular forms.

We introduce and study three basic distributions with values in $\mathds{Q}$,
in the group $\mu(\overline{K})$ of roots of unity in the algebraic closure
$\overline{K}$ of $K$, and in the group $U^{(1)}(C_{\infty})$ of $1$-units of
the completed algebraic closure $C_{\infty}$ of $K_{\infty}$, respectively.

There result product formulas for division points and discriminants that
encompass known results (e.g. analogues of Wallis’ formula for
$(2\pi\imath)^{2}$ in the rank-$1$ case, of Jacobi’s formula
$\Delta=(2\pi\imath)^{12}q\prod(1-q^{n})^{24}$ in the rank-$2$ case, and
similar boundary expansions for $r>2$) and several new ones: the definition of
a canonical discriminant for the most general case of Drinfeld modules and the
description of the sizes of division and discriminant forms.

In the now classical case where $(K,\infty)=(\mathds{F}_{q}(T),\infty)$ and
$r=1$, $2$ or $3$, we give explicit values for the logarithms of such forms.

###### Key words and phrases:

Drinfeld modules, Drinfeld modular forms, Distributions and derived
distributions, Product formulas

###### 1991 Mathematics Subject Classification:

MSC 11G09, 11F52, 11R58

##  1\. Introduction and Notation

### 1.1.

This paper originates in the author’s attempt to find a common framework for
work of Carlitz [6], Hayes [22], [23], Galovich-Rosen [8], and the author
[11], [10], [12] Ch. IV and VI about division points of Drinfeld modules of
rank $1$ and of higher rank. Since long time it appeared that the product
formulas for periods of Drinfeld modules of rank $1$ given in [12] Ch. IV, for
discriminants in the higher rank case ([12] Ch. VI, [19], see also [1]), and
similar expressions for the respective division points, should have a common
source.

Ideally, one would hope for a representation of the discriminant as a
(suitably regularized as in [12] Ch. IV) product over the elements of the
period lattice, and such that the division points are described by partial
products.

That this is essentially true is the content of Theorem 7.4 along with its
corollaries.

### 1.2.

Let us introduce some notation. Throughout, $K$ is a global function field
with $\mathds{F}_{q}$ as its field of constants, where $q$ is a power of the
prime number $p$, and “$\infty$” is a fixed place of $K$, of degree
$d_{\infty}$ over $\mathds{F}_{q}$. We let $A\subset K$ be the Dedekind ring
of elements of $K$ regular away from $\infty$. Equivalently, $K$ is the
function field of a smooth projective geometrically connected curve
$\mathcal{C}$ over $\mathds{F}_{q}$, and
$A=\mathcal{O}_{\mathcal{C}}(\mathcal{C}\smallsetminus\\{\infty\\})$. Such
rings are called Drinfeld coefficient rings for short. As $q$ is fixed
throughout, we omit it from notation and write $\mathds{F}$ for
$\mathds{F}_{q}$. Let further $K_{\infty}$ be the completion of $K$ at
$\infty$, and $\pi\in K$ a uniformizer at $\infty$. We normalize the absolute
value $\lvert\mathrel{\;.\;}\rvert=\lvert\mathrel{\;.\;}\rvert_{\infty}$ by

(1.2.1) $\lvert\pi\rvert^{-1}=q_{\infty}\vcentcolon=q^{d_{\infty}},$

and let $C_{\infty}$ be the completed algebraic closure of $K_{\infty}$ with
respect to $\lvert\mathrel{\;.\;}\rvert$.

#### 1.2.2.

The most simple example is where $\mathcal{C}$ is the projective line and
“$\infty$” the usual place at infinity, so $K=\mathds{F}(T)$ with an
indeterminate $T$, $A=\mathds{F}[T]$, and $K_{\infty}=\mathds{F}((T))^{-1}$.
For $x\in C_{\infty}$, we let

(1.2.3) $\log x=\log_{q}\lvert x\rvert\quad(x\neq 0)\qquad\text{and}\qquad\log
0={-}\infty.$

If $0\neq x\in A$, then $\log x$ agrees with $\deg x=\dim_{\mathds{F}}A/(x)$.

### 1.3.

An $A$-lattice in $C_{\infty}$ is a finitely generated (hence projective of
some rank $r$) $A$-submodule $\Lambda$ of $C_{\infty}$ which is discrete in
the sense that it has finite intersection with each ball of finite diameter in
$C_{\infty}$.

Fix $r\in\mathds{N}=\\{1,2,3,\dots\\}$ and put $V\vcentcolon=K^{r}$,
$V_{\infty}\vcentcolon=K_{\infty}^{r}$. An $A$-lattice in $V$ is a finitely
generated $A$-submodule $Y$ of $V$ of full rank $r$. Let

(1.3.1)
$\Psi=\Psi^{r}\vcentcolon=\\{\boldsymbol{\omega}=(\omega_{1},\dots,\omega_{r})\in
C_{\infty}^{r}\mid\omega_{1},\dots,\omega_{r}\text{ $K_{\infty}$-linearly
independent}\\}$

be the Drinfeld space of dimension $r$. To each $\boldsymbol{\omega}\in\Psi$
there corresponds an embedding

(1.3.2) $\begin{split}i_{\boldsymbol{\omega}}\colon
V&\lhook\joinrel\longrightarrow C_{\infty}.\\\
\boldsymbol{v}=(v_{1},\dots,v_{r})&\longmapsto\boldsymbol{v}\boldsymbol{\omega}\vcentcolon=\sum_{1\leq
i\leq r}v_{i}\omega_{i}.\end{split}$

It has the property: Given a lattice $Y$ in $V$ and
$\boldsymbol{\omega}\in\Psi$, the set
$\Lambda=Y_{\boldsymbol{\omega}}\vcentcolon=i_{\boldsymbol{\omega}}(Y)$ is an
$A$-lattice in $C_{\infty}$. Vice versa, given a lattice $\Lambda\subset
C_{\infty}$, the choice of a $K$-basis of $K\Lambda$ yields an
$\boldsymbol{\omega}\in\Psi$ and $Y\subset V$ such that
$\Lambda=Y_{\boldsymbol{\omega}}$. We note that

(1.3.3) $\Omega=\Omega^{r}\vcentcolon=C_{\infty}^{\times}\backslash\Psi$

is the classical Drinfeld symmetric space as defined in [7].

### 1.4.

To each $\Lambda$ as above, we let

(1.4.1) $\begin{split}e^{\Lambda}\colon C_{\infty}&\longrightarrow
C_{\infty}\\\ z&\longmapsto
z\sideset{}{{}^{\prime}}{\prod}_{\lambda\in\Lambda}(1-z/\lambda)\end{split}$

be its exponential function. Here and in the sequel we use the

### Convention

$\sideset{}{{}^{\prime}}{\prod}(\cdots)$ resp.
$\sideset{}{{}^{\prime}}{\sum}(\cdots)$ is the product resp. sum over the non-
zero elements of the respective index set.

The main properties of $e^{\Lambda}$ are easily established: The product
converges and defines an $\mathds{F}$-linear map onto $C_{\infty}$ with kernel
$\Lambda$. Furthermore, for each $0\neq a\in A$, there exists a polynomial
$\phi_{a}^{\Lambda}\in C_{\infty}[X]$ of shape

(1.4.2) $\phi_{a}^{\Lambda}(X)=aX+\sum_{1\leq i\leq r\deg
a}{}_{a}\ell_{i}X^{q^{i}}$

such that the diagram

(1.4.3)
${0}$${\Lambda}$${C_{\infty}}$${C_{\infty}}$${0}$${0}$${\Lambda}$${C_{\infty}}$${C_{\infty}}$${0}$$\scriptstyle{a}$$\scriptstyle{e^{\Lambda}}$$\scriptstyle{a}$$\scriptstyle{\phi_{a}^{\Lambda}}$$\scriptstyle{e^{\Lambda}}$

is commutative. Its coefficients
${}_{a}\ell_{i}={}_{a}\ell_{i}(\Lambda)={}_{a}\ell_{i}^{Y}(\boldsymbol{\omega})$
if $\Lambda=Y_{\boldsymbol{\omega}}$ depend on $\Lambda$ or on
$\boldsymbol{\omega}\in\Psi$, and the discriminant

(1.4.4)
$\Delta_{a}(\Lambda)=\Delta_{a}^{Y}(\boldsymbol{\omega})\vcentcolon={}_{a}\ell_{r\deg
a}(\Lambda)$

doesn’t vanish. It is well-known that the collection
$\\{\phi_{a}^{\Lambda}\mid a\in A\\}$ (where $\phi_{0}^{\Lambda}=0$) defines a
Drinfeld $A$-module $\phi^{\Lambda}$ over $C_{\infty}$, which establishes a
bijective correspondence (see, e.g., [7], [21], [27])

(1.4.5) $\\{\text{$A$-lattices in $C_{\infty}$ of rank
$r$}\\}\longleftrightarrow\\{\text{Drinfeld $A$-modules over $C_{\infty}$ of
rank $r$}\\}.$

We note that for $0\neq c\in C_{\infty}$

(1.4.6) ${}_{a}\ell_{i}(c\Lambda)=c^{1-q^{i}}{}_{a}\ell_{i}(\Lambda)$

holds. In fact, the ${}_{a}\ell_{i}$ are modular forms for the group
$\Gamma_{Y}=\operatorname{GL}(Y)$ of weight $q^{i}-1$ and type $0$ ([5],
[19]).

### 1.5.

Assume that $\Lambda=Y_{\boldsymbol{\omega}}$ and $0\neq a\in A$ as before. By
(1.4.3) and the properties of $e^{\Lambda}$, the

(1.5.1)
$d_{\boldsymbol{u}}^{Y}(\boldsymbol{\omega})\vcentcolon=e^{Y_{\boldsymbol{\omega}}}(\mathbf{u}\boldsymbol{\omega})$

with $\mathbf{u}\in a^{-1}Y\smallsetminus Y$ are the non-trivial zeroes of
$\phi_{a}^{\Lambda}(X)$, i.e., the $a$-division points of $\phi^{\Lambda}$,
and depend only on the class of $\mathbf{u}$ in $a^{-1}Y/Y$. Hence we may
write

(1.5.2)
$\phi_{a}^{Y_{\boldsymbol{\omega}}}(X)=aX\sideset{}{{}^{\prime}}{\prod}_{\mathbf{u}\in
a^{-1}Y/Y}(1-d_{\boldsymbol{u}}^{Y}(\boldsymbol{\omega})^{-1}X).$

In particular,

(1.5.3) $\Delta_{a}^{Y}=a\Big{(}\sideset{}{{}^{\prime}}{\prod}_{\mathbf{u}\in
a^{-1}Y/Y}d_{\mathbf{u}}^{Y}\Big{)}^{-1}=a\Delta_{(a)}^{Y}$

as functions of $\boldsymbol{\omega}$. That is, $\Delta_{(a)}^{Y}$ is
$\Delta_{a}^{Y}$ deprived of the factor $a$.

### 1.6.

Given two lattices $\Lambda\subset\Lambda^{\prime}$ of rank $r$ in
$C_{\infty}$ (we call these a lattice pair of rank $r$ in $C_{\infty}$, and
also write $\Lambda^{\prime}|\Lambda$ for this data), the index
$[\Lambda^{\prime}:\Lambda]$ is finite, say,
$[\Lambda^{\prime}:\Lambda]=q^{d}$. In analogy with (1.4.2) and (1.4.3), there
exists an $\mathds{F}$-linear polynomial $\varphi$ (that is, a polynomial
where only $1$, $q$, $q^{2}$, …appear as exponents) with linear coefficient 1
(i.e., its derivative $\varphi^{\prime}$ equals $1$) and of degree $q^{d}$
such that

(1.6.1) $e^{\Lambda^{\prime}}=\varphi\circ e^{\Lambda}.$

If $\phi=\phi^{\Lambda}$ and $\phi^{\prime}=\phi^{\Lambda^{\prime}}$ are the
associated Drinfeld modules, then $\varphi$ describes an isogeny from $\phi$
to $\phi^{\prime}$. More generally, if $\phi$, $\phi^{\prime}$ are arbitrary
Drinfeld $A$-modules of rank $r$, where $\phi=\phi^{\Lambda}$,
$\phi^{\prime}=\phi^{\Lambda^{\prime}}$, and
$\varphi\colon\phi\to\phi^{\prime}$ is an isogeny with $\varphi^{\prime}=c\in
C_{\infty}$, then $c\neq 0$, $\Lambda\subset c^{-1}\Lambda^{\prime}$, and
replacing $\Lambda^{\prime}$ with $c^{-1}\Lambda^{\prime}$ yields a lattice
pair
$\Lambda^{\prime}|\Lambda=Y_{\boldsymbol{\omega}}^{\prime}|Y_{\boldsymbol{\omega}}$,
which means that $\varphi^{\prime}=1$. Hence the collection of lattice pairs
$Y^{\prime}|Y$ in $V$ controls the collection of all possible isogenies of
Drinfeld $A$-modules of rank $r$. In the above situation, we define

(1.6.2)
$\Delta(\Lambda^{\prime}|\Lambda)=\Delta^{Y^{\prime}|Y}(\boldsymbol{\omega})\vcentcolon=\text{the
leading coefficient of $\varphi$},$

and call it the discriminant of $\varphi$, or of $\Lambda^{\prime}|\Lambda$.
Hence $\Delta_{(a)}$ as defined in (1.5.3) equals
$\Delta(a^{-1}\Lambda|\Lambda)=\Delta^{a^{-1}Y|Y}(\boldsymbol{\omega})$.
Generalizing (1.5.3), we have

(1.6.3)
$\Delta^{Y^{\prime}|Y}(\boldsymbol{\omega})=\Big{(}\sideset{}{{}^{\prime}}{\prod}_{\mathbf{u}\in
Y^{\prime}/Y}d_{\mathbf{u}}^{Y}(\boldsymbol{\omega})\Big{)}^{-1}.$

### 1.7.

The family of all $d_{\mathbf{u}}^{Y}$ and $\Delta^{Y^{\prime}|Y}$, where
$\mathbf{u}$ runs through $V=K^{r}$ and $Y^{\prime}|Y$ through the set of
lattice pairs in $V$, has some formal properties which are close to defining a
distribution on the set

(1.7.1) $\mathfrak{Y}=\\{(\mathbf{u},Y)\mid\mathbf{u}\in V,Y\text{ an
$A$-lattice in }V\\}$

(see Section 2 for definitions). Our Theorem 7.4 along with its corollaries
may be stated in simplified form as follows.

######  Main Theorem:

  1. $\mathrm{(i)}$

Given a lattice $\Lambda$ of rank $r$ in $C_{\infty}$, there exists a
canonical discriminant $\Delta(\Lambda)$ such that for each $0\neq a\in A$,

(1.7.2) $\Delta_{a}(\Lambda)=\operatorname{sgn}(a)\Delta(\Lambda)^{(q^{r\deg
a}-1)/w_{r}}$

holds. Here $w_{r}=q_{\infty}^{r}-1$, and $\operatorname{sgn}(a)$ is a
$(q_{\infty}-1)$-th root of unity (see 5.1). If
$\Lambda=Y_{\boldsymbol{\omega}}$ with $\boldsymbol{\omega}$ varying through
$\Psi$, then $\Delta^{Y}(\boldsymbol{\omega})=\Delta(Y_{\boldsymbol{\omega}})$
defines a modular form of weight $w_{r}$ and type $0$ for
$\Gamma_{Y}=\operatorname{GL}(Y)$.

  2. $\mathrm{(ii)}$

Let $\mathcal{O}(\Psi)^{*}$ be the multiplicative group of invertible
holomorphic functions on the Drinfeld space $\Psi=\Psi^{r}$. Then

(1.7.3)
$\begin{split}F\colon\mathfrak{Y}&\longrightarrow\mathcal{O}(\Psi)^{*}\\\
(\boldsymbol{u},Y)&\longmapsto\begin{cases}(d_{\mathbf{u}}^{Y})^{w_{r}}\Delta^{Y}&(\mathbf{u}\notin
Y)\\\ \Delta^{Y}&(\mathbf{u}\in Y)\end{cases}\end{split}$

is a distribution on $\mathfrak{Y}$.

  3. $\mathrm{(iii)}$

Both sorts of functions, $d_{\mathbf{u}}^{Y}(\boldsymbol{\omega})$ and
$\Delta^{Y}(\boldsymbol{\omega})$, may be evaluated through simple product
formulas in terms of $Y$, $\mathbf{u}$, and $\boldsymbol{\omega}$ (see Theorem
7.4).

As a by-product, we get the sizes
$\lvert\Delta(\Lambda^{\prime}|\Lambda)\rvert$, $\lvert\Delta(\Lambda)\rvert$,
$\lvert d_{\mathbf{u}}^{Y}(\boldsymbol{\omega})\rvert$ of the involved
functions (see Corollary 7.8 and Section 8).

### 1.8.

We start in Section 2 with defining the domain $\mathfrak{Y}$ and the notions
of distributions and derived distributions on it. In Section 3, the basic
relations between the functions $d_{\mathbf{u}}^{Y}$ and
$\Delta^{Y^{\prime}|Y}$ are investigated. These are such that they form a
derived distribution on $\mathfrak{Y}$ with values in the group
$\mathcal{O}(\Psi)^{*}$, in fact, with invertible modular forms as values. As
a consequence of the relations, we describe in Theorem 3.5 the multiplicative
action of Hecke correspondences on the discriminant forms
$\Delta^{Y^{\prime}|Y}$. This is analogous with (but in view of the unbounded
rank $r$ more general than) the results of Gilles Robert [25], [26] about the
classical elliptic discriminant.

In view of the decomposition of the multiplicative group (see (5.1.3))

$C_{\infty}^{*}=\pi^{\mathds{Q}}\times\mu(C_{\infty})\times
U^{(1)}(C_{\infty}),$

where $\mu(C_{\infty})$ and $U^{(1)}(C_{\infty})$ are the roots of unity and
the $1$-units in $C_{\infty}$, respectively, we next study certain
distributions with values in $\mathds{Q}$, in $U^{(1)}(C_{\infty})$, and in
$\mu(C_{\infty})$, in Sections 4, 5, and 6. The $\mu(C_{\infty})$-valued
distributions, treated in Section 6, behave differently and are delicate to
handle. For the $\mathds{Q}$-valued distributions, we introduce the
$Z$-function of a lattice $\Lambda$, an invariant analogous with the zeta
function of a variety over a finite field.

This enables us to show in Section 7 our main result Theorem 7.4 (essentially
item (iii) of the Main Theorem announced above) and some of its consequences,
among which the definition of the canonical discriminant $\Delta$. We should
note here that such a discriminant has been defined in [5] and [19], but in
each case only up to roots of unity. Here, due to the analysis in Section 6,
we get the precise value of that root of unity, and thereby the canonical
$\Delta(\Lambda)=\Delta^{Y}(\boldsymbol{\omega})$.

We conclude in Section 8 with an application to the situation of 1.2.2, where
$A=\mathds{F}[T]$. This connects the present results with existing ones (cases
$r=1$ and $2$), and produces new ones about sizes of modular forms ($r=2$ and
$3$).

### Notation

  * •

$\mathds{F}=\mathds{F}_{q}$ the finite field with $q$ elements, of
characteristic $p$

  * •

$K$ a global function field with $\mathds{F}$ as field of constants

  * •

$\infty$ a fixed place of $K$, of degree $d_{\infty}$ over $\mathds{F}$

  * •

$A$ the Dedekind subring of $K$ of elements regular off $\infty$

  * •

$K_{\infty}$ the completion of $K$ at $\infty$, with a uniformizer $\pi\in K$

  * •

$C_{\infty}$ the completed algebraic closure of $K_{\infty}$

  * •

$\lvert\mathrel{\;.\;}\rvert$ the absolute value on $C_{\infty}$ with value
group $q^{\mathds{Q}}$, normalized by
$\lvert\pi\rvert^{-1}=q_{\infty}=q^{d_{\infty}}$

  * •

$\log x=\log_{q}\lvert x\rvert$ ($0\neq x\in C_{\infty}$), $\log 0={-}\infty$

  * •

$w\vcentcolon=q_{\infty}-1$, $w_{r}\vcentcolon=q_{\infty}^{r}-1$

  * •

$r\in\mathds{N}=\\{1,2,3,\dots\\}$ a fixed natural number, the rank of our
situation (usually omitted from notation)

  * •

$V=K^{r}$, $V_{\infty}=K_{\infty}^{r}$, $Y$, $Y^{\prime}$ $A$-lattices in $V$

  * •

$\mathfrak{Y}=\\{(\mathbf{u},Y)\mid\mathbf{u}\in V,Y\subseteq V\text{ a
lattice}\\}$

  * •

$\Psi=\Psi^{r}$, $\Omega=C_{\infty}^{*}\backslash\Psi$ the Drinfeld spaces
with groups $\mathcal{O}(\Psi)^{*}$ resp. $\mathcal{O}(\Omega)^{*}$ of
invertible holomorphic functions

  * •

$\Lambda$, $\Lambda^{\prime}$ $A$-lattices of rank $r$ in $C_{\infty}$

  * •

$\Lambda_{N}=\\{\lambda\in\Lambda\mid\log\lambda\leq N\\}$,
$\Lambda_{N,N^{\prime}}=\Lambda_{N^{\prime}}\smallsetminus\Lambda_{N}$ ($N\leq
N^{\prime}$)

  * •

$e^{\Lambda},\phi^{\Lambda}$ exponential function and Drinfeld $A$-module
associated with $\Lambda$

  * •

$d_{\mathbf{u}}^{Y}(\boldsymbol{\omega})$ division point/function of
$\phi^{\Lambda}$, where $\Lambda=Y_{\boldsymbol{\omega}}$ with
$\boldsymbol{\omega}\in\Psi$

  * •

$\Delta_{a}^{Y}$, $\Delta_{(a)}^{Y}$, $\Delta_{\mathfrak{n}}^{Y}$,
$\Delta^{Y^{\prime}|Y}$ various discriminants/discriminant functions

##  2\. Distributions and Derived Distributions on $\mathfrak{Y}$

For some background and motivation on distributions in the number field case,
see [24] Ch. I. Throughout, $A$ is a fixed Drinfeld coefficient ring as
described in 1.2, and $r$ a fixed natural number.

######  Definition 2.1:

We let $\mathfrak{Y}=\mathfrak{Y}^{r}$ be the set of pairs $(\mathbf{u},Y)$ as
in (1.7.1), where $\mathbf{u}\in V=K^{r}$ and $Y\subset V$ is an $A$-lattice.
We call $\mathfrak{Y}$ the distribution domain for the fixed data $A$ and $r$.
Given an additively written abelian group $M$, an $M$-valued distribution on
$\mathfrak{Y}$ is a function $f\colon\mathfrak{Y}\to M$ subject to

(2.1.1) $f(\mathbf{u},Y)\text{ depends only on the class of $\mathbf{u}$
modulo $Y$};$

if $Y^{\prime}|Y$ is a lattice pair in $V$ and $\mathbf{v}\in V$, then

(2.1.2) $\sum_{\begin{subarray}{c}\mathbf{u}\in V/Y\\\
\mathbf{u}\equiv\mathbf{v}\leavevmode\nobreak\
(\mathrm{mod}Y^{\prime})\end{subarray}}f(\mathbf{u},Y)=f(\mathbf{v},Y^{\prime}).$

### 2.2.

The prototype of a distribution on $\mathfrak{Y}$ comes out as follows. Assume
that $m\colon V\to M$ is a function such that for each
$(\mathbf{u},Y)\in\mathfrak{Y}$, the infinite sum

(2.2.1) $f(\mathbf{u},Y)\vcentcolon=\sum_{\begin{subarray}{c}\mathbf{x}\in
V\\\ \mathbf{x}\equiv\mathbf{u}\leavevmode\nobreak\
(\mathrm{mod}Y)\end{subarray}}m(\mathbf{x})$

converges (which requires a suitable topological structure on $M$). Then
(2.1.1) and (2.1.2) are trivially fulfilled, and $f$ defines a distribution.
This remains true if the sum in (2.2.1) is replaced by
$\sideset{}{{}^{\prime}}{\sum}$.

### 2.3.

More specifically, for each $\boldsymbol{\omega}\in\Psi$,
$(\mathbf{u},Y)\in\mathfrak{Y}$, and $k\in\mathds{N}$, define

(2.3.1)
$E_{k,\mathbf{u}}^{Y}(\boldsymbol{\omega})\vcentcolon=\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\mathbf{x}\in
V\\\ \mathbf{x}\equiv\mathbf{u}\leavevmode\nobreak\
(\mathrm{mod}Y)\end{subarray}}(\mathbf{x}\boldsymbol{\omega})^{{-}k},$

where $\mathbf{x}\boldsymbol{\omega}=\sum_{1\leq i\leq r}x_{i}\omega_{i}\in
C_{\infty}$. The sum converges and defines a holomorphic function
$E_{k,\mathbf{u}}^{Y}$ on $\Psi$, the Eisenstein series of weight $k$ and
shape $(\mathbf{u},Y)$. In fact, $E_{k,\mathbf{u}}^{Y}$ is a modular form of
weight $k$ for a suitable congruence subgroup of
$\Gamma_{Y}=\operatorname{GL}(Y)$. For details, see e.g. [20], [5], or [19].
Therefore, the system $\\{E_{k,\mathbf{u}}^{Y}\\}$ with $k$ fixed describes a
distribution with values in the $C_{\infty}$-algebra of modular forms.

### 2.4.

Let $f\colon\mathfrak{Y}\to M$ be a distribution. Its derivative $Df=g$ is the
function $g\colon\mathfrak{Y}\to M$ defined by

(2.4.1) $g(\mathbf{u},Y)\vcentcolon=f(\mathbf{u},Y)-f(0,Y).$

Since $g\equiv 0$ on $\mathfrak{Y}\smallsetminus\mathfrak{Y}^{*}$, we regard
it as a function on

(2.4.2)
$\mathfrak{Y}^{*}\vcentcolon=\\{(\mathbf{u},Y)\in\mathfrak{Y}\mid\mathbf{u}\notin
Y\\}.$

If $Y^{\prime}|Y$ is a lattice pair in $V$, we define the discriminant
$\operatorname{Disc}(g)^{Y^{\prime}|Y}$ of $Y^{\prime}|Y$ with respect to $g$
as

(2.4.3)
$\operatorname{Disc}(g)^{Y^{\prime}|Y}\vcentcolon={-}\sideset{}{{}^{\prime}}{\sum}_{\mathbf{u}\in
Y^{\prime}/Y}g(\mathbf{u},Y).\qquad\text{(Note the minus sign!)}$

By direct calculation, we find for $\mathbf{v}\in V$:

(2.4.4) $\sum_{\begin{subarray}{c}\mathbf{u}\in V/Y\\\
\mathbf{u}\equiv\mathbf{v}\leavevmode\nobreak\
(\mathrm{mod}Y^{\prime})\end{subarray}}g(\mathbf{u},Y)+\operatorname{Disc}(g)^{Y^{\prime}|Y}=g(\mathbf{v},Y^{\prime}),$

and for a tower $Y^{\prime\prime}|Y^{\prime}|Y$:

(2.4.5)
$\operatorname{Disc}(g)^{Y^{\prime\prime}|Y}=\operatorname{Disc}(g)^{Y^{\prime\prime}|Y}+[Y^{\prime\prime}:Y^{\prime}]\operatorname{Disc}(g)^{Y^{\prime}|Y}.$

Moreover,

(2.4.6)
$\operatorname{Disc}(g)^{Y^{\prime}|Y}=[Y^{\prime}:Y]f(0,Y)-f(0,Y^{\prime}).$

Note that the definition (2.4.3) of $\operatorname{Disc}(g)$ as well as
(2.4.4) and (2.4.5) involve only $g$ but not $f$, and (2.4.5) is a consequence
of (2.4.4) and (2.1.1) for $g$ without reference to $f$.

######  Definition 2.5:

A derived distribution on $\mathfrak{Y}$ with values in $M$ is a function
$g\colon\mathfrak{Y}^{*}\to M$ subject to (2.1.1) , i.e., $g(\mathbf{u},Y)$
depends only on the class of $\mathbf{u}$ modulo $Y$, and (2.4.4), where
$\operatorname{Disc}(g)^{Y^{\prime}|Y}$ is defined in (2.4.3) (and thus to
(2.4.5), by the remark above).

It is a major problem to decide whether a given derived distribution $g$ comes
in fact as the derivative of a distribution $f$ (a primitive of $g$) as above.
We note the trivial observation:

######  Lemma 2.6:

  1. $\mathrm{(i)}$

Both the sets of distributions $f$ and of derived distributions $g$ are
abelian groups, and $f\mapsto Df=g$ is a homomorphism.

  2. $\mathrm{(ii)}$

If $M$ has no $p$-torsion then a primitive $f$ of $g$ is unique up to adding a
constant distribution. Here a distribution $f$ is constant if
$f(\mathbf{u},Y)=f(0,Y)$ and $f(0,Y^{\prime})=[Y^{\prime}:Y]f(0,Y)$ for all
$\mathbf{u}\in V$ and lattice pairs $Y^{\prime}|Y$.

For the next definition, we suppose that $M$ is a Hausdorff topological group,
so that the occurring limits are meaningful.

######  Definition 2.7:

  1. $\mathrm{(i)}$

Let $f\colon\mathfrak{Y}\to M$ be a distribution. It is motivated by $m\colon
V\smallsetminus\\{0\\}\to M$ if there exists a norm
$\lVert\mathrel{\;.\;}\rVert$ on $V_{\infty}$ such that

(2.7.1)
$f(\mathbf{u},Y)=\lim_{N\to\infty}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\mathbf{x}\in
V\\\ \mathbf{x}\equiv\mathbf{u}\leavevmode\nobreak\ (\mathrm{mod}Y)\\\
\log_{q}\lVert\mathbf{x}\rVert\leq N\end{subarray}}m(\mathbf{x}).$

  2. $\mathrm{(ii)}$

A derived distribution $g\colon\mathfrak{Y}^{*}\to M$ is motivated by $m$ if

(2.7.2)
$g(\mathbf{u},Y)=\lim_{N\to\infty}\Big{(}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\mathbf{x}\in
V\\\ \mathbf{x}\equiv\mathbf{u}\leavevmode\nobreak\ (\mathrm{mod}Y)\\\
\log_{q}\lVert\mathbf{x}\rVert\leq
N\end{subarray}}m(\mathbf{x})-\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\mathbf{y}\in
Y\\\ \log_{q}\lVert\mathbf{y}\rVert\leq N\end{subarray}}m(\mathbf{y})\Big{)}.$

######  Remarks:

  1. (i)

As one easily verifies, the choice of the norm $\lVert\mathrel{\;.\;}\rVert$
on the finite-dimensional $K_{\infty}$-vector space $V_{\infty}$ is
irrelevant, as all norms are equivalent. Typical norms on $V_{\infty}$ are the
$\lVert\mathrel{\;.\;}\rVert_{\boldsymbol{\omega}}$ with
$\boldsymbol{\omega}\in\Psi$, where
$\lVert\mathbf{x}\rVert_{\boldsymbol{\omega}}=\lvert\mathbf{x}\boldsymbol{\omega}\rvert$.

  2. (ii)

If the distribution $f$ is motivated by $m$, so is its derivative $g=Df$.

  3. (iii)

Given $m\colon V\smallsetminus\\{0\\}\to M$ such that the limits in (2.7.1)
(resp. (2.7.2)) always exist, it defines a distribution (resp. derived
distribution) on $\mathfrak{Y}$.

  4. (iv)

If the topology on $M$ is discrete, each limit $\lim_{N\to\infty}x_{N}$ is
stationary, that is, $\lim_{N\to\infty}x_{N}=x_{N}$ for $N$ sufficiently
large.

We will see at once that the division forms
$d_{\mathbf{u}}^{Y}(\boldsymbol{\omega})$ define a derived distribution. Later
it will come out that it is motivated by
$m\colon\mathbf{x}\mapsto\mathbf{x}\boldsymbol{\omega}$, but in general, it
isn’t the derivative of a distribution.

### 2.8.

At several occasions, we will make use of the following easily proved facts.
Let $a,b$ be natural numbers with $c\vcentcolon=\gcd(a,b)$. Then

(2.8.1) $\gcd(q^{a}-1,q^{b}-1)=q^{c}-1.$

Let $a_{i}\in C_{\infty}^{*}$ be finitely many elements with assigned weights
$k_{i}\in\mathds{Z}$ and the property that

$\prod a_{i}^{n_{i}}=1\qquad\text{whenever}\qquad\sum
n_{i}k_{i}=0\quad(n_{i}\in\mathds{Z}).$

Then there exists a unique $b\in C_{\infty}^{*}$ such that

(2.8.2) $a_{i}=b^{k_{i}/k}\quad(k\vcentcolon=\gcd(k_{i})),$

and $b$ lies in the group generated by $\\{a_{i}\\}$.

##  3\. Basic properties of the functions $d_{\mathbf{u}}^{Y}$ and
$\Delta^{Y^{\prime}|Y}$

######  Proposition 3.1:

Let a lattice pair $Y^{\prime}|Y$ and $\mathbf{v}\in V\smallsetminus
Y^{\prime}$ be given. Then the identity

(3.1.1) $\Delta^{Y^{\prime}|Y}\prod_{\begin{subarray}{c}\mathbf{u}\in V/Y\\\
\mathbf{u}\equiv\mathbf{v}\leavevmode\nobreak\
(\mathrm{mod}Y^{\prime})\end{subarray}}d_{\mathbf{u}}^{Y}=d_{\mathbf{v}}^{Y^{\prime}}$

holds.

###### Proof.

Fix $\boldsymbol{\omega}\in\Psi$ and let $\varphi$ be the unique normalized
isogeny from the Drinfeld module $\phi=\phi^{Y_{\boldsymbol{\omega}}}$ to
$\phi^{\prime}=\phi^{Y_{\boldsymbol{\omega}}^{\prime}}$. Then

(3.1.2) $e^{Y_{\boldsymbol{\omega}}^{\prime}}=\varphi\circ
e^{Y_{\boldsymbol{\omega}}}$

and, according to 1.5 and 1.6,

(3.1.3) $\varphi(X)=X\sideset{}{{}^{\prime}}{\prod}_{\mathbf{w}\in
Y^{\prime}/Y}\big{(}1-(d_{\mathbf{w}}^{Y})^{-1}X\big{)}=\Delta^{Y^{\prime}|Y}(\boldsymbol{\omega})\prod_{\mathbf{w}\in
Y^{\prime}/Y}(X-d_{\mathbf{w}}^{Y})$

(where $d_{0}^{Y}=0$). The $\mathbf{u}\in V/Y$ with
$\mathbf{u}\equiv\mathbf{v}\pmod{Y^{\prime}}$ are obtained by
$\mathbf{u}=\mathbf{v}-\mathbf{w}$ with $\mathbf{w}\in Y^{\prime}/Y$. Hence,
upon inserting
$d_{\mathbf{v}}^{Y}(\boldsymbol{\omega})=e^{Y_{\boldsymbol{\omega}}}(\mathbf{v}\boldsymbol{\omega})$,
the right hand side gives

$\displaystyle\Delta^{Y^{\prime}|Y}(\boldsymbol{\omega})\prod_{\mathbf{w}\in
Y^{\prime}/Y}\big{(}e^{Y_{\boldsymbol{\omega}}}(\mathbf{v}\boldsymbol{\omega})-e^{Y_{\boldsymbol{\omega}}}(\mathbf{w}\boldsymbol{\omega})\big{)}$
$\displaystyle=\Delta^{Y^{\prime}|Y}(\boldsymbol{\omega})\prod_{\begin{subarray}{c}\mathbf{u}\in
V\\\ \mathbf{u}\equiv\mathbf{v}\leavevmode\nobreak\
(\mathrm{mod}Y^{\prime})\end{subarray}}e^{Y_{\boldsymbol{\omega}}}(\mathbf{u}\boldsymbol{\omega})$
$\displaystyle=\Delta^{Y^{\prime}|Y}(\boldsymbol{\omega})\prod_{\mathbf{u}}d_{\mathbf{u}}^{Y_{\boldsymbol{\omega}}},$

while

$\varphi(e^{Y_{\boldsymbol{\omega}}}(\mathbf{v}\boldsymbol{\omega}))=e^{Y_{\boldsymbol{\omega}}^{\prime}}(\mathbf{v}\boldsymbol{\omega})=d_{\mathbf{v}}^{Y^{\prime}}(\boldsymbol{\omega}).$

∎

### 3.2.

The proposition states that $g(\mathbf{u},Y)\vcentcolon=d_{\mathbf{u}}^{Y}$
describes a derived distribution with values in the multiplicative group
$C_{\infty}^{*}$ (if the argument $\boldsymbol{\omega}$ of
$d_{\mathbf{u}}^{Y}$ is fixed), or in the multiplicative group of holomorphic
invertible functions $\mathcal{O}(\Psi)^{*}$ on $\Psi$ (if
$d_{\mathbf{u}}^{Y}$ is regarded as a function on $\Psi$).

We keep this dichotomy in what follows (regarding $\boldsymbol{\omega}$ either
as fixed or as a variable on $\Psi$), but write in general for simplicity only
the case of a constant $\boldsymbol{\omega}$. Then
$\Delta^{Y^{\prime}|Y}=(\sideset{}{{}^{\prime}}{\prod}_{\mathbf{u}\in
Y^{\prime}/Y}d_{\mathbf{u}}^{Y})^{-1}$ becomes the discriminant
$\operatorname{Disc}(g)^{Y^{\prime}|Y}$ as formally defined in (2.4.3), and
from (2.4.4) we find

(3.2.1)
$\Delta^{Y^{\prime\prime}|Y}=\Delta^{Y^{\prime\prime}|Y^{\prime}}\cdot(\Delta^{Y^{\prime}|Y})^{[Y^{\prime\prime}:Y]}$

if $Y^{\prime\prime}|Y^{\prime}|Y$.

Next we give an application of the derived distribution property of the
$d_{*}^{*}$ to the multiplicative Hecke action on the forms
$\Delta^{Y^{\prime}|Y}$.

### 3.3.

Let $\mathfrak{p}$ be a prime ideal of $A$ coprime with the lattice pair
$Y^{\prime}|Y$, which means that $\mathfrak{p}$ doesn’t divide the Euler-
Poincaré characteristic $\chi(Y^{\prime}|Y)$ of the finite $A$-module
$Y^{\prime}|Y$. We write $(\mathfrak{p},\chi(Y^{\prime}|Y))=1$ for this
property. Then canonically,

(3.3.1)
$Y/\mathfrak{p}Y\overset{\cong}{\longrightarrow}Y^{\prime}/\mathfrak{p}Y^{\prime},$

which is an $r$-dimensional vector space over the finite field
$\mathds{F}_{\mathfrak{p}}\vcentcolon=A/\mathfrak{p}$. An $A$-lattice $Z$ with
$\mathfrak{p}Y\subset Z\subset Y$ has type $i$ for $(Y,\mathfrak{p})$ if

(3.3.2) $\dim_{\mathds{F}_{\mathfrak{p}}}(Z/\mathfrak{p}Y)=i\quad(0\leq i<r).$

Then

(3.3.3) $Z\longmapsto Z^{\prime}\vcentcolon=Z+\mathfrak{p}Y^{\prime}$

is a well-defined bijection from the set $\mathcal{L}(Y,\mathfrak{p},i)$ of
lattices of type $i$ for $(Y,\mathfrak{p})$ to
$\mathcal{L}(Y^{\prime},\mathfrak{p},i)$, with inverse $Z^{\prime}\mapsto
Z\vcentcolon=Z^{\prime}\cap Y$. We have

(3.3.4)
$\\#\mathcal{L}(Y,\mathfrak{p},i)=\\#\operatorname{Gr}_{\mathds{F}_{\mathfrak{p}}}(r,i)=\vcentcolon
c_{i}^{(r)}(\mathfrak{p}),$

where $\operatorname{Gr}_{\mathds{F}_{\mathfrak{p}}}(r,i)$ is the Grassmannian
of the $i$-subspaces of $\mathds{F}_{\mathfrak{p}}^{r}$. Its cardinality
$c_{i}^{(r)}(\mathfrak{p})$ is given by a well-known formula ([28] Ch. III
Proposition 3.18; replace $p$ with
$q_{\mathfrak{p}}\vcentcolon=\\#\mathds{F}_{\mathfrak{p}}$), for which we
currently have no use.

The $\mathfrak{p}$-th Hecke correspondence of type $i$ ($0<i<r$) is the set-
valued map $Y\mapsto\mathcal{L}(Y,\mathfrak{p},i)$. Regarding a modular form
$f$ for $\Gamma_{Y}=\operatorname{GL}(Y)$ as a certain function
$f(\boldsymbol{\omega})=f(Y_{\boldsymbol{\omega}})$ on the set of lattices
$\Lambda=Y_{\boldsymbol{\omega}}$ isomorphic with $Y$, the $\mathfrak{p}$-th
Hecke Operator of type $i$ acts on $f$ through

(3.3.5)
$T_{\mathfrak{p},i}(f)(Y_{\boldsymbol{\omega}})=\sum_{Z\in\mathcal{L}(Y,\mathfrak{p},i)}f(Z_{\boldsymbol{\omega}}).$

(In the notation of Shimura [28] III, $T_{\mathfrak{p},i}$ would be written as
$T(1,\dots,1,\mathfrak{p},\dots,\mathfrak{p})$ with $i$ 1’s and $(r-i)$
$\mathfrak{p}$’s.)

### 3.4.

We postpone the study of Hecke operators in the above sense to possible future
work. Instead, we investigate products analogous with (3.3.5), and where $f$
is a discriminant $\Delta^{Y^{\prime}|Y}$. That is, we are concerned with a
multiplicative Hecke operator $T_{\mathfrak{p},i}^{*}$, where

(3.4.1)
$T_{\mathfrak{p},i}^{*}(f)(Y_{\boldsymbol{\omega}})=\prod_{Z\in\mathcal{L}(Y,\mathfrak{p},i)}f(Z_{\boldsymbol{\omega}}).$

Assuming the framework of 3.3, where $\mathfrak{p}$ is a prime of degree
$d\in\mathds{N}$, there is the following result.

######  Theorem 3.5:

The action of $T_{\mathfrak{p},i}^{*}$ on $\Delta^{Y^{\prime}|Y}$ is given by

(3.5.1)
$T_{\mathfrak{p},i}^{*}(\Delta^{Y^{\prime}|Y})=\left(\frac{\Delta^{\mathfrak{p}Y^{\prime}|\mathfrak{p}Y}}{\Delta^{Y^{\prime}|Y}}\right)^{e}(\Delta^{Y^{\prime}|Y})^{c_{i}^{(r)}(\mathfrak{p})}$

with the exponent $c_{i}^{(r)}(\mathfrak{p})$ of (3.3.4) and

(3.5.2) $e=c_{i}^{(r)}(\mathfrak{p})q^{d_{i}}(q^{d(r-i)}-1)(q^{dr}-1)^{{-}1}.$

###### Proof.

  1. (i)

We will use the so-far established properties of the $d_{*}^{*}$ and
$\Delta^{**}$ and the geometry of
$Y/\mathfrak{p}Y\cong\mathds{F}_{\mathfrak{p}}^{r}$. The frequently occurring
$c_{i}^{(r)}(\mathfrak{p})$ will be abbreviated by $c$. For
$Z\in\mathcal{L}(Y,\mathfrak{p},i)$, we let
$Z^{\prime}\in\mathcal{L}(Y^{\prime},\mathfrak{p},i)$ be as in (3.3.3). The
diagram of inclusions of lattices

(3.5.3)
${\mathfrak{p}Y^{\prime}}$${Z^{\prime}}$${Y^{\prime}}$${\mathfrak{p}Y}$${Z}$${Y}$

will be crucial.

  2. (ii)

We calculate

(3.5.4)
$\frac{T_{\mathfrak{p},i}^{*}(\Delta^{Y^{\prime}|Y})}{(\Delta^{Y^{\prime}|Y})^{c}}=\prod_{Z\in\mathcal{L}(Y,\mathfrak{p},i)}\left(\frac{\Delta^{Z^{\prime}|Z}}{\Delta^{Y^{\prime}|Y}}\right),$

taking into account that $\Delta^{Z^{\prime}|Z}$ and $\Delta^{Y^{\prime}|Y}$
may be written as products of division forms $d_{\mathbf{u}}^{\mathfrak{p}Y}$.

  3. (iii)

First, we choose representatives for $Y^{\prime}/\mathfrak{p}Y$ as follows.
From $(\mathfrak{p},\chi(L^{\prime}/L))=1$, the $A$-module
$Y^{\prime}/\mathfrak{p}Y$ splits canonically into its $\mathfrak{p}$-primary
part $Y/\mathfrak{p}Y$ and its non-$\mathfrak{p}$-part $\overline{Q}$, which
maps isomorphically onto $Y^{\prime}/Y$. Let $P$ be an $\mathds{F}$-subvector
space of $Y$ complementary with $\mathfrak{p}Y$, which we endow via
$P\overset{\simeq}{\to}Y/\mathfrak{p}Y$ with an
$\mathds{F}_{\mathfrak{p}}$-structure. In particular,
$Z\in\mathcal{L}(Y,\mathfrak{p},i)$ corresponds to an
$\mathds{F}_{\mathfrak{p}}$-subspace, labelled $\overline{Z}$, of dimension
$i$ of $P\cong\mathds{F}_{\mathfrak{p}}^{r}$. Further, we let $Q$ be an
$\mathds{F}$-vector space system of representatives for $\overline{Q}$ in
$Y^{\prime}$. Then

(3.5.5) $Y^{\prime}=\mathfrak{p}Y\oplus P\oplus Q,$

and for each $Z$, its associated $Z^{\prime}$ is $Z\oplus Q$.

  4. (iv)

Now

$(\Delta^{Y^{\prime}|Y})^{-1}=\sideset{}{{}^{\prime}}{\prod}_{\mathbf{u}\in
Q}d_{\mathbf{u}}^{Y}=\Big{(}\sideset{}{{}^{\prime}}{\prod}_{\mathbf{u}\in
Q}\prod_{\mathbf{v}\in
P}d_{\mathbf{u}+\mathbf{v}}^{\mathfrak{p}Y}\Big{)}(\Delta^{Y|\mathfrak{p}Y})^{\\#Q-1}.$

For each subset $W$ of $P\oplus Q\smallsetminus\\{0\\}$, put

(3.5.6) $\pi(W)\vcentcolon=\prod_{\mathbf{w}\in
W}d_{\mathbf{w}}^{\mathfrak{p}Y}.$

Thus the above becomes

(3.5.7) $(\Delta^{Y^{\prime}|Y})^{-1}=\pi(P\oplus Q\smallsetminus
P)\pi(P\smallsetminus\\{0\\})^{1-\\#Q}.$

  5. (v)

We aim to write $\prod_{Z}\Delta^{Z^{\prime}|Z}$ in the same format. For
$\mathbf{u}\in Q\smallsetminus\\{0\\}$,

$d_{\mathbf{u}}^{Z}=\Big{(}\prod_{\begin{subarray}{c}\mathbf{v}\in P\oplus
Q\\\ \mathbf{v}\equiv\mathbf{u}\leavevmode\nobreak\
(\mathrm{mod}\mathfrak{p}Y)\end{subarray}}d_{\mathbf{v}}^{\mathfrak{p}Y}\Big{)}\Delta^{Z|\mathfrak{p}Y}=\prod_{\mathbf{v}\in\overline{Z}}d_{\mathbf{u}+\mathbf{v}}^{\mathfrak{p}Y}(\pi(\overline{Z}\smallsetminus\\{0\\}))^{-1},$

and so

(3.5.8)
$\displaystyle\Big{(}\prod_{Z\in\mathcal{L}(Y,\mathfrak{p},i)}\Delta^{Z^{\prime}|Z}\Big{)}^{-1}$
$\displaystyle=\prod_{Z}\sideset{}{{}^{\prime}}{\prod}_{\mathbf{u}\in
Z^{\prime}/Z}d_{\mathbf{u}}^{Z}$
$\displaystyle=\prod_{Z}\sideset{}{{}^{\prime}}{\prod}_{\mathbf{u}\in
Q}\prod_{\mathbf{v}\in\overline{Z}}d_{\mathbf{u}+\mathbf{v}}^{\mathfrak{p}Y}/\Big{(}\prod_{Z}\pi(\overline{Z}\smallsetminus\\{0\\})^{\\#Q-1}\Big{)}.$

  6. (vi)

Let us first treat the denominator of (3.5.8). In the product

$\prod_{Z}\pi(\overline{Z}\smallsetminus\\{0\\})=\prod_{Z}\sideset{}{{}^{\prime}}{\prod}_{\mathbf{v}\in\overline{Z}}d_{\mathbf{v}}^{\mathfrak{p}Y}=\sideset{}{{}^{\prime}}{\prod}_{\mathbf{w}\in
P}(d_{\mathbf{w}}^{\mathfrak{p}Y})^{m},$

each subscript $0\neq\mathbf{w}\in P$ appears with multiplicity

(3.5.9) $m=c\cdot(q^{d_{i}}-1)/(\\#P-1)$

independently of $\mathbf{w}$. Here $c=c_{i}^{(r)}(\mathfrak{p})=\\#\\{Z\\}$,
$q^{d_{i}}=\\#\overline{Z}$, $\\#P=q^{d_{r}}$. Hence the denominator of
(3.5.8) is

(3.5.10)
$\prod_{Z}\pi(\overline{Z}\smallsetminus\\{0\\})^{\\#Q-1}=\pi(P\smallsetminus\\{0\\})^{m\cdot(\\#Q-1)}.$

  7. (vii)

The numerator of (3.5.8) is

$\prod_{Z}\sideset{}{{}^{\prime}}{\prod}_{\mathbf{u}\in
Q}\prod_{\mathbf{v}\in\overline{Z}}d_{\mathbf{u}+\mathbf{v}}^{\mathfrak{p}Y}=\sideset{}{{}^{\prime}}{\prod}_{\mathbf{w}\in
P\oplus Q}(d_{\mathbf{w}}^{\mathfrak{p}Y})^{m(\mathbf{w})}$

with certain multiplicities $m(\mathbf{w})$. Since $\mathbf{u}\neq 0$, the
subscript $\mathbf{w}=\mathbf{u}+\mathbf{v}$ always lies in $P\oplus
Q\smallsetminus P$. The value $\mathbf{w}=\mathbf{u}$ with $\mathbf{v}=0$
appears precisely once for each $Z$. Hence $m(\mathbf{w})=c=\\#\\{Z\\}$ if
$\mathbf{w}=\mathbf{u}\in Q\smallsetminus\\{0\\}$. On the other hand, if
$\mathbf{w}=\mathbf{u}+\mathbf{v}$ with $\mathbf{v}\neq 0$ (i.e.,
$\mathbf{w}\in P\oplus Q\smallsetminus P\smallsetminus Q$), then
$m(\mathbf{w})=c\cdot(q^{di}-1)/(q^{dr}-1)$, since the
$\\#\\{Z\\}(\\#(Q)-1)(q^{di}-1)$ values of $\mathbf{w}=\mathbf{u}+\mathbf{v}$
with $\mathbf{v}\neq 0$ in the triple product are evenly distributed over the
$\\#(P\oplus Q\smallsetminus P\smallsetminus Q)=(\\#Q-1)(\\#P-1)$ elements of
$P\oplus Q\smallsetminus P\smallsetminus Q$. So the numerator is

(3.5.11) $\prod_{Z}\sideset{}{{}^{\prime}}{\prod}_{\mathbf{u}\in
Q}\prod_{\mathbf{v}\in\overline{Z}}d_{\mathbf{u}+\mathbf{v}}^{\mathfrak{p}Y}=\pi(Q\smallsetminus\\{0\\})^{c}\pi(P\oplus
Q\smallsetminus P\smallsetminus Q)^{c(q^{di}-1)/(q^{dr}-1)}$

with $c=c_{i}^{(r)}(\mathfrak{p})$.

  8. (viii)

Combining formulas (3.5.7) to (3.5.11), the obvious

(3.5.12)
$\pi(W_{1}\mathbin{\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}}W_{2})=\pi(W_{1})\pi(W_{2})$

for $W_{1}=P\oplus Q\smallsetminus P\smallsetminus Q$,
$W_{2}=P\smallsetminus\\{0\\}$, and simplifying, one finds

(3.5.13)
$\prod_{Z\in\mathcal{L}(Y,\mathfrak{p},i)}\frac{\Delta^{Z^{\prime}|Z}}{\Delta^{Y^{\prime}|Y}}=\pi(P\oplus
Q\smallsetminus P\smallsetminus
Q)^{c\ell}\pi(P\smallsetminus\\{0\\})^{{-}c\ell(\\#Q-1)}$

with
$\ell\vcentcolon=1-(q^{d_{i}}-1)/(q^{d_{r}}-1)=q^{di}(q^{d(r-i)}-1)/(q^{dr}-1)$.

  9. (ix)

It remains to evaluate

$\displaystyle\pi(P\oplus Q\smallsetminus P\smallsetminus
Q)/\pi(P\smallsetminus\\{0\\})^{\\#Q-1}$ $\displaystyle=\frac{\pi(P\oplus
Q\smallsetminus\\{0\\})\pi^{-1}(Q\smallsetminus\\{0\\})}{\pi(P\smallsetminus\\{0\\})^{\\#Q}}$
$\displaystyle=(\Delta^{Y|\mathfrak{p}Y})^{\\#Q}(\Delta^{Y^{\prime}|\mathfrak{p}Y^{\prime}})^{-1}\Delta^{\mathfrak{p}Y^{\prime}|\mathfrak{p}Y}$
$\displaystyle=\Delta^{\mathfrak{p}Y^{\prime}|\mathfrak{p}Y}/\Delta^{Y^{\prime}|Y}$

by (3.2.1) and $[Y^{\prime}:Y]=\\#Q$. Together with (3.5.13), this gives

$\prod_{Z}\frac{\Delta^{Z^{\prime}|Z}}{\Delta^{Y^{\prime}|Y}}=\left(\frac{\Delta^{\mathfrak{p}Y^{\prime}|\mathfrak{p}Y}}{\Delta^{Y^{\prime}|Y}}\right)^{c\ell}$

with $c=c_{i}^{(r)}(\mathfrak{p})$ and $c\ell=e$ as stated in (3.5.1).

∎

######  Corollary 3.6:

In the situation of the theorem, assume that $\mathfrak{p}$ is principal,
generated by $n\in A$.111The letter $p$ is occupied by
$p=\operatorname{char}(\mathds{F})$. Then

(3.6.1)
$T_{\mathfrak{p},i}^{*}(\Delta^{Y^{\prime}|Y})=n^{e^{\prime}}(\Delta^{Y^{\prime}|Y})^{c}$

with the exponent $e^{\prime}=(1-[Y^{\prime}:Y])e$ and $e$ and
$c=c_{i}^{(r)}(\mathfrak{p})$ as in (3.5.1).

###### Proof.

We use $\mathfrak{p}Y^{\prime}=nY^{\prime}$, $\mathfrak{p}Y=nY$ and the fact
that $\Delta^{Y^{\prime}|Y}$ as a modular form has weight $[Y^{\prime}:Y]-1$.
This gives
$\Delta^{nY^{\prime}|nY}=n^{1-[Y^{\prime}:Y]}\Delta^{Y^{\prime}|Y}$, and thus
the assertion. ∎

######  Remarks 3.7:

  1. (i)

The assumption in 3.3 that $\mathfrak{p}$ is a prime ideal is not essential,
and was only made for reasons of presentation. Results generalizing Theorem
3.5 can be obtained for multiplicative Hecke operators
$T_{\mathfrak{n},i}^{*}$ for arbitrary ideals $\mathfrak{n}\subset A$.

  2. (ii)

Consider $\Delta^{Y^{\prime}|Y}$ as a modular form for the group
$\Gamma_{Y^{\prime}|Y}=\operatorname{GL}(Y)\cap\operatorname{GL}(Y^{\prime})$,
defined on $\Omega=C_{\infty}^{*}\backslash\Psi$. Let

$P\colon\mathcal{O}(\Omega)^{*}\longrightarrow\mathbf{H}(\mathcal{BT},\mathds{Z})$

be the van der Put transform from the multiplicative group of invertible
holomorphic functions on $\Omega$ to the additive group of $\mathds{Z}$-valued
harmonic $1$-cochains on the Bruhat-Tits building
$\mathcal{BT}=\mathcal{BT}^{r}$ of $\operatorname{PGL}(r,K_{\infty})$, see
[17] for details. It is $\operatorname{GL}(r,K_{\infty})$-equivariant and
defines a short exact sequence

${1}$${C_{\infty}^{*}}$${\mathcal{O}(\Omega)^{*}}$${\mathbf{H}(\mathcal{BT},\mathds{Z})}$${0.}$$\scriptstyle{P}$

Then $P(\Delta^{Y^{\prime}|Y})$ belongs to the invariants
$\mathbf{H}(\mathcal{BT},\mathds{Z})^{\Gamma_{Y^{\prime}|Y}}$. As
$T_{\mathfrak{p},i}^{*}(\Delta^{Y^{\prime}|Y})=\mathrm{const.}(\Delta^{Y^{\prime}|Y})^{c}$
if $\mathfrak{p}=(n)$ is principal, $P(\Delta^{Y^{\prime}|Y})$ is an eigenform
under the additive Hecke operator $T_{\mathfrak{p},i}$ that acts on
$\mathbf{H}(\mathcal{BT},\mathds{Z})^{\Gamma_{Y^{\prime}|Y}}$, with eigenvalue
$c=c_{i}^{(r)}(\mathfrak{p})$. It is a remarkable fact that this eigenvalue
depends only on $\lvert\mathfrak{p}\rvert=q^{d}$ and $i$, but neither on
$\mathfrak{p}$ itself nor on $Y^{\prime}|Y$.

######  Example 3.8:

We present the most simple example to which Theorem 3.5 applies, namely the
case where as in 1.2.2

(3.8.1) $A=\mathds{F}[T],\quad r=2,\quad Y=A^{2}\quad\text{and}\quad
Y^{\prime}=T^{-1}Y.$

Then $\Delta^{Y^{\prime}|Y}=\Delta_{(T)}=T^{-1}\Delta$, where
$\Delta=\Delta_{T}$ is the “usual” discriminant studied, e.g., in [13]. Let
$\mathfrak{p}=(n)$ be a prime different from $(T)$ (which is no restriction as
$\Delta_{T}=\Delta_{T-1}$), of degree $d=\deg\mathfrak{p}$. Let
$\\{\mathbf{e}_{1},\mathbf{e}_{2}\\}$ be the standard basis of $Y$. The
$c_{1}^{(2)}(\mathfrak{p})=q^{d}+1$ elements $Z$ of
$\mathcal{L}(Y,\mathfrak{p})\vcentcolon=\mathcal{L}(Y,\mathfrak{p},1)$ are the
sublattices $Z_{a}$ ($a\in A\mid\deg a<d$) and $Z_{\infty}$, where

(3.8.2) $Z_{a}=A(\mathbf{e}_{1}+a\mathbf{e}_{2})\oplus An\mathbf{e}_{2},\quad
Z_{\infty}=An\mathbf{e}_{1}\oplus A\mathbf{e}_{2}.$

By Corollary 3.6, the operator
$T_{\mathfrak{p}}^{*}\vcentcolon=T_{\mathfrak{p},1}^{*}$ acts as

(3.8.3)
$T_{\mathfrak{p}}^{*}(\Delta)=n^{e}\Delta^{q^{d}+1}\quad\text{with}\quad
e={-}(q^{2}-1)q^{d}.$

Put $\boldsymbol{\omega}=(\omega,1)$, where
$\omega\in\Omega=\Omega^{2}=C_{\infty}\smallsetminus K_{\infty}$, and write
$Z_{a,\boldsymbol{\omega}}$ for
$(Z_{a})_{\boldsymbol{\omega}}=A(\omega+a)\oplus An$ and
$Z_{\infty,\boldsymbol{\omega}}=(Z_{\infty})_{\boldsymbol{\omega}}=An\omega\oplus
A$. Then

$\displaystyle\Delta(Z_{a,\boldsymbol{\omega}})$
$\displaystyle=\Delta(nZ^{\prime}_{a,\boldsymbol{\omega}}),$
$\displaystyle\text{where
}Z_{a,\boldsymbol{\omega}}^{\prime}=n^{-1}Z_{a,\boldsymbol{\omega}}=A\left(\frac{\omega+a}{n}\right)\oplus
A$ $\displaystyle=n^{1-q^{2}}\Delta(Z_{a,\boldsymbol{\omega}}^{\prime})$
$\displaystyle=n^{1-q^{2}}\Delta\left(\frac{\omega+a}{n}\right)$ and
$\displaystyle\Delta(Z_{\infty,\boldsymbol{\omega}})$
$\displaystyle=\Delta(n\omega),$

translating between functions on $\\{\text{lattices in }C_{\infty}\\}$ and on
$\Omega$. Hence, as functions on $\Omega$,

$\displaystyle n^{(1-q^{2})q^{d}}\Delta^{q^{d}+1}(\omega)$
$\displaystyle=T_{\mathfrak{p}}^{*}(Y_{\boldsymbol{\omega}})$
$\displaystyle=\Delta(Z_{\infty,\boldsymbol{\omega}})\prod_{a}\Delta(Z_{a,\boldsymbol{\omega}})$
$\displaystyle=\Delta(n\omega)\prod_{a}n^{(1-q^{2})}\Delta\left(\frac{\omega+a}{n}\right)=n^{(1-q^{2})q^{d}}\Delta(n\omega)\prod_{a}\Delta\left(\frac{\omega+a}{n}\right)$

and finally the functional equation

(3.8.4)
$\Delta^{q^{d}+1}(\omega)=\Delta(n\omega)\prod_{\begin{subarray}{c}a\in A\\\
\deg a<n\end{subarray}}\Delta\left(\frac{\omega+a}{n}\right)$

for $\Delta$ found in [14] Corollary 2.11. Note that this formula was then a
consequence of deep results about conditionally convergent Eisenstein series
on the Bruhat-Tits tree $\mathcal{T}=\mathcal{BT}^{2}$ and their behavior
under the Hecke operators. Here, the other way round, the far more general
Theorem 3.5 implies certain Hecke eigenvalues, cf. Remarks 3.7(ii). The
functional equation (3.8.4) is analogous with the formula

(3.8.5) $\Delta^{p+1}(z)=\varepsilon(p)\Delta(pz)\prod_{0\leq
i<p}\Delta\left(\frac{z+i}{p}\right)$

for the classical elliptic discriminant $\Delta(z)$ on the complex upper half-
plane, where $p\in\mathds{N}$ is a prime number and $\varepsilon(p)={-}1$ if
$p=2$ and $\varepsilon(p)=1$ otherwise. It may be derived from Jacobi’s
product formula $\Delta=(2\pi\imath)^{12}q\prod_{n\geq 1}(1-q^{n})^{24}$;
unfortunately, the author was unable to find a reference to the certainly
well-known (3.8.5).

##  4\. $Z$-functions of lattices in $C_{\infty}$

In the whole section, $\Lambda$ is a fixed $A$-lattice of rank
$r\in\mathds{N}$ in $C_{\infty}$.

### 4.1.

For $N\in\mathds{Q}$, let

(4.1.1) $\Lambda_{N}\vcentcolon=\\{\lambda\in\Lambda\mid\log\lambda\leq N\\},$

a finite-dimensional $\mathds{F}$-vector space. There are various methods to
encode the shape of $\Lambda$, among which the $\mathds{F}$-spectrum
$\operatorname{Spec}_{\mathds{F}}(\Lambda)$ as in [18] 1.9, the growth
function $s\mapsto\dim_{\mathds{F}}(\Lambda_{s})$, or the $Z$-function
$Z_{\Lambda}$ of $\Lambda$ defined below, which is particularly well-suited
for our present purposes. All these are equivalent.

### 4.2.

Let $S$ be an indeterminate, and choose a compatible system $\\{S^{1/n}\mid
n\in\mathds{N}\\}$ of $n$-th roots of $S$, that is, $(S^{1/n})^{n/m}=S^{1/m}$
if $m\mid n$. This is possible by Zorn’s lemma. Define

(4.2.1)
$\mathds{Z}\\{\\{S\\}\\}\vcentcolon=\bigcup_{n\in\mathds{N}}\mathds{Z}((S^{1/n}))$

as the ring of formal Puiseux series in $S$ over $\mathds{Z}$. Here, as usual,
$\mathds{Z}((X))$ is the ring of formal Laurent series in $X$.

As $\Lambda$ is finitely generated as an $A$-module, there exists a common
denominator $n\in\mathds{N}$ for the set $\\{\log\lambda\mid
0\neq\lambda\in\Lambda\\}\subset\mathds{Q}$. Hence $S^{\log\lambda}$ is a
well-defined power of $S^{1/n}$. We define the $Z$-function of  $\Lambda$ as

(4.2.2)
$Z_{\Lambda}(S)=\sideset{}{{}^{\prime}}{\sum}_{\lambda\in\Lambda}S^{\log\lambda}$

and, more generally, for $u\in K\Lambda$,

$Z_{u,\Lambda}(S)=\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\lambda\in
K\Lambda\\\ \lambda\equiv u\leavevmode\nobreak\
(\mathrm{mod}\Lambda)\end{subarray}}S^{\log\lambda}.$

Then $Z_{0,\Lambda}=Z_{\Lambda}$, $Z_{\Lambda}$ and $Z_{u,\Lambda}$ are well-
defined formal Laurent series in some $S^{1/n}$, and therefore Puiseux series
in $S$. Furthermore, $Z_{\Lambda}$ and $Z_{u,\Lambda}$ agree up to a finite
number of terms, since almost all $\lambda\in\Lambda$ satisfy
$\log\lambda>\log u$.

### 4.3.

Writing $\Lambda=Y_{\boldsymbol{\omega}}$, where $\boldsymbol{\omega}\in\Psi$
is fixed, $u\in K\Lambda$ equals $\mathbf{u}\boldsymbol{\omega}$ with
$\mathbf{u}\in V$, and the map

(4.3.1) $(\mathbf{u},Y)\longmapsto
Z_{\mathbf{u},Y}(\boldsymbol{\omega},S)\vcentcolon=Z_{\mathbf{u}\boldsymbol{\omega},Y_{\boldsymbol{\omega}}}(S)$

satisfies the rules (2.1.1) and (2.1.2). Hence it defines a distribution on
$\mathfrak{Y}$ with values in $\mathds{Z}((S^{1/n}))$. (If we allow
$\boldsymbol{\omega}$ to vary over $\Psi$, then the denominator $n$ is
unbounded, and we must replace $\mathds{Z}((S^{1/n}))$ with
$\mathds{Z}\\{\\{S\\}\\}$. This will however play no role in the present
paper.) We also note the homogeneity property

(4.3.2) $Z_{cu,c\Lambda}(S)=S^{\log c}Z_{u,\Lambda}(S)$

for $0\neq c\in C_{\infty}$.

Let $\Lambda$ and $u\in K\Lambda$ be given, and assume that $n\in\mathds{N}$
is a common denominator of $\log u$ and $\log\lambda$
($0\neq\lambda\in\Lambda$). After some preparation, we will show:

######  Theorem 4.4:

  1. $\mathrm{(i)}$

$Z_{u,\Lambda}(S)\in\mathds{Z}((S^{1/n}))$ is in fact a rational function in
$S^{1/n}$. More precisely,

(4.4.1)
$Z_{u,\Lambda}(S)(1-q_{\infty}^{r}S^{d_{\infty}})\in\mathds{Z}[S^{1/n},S^{{-}1/n}]$

is a Laurent polynomial with non-negative coefficients.

  2. $\mathrm{(ii)}$

$Z_{u,\Lambda}(1)={-}1$ if $u\in\Lambda$ and $0$ otherwise.

  3. $\mathrm{(iii)}$

For $N\in\mathds{Q}$ large enough, we have the identity

(4.4.2) $\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\lambda\in
K\Lambda\\\ \lambda\equiv u\leavevmode\nobreak\ (\mathrm{mod}\Lambda)\\\
\log\lambda\leq
N\end{subarray}}\log\lambda-\sideset{}{{}^{\prime}}{\sum}_{\lambda\in\Lambda_{N}}\log\lambda=Z_{u,\Lambda}^{\prime}(1)-Z_{0,\Lambda}^{\prime}(1),$

where $Z_{*,*}^{\prime}=\frac{\mathop{d}}{\mathop{dS}}Z_{**}$.

  4. $\mathrm{(iv)}$

For $c\in C_{\infty}^{*}$,

(4.4.3) $Z_{cu,c\Lambda}^{\prime}(1)\text{ equals
}Z_{u,\Lambda}^{\prime}(1)\text{ if }u\notin\Lambda\text{ and
}Z_{u,\Lambda}^{\prime}(1)-\log c\text{ if }u\in\Lambda.$

######  Remark:

In view of (i), $Z_{u,\Lambda}$ and its derivative may be evaluated at $S=1$,
i.e., at $S^{1/n}=1$.

### 4.5.

For a given rational number $N$ and $d\in\mathds{N}$, let

$N<x_{1}<x_{2}<\cdots<x_{s}\leq N+d$

be the rational numbers that appear in the half-open interval $(N,N+d]$ as
$x_{i}=\log\lambda$ for some $\lambda\in\Lambda$. Let $m_{1},\dots,m_{s}$ be
the respective multiplicities,
$m_{i}\vcentcolon=\\#\\{\lambda\in\Lambda\mid\log\lambda=x_{i}\\}$. We call
$((x_{1},m_{1}),\cdots,(x_{s},m_{s}))$ the $(N,d)$-spectrum
$\operatorname{Spec}_{(N,d)}(\Lambda)$ of $\Lambda$. We say that $\Lambda$ is
ultimately quasi-periodic with quasi-period $d$ (brief: uqp-$d$) if there is
some $N_{0}\in\mathds{Q}$ such that for all $N\geq N_{0}$, the following
relation holds:

(4.5.1) $\text{If
}\operatorname{Spec}_{(N,d)}(\Lambda)=((x_{1},m_{1}),\dots,(x_{s},m_{s})),\\\
\text{ then
}\operatorname{Spec}_{(N+d,d)}(\Lambda)=((x_{1}+d,q^{rd}m_{1}),\dots,(x_{s}+d,q^{rd}m_{s})).$

If this is the case for some $d$, then we call $N_{0}$ sufficiently large for
$\Lambda$, written $N_{0}\gg_{\Lambda}0$, or simply $N_{0}\gg 0$ if $\Lambda$
is fixed. The following are easy to verify.

#### 4.5.2.

If $\Lambda$ is uqp-$d$, then the occurring spectral values
$x_{1},\dots,x_{s}>N_{0}$ are well-defined modulo $d\mathds{Z}$ (however, the
numbering may depend on $N$);

#### 4.5.3.

if $\Lambda$ is upq-$d$ for $d=d_{1},d_{2}\in\mathds{N}$, then it is uqp-$g$
for $g\vcentcolon=\gcd(d_{1},d_{2})$;

#### 4.5.4.

if $r=\operatorname{rk}_{A}(\Lambda)=1$, then $\Lambda$ is uqp-$d$ for each
$d=\deg a$, where $a\in A\smallsetminus\mathds{F}$;

#### 4.5.5.

being uqp-$d$ is stable under taking orthogonal direct sums of lattices in
$C_{\infty}$.

(The direct sum $\Lambda=\oplus\Lambda^{(i)}$ of lattices $\Lambda^{(i)}$ is
orthogonal, written $\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}}
}}}\displaylimits\Lambda^{(i)}$, if for each $\lambda\in K_{\infty}\Lambda$,
$\lambda=\sum\lambda_{i}$ with $\lambda_{i}\in K_{\infty}\Lambda^{(i)}$, the
rule $\lvert\lambda\rvert=\max_{i}\lvert\lambda_{i}\rvert$ holds.)

Here 4.5.4 is a consequence of the Riemann-Roch theorem, which implies that
$\ell(N)\vcentcolon=\dim_{\mathds{F}}(\Lambda_{N})$ grows as
$\ell(N+d)=\ell(N)+d$ for $N\gg 0$. Hence

#### 4.5.6.

in case $r=1$, $\Lambda$ is uqp-$d_{\infty}$, as $d_{\infty}=\gcd\\{\deg a\mid
0\neq a\in A\\}$.

### 4.6.

We want to show that $\Lambda$ is always uqp-$d_{\infty}$. By the preceding,
this was obvious if we could write $\Lambda$ as an orthogonal sum
$\Lambda^{(1)}\perp\cdots\perp\Lambda^{(r)}$ of one-dimensional
$\Lambda^{(i)}$. This is always possible if the base ring $A$ is a polynomial
ring $A=\mathds{F}[T]$, due to the existence of successive minimum bases (see
[16] Sect. 3), but is delicate in general. So we reduce the wanted assertion
to the polynomial case.

### 4.7.

Let $a\in A$ be non-constant, of degree $d\in\mathds{N}$, and put
$A_{0}\vcentcolon=\mathds{F}[a]\subset A$,
$K_{0}\vcentcolon=\mathds{F}(a)=\operatorname{Quot}A_{0}$,
$K_{0,\infty}\vcentcolon=\mathds{F}((a^{-1}))=$ the completion of the rational
function field $K_{0}$ at its infinite place $\infty_{0}$. Consider the
diagram of ring/field extensions

(4.7.1) ${A}$${K}$${K_{\infty}}$${A_{0}}$${K_{0}}$${K_{0,\infty}.}$

It has the following properties:

#### 4.7.2.

“$\infty$” is the unique place of $K$ above the place “$\infty_{0}$” of
$K_{0}$;

#### 4.7.3.

$d=\operatorname{rk}_{A_{0}}(A)=[K:K_{0}]=[K_{\infty}:K_{0,\infty}]=e\cdot f$,
where $e={-}v_{\infty}(a)=d/d_{\infty}$ is the ramification index and
$f=d_{\infty}$ the residue class degree at $\infty_{0}$.

Let $\lvert\mathrel{\;.\;}\rvert_{0}$ be the absolute value on $K_{0,\infty}$
with its natural normalization $\lvert a\rvert_{0}=q$ and $\log_{0}$ the
logarithm function $\log_{0}x=\log_{q}\lvert x\rvert_{0}$ on $C_{\infty}$.
Then

(4.7.4) $\log=d\cdot\log_{0}.$

Now $\operatorname{rk}_{A_{0}}(\Lambda)=d\operatorname{rk}_{A}(\Lambda)=dr$,
and the $A_{0}$-module $\Lambda$ may be written as
$\Lambda=\mathop{\mathchoice{\vbox{
\hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{1\leq
i\leq dr}\Lambda^{(i)}$ with one-dimensional $\Lambda^{(i)}$. We have

(4.7.5) $\Lambda_{N}=\\{\lambda\in\Lambda\mid\log\lambda\leq
N\\}=\\{\lambda\in\Lambda\mid\log_{0}\lambda\leq
N/d\\}=\mathop{\mathchoice{\vbox{ \hbox{\scalebox{2.1}{$\displaystyle\perp$}}
}}{\vbox{ \hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{
\hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}}
}}}\displaylimits\Lambda_{N/d}^{(i)},$

as the sum of $A_{0}$-modules is orthogonal. Since $\Lambda^{(i)}$ is uqp-1,
$\Lambda$ is uqp-1 as $A_{0}$-module, and then uqp-$d$ as an $A$-module.

Together with the preceding considerations, we find:

######  Proposition 4.8:

Each $A$-lattice $\Lambda$ of rank $r$ in $C_{\infty}$ is ultimately quasi-
periodic with quasi-period $d_{\infty}$.

### 4.9.

Now we come to the

###### Proof of Theorem 4.4.

  1. (i)

Let $N_{0}\geq\log u$ be a rational number such that (4.5.1) is fulfilled for
$N\geq N_{0}$ and $d=d_{\infty}$. Write

(4.9.1) $Z_{u,\Lambda}(S)=f_{u,N}(S^{1/n})+g_{N}(S^{1/n})$

with

(4.9.2)
$f_{u,N}(S^{1/n})\vcentcolon=\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\lambda\in
K\Lambda\\\ \lambda\equiv u\leavevmode\nobreak\ (\mathrm{mod}\Lambda)\\\
\log\lambda\leq N\end{subarray}}S^{\log\lambda},$

a Laurent polynomial in $S^{1/n}$, and the tail $g_{N}(S^{1/n})$, which
doesn’t depend on $u$. Here $N\geq N_{0}$, and $n$ is a common denominator for
the appearing $\log\lambda$.

Let
$\operatorname{Spec}_{(N,d_{\infty})}(\Lambda)=((x_{1},m_{1}),\dots,(x_{s},m_{s}))$.
Due to (4.5.1), we have

(4.9.3)
$\begin{split}g_{N}(S^{1/n})&=(m_{1}S^{x_{1}}+\cdots+m_{s}S^{x_{s}})(1+q_{\infty}^{r}S^{d_{\infty}}+q^{2r}S^{2d_{\infty}}+\cdots)\\\
&=(m_{1}S^{x_{1}}+\cdots+m_{s}S^{x_{s}})(1-q_{\infty}^{r}S^{d_{\infty}})^{-1},\end{split}$

which gives (i).

  2. (ii)

Assume $u\in\Lambda$. Then

$Z_{\Lambda}(1)=f_{0,N}(1)+g_{N}(1)=\\#(\Lambda_{N}\smallsetminus\\{0\\})+\\#(\Lambda_{N+d_{\infty}}\smallsetminus\Lambda_{N})(1-q_{\infty}^{r})^{-1}={-}1,$

as
$\dim_{\mathds{F}}\Lambda_{N+d_{\infty}}=rd_{\infty}+\dim_{\mathds{F}}\Lambda_{N}$.

Let now $u\notin\Lambda$. Since $\lambda\mapsto u+\lambda$ is a bijection of
$\Lambda_{N}$ with the set $\\{\lambda^{\prime}\in
K\Lambda\mid\lambda^{\prime}\equiv u\pmod{\Lambda}\text{ and
}\log\lambda^{\prime}\leq N\\}$, the Laurent polynomial $f_{u,N}$ contains
exactly one term more than the corresponding polynomial $f_{0,N}$ for
$Z_{0,\Lambda}=Z_{\Lambda}$, viz, the term $S^{\log u}$. Hence
$Z_{u,\Lambda}(1)=Z_{0,\Lambda}(1)+1=0$.

  3. (iii)

We have $\frac{\mathop{d}}{\mathop{dS}}(S^{\log\lambda})|_{S=1}=\log\lambda$,
and the stated formula (4.4.2) follows from
$Z_{u,\Lambda}(S)-Z_{0,\Lambda}(S)=f_{u,N}(S^{1/n})-f_{0,N}(S^{1/n})$ for
$N\gg 0$.

  4. (iv)

follows from (4.3.2) and (ii).

∎

######  Remark 4.10:

By (4.3.1) for each $\boldsymbol{\omega}\in\Psi$, the rule
$f\colon(\mathbf{u},Y)\mapsto
Z_{\mathbf{u}\boldsymbol{\omega},Y_{\boldsymbol{\omega}}}^{\prime}(1)$ yields
a $\mathds{Q}$-valued distribution. The formula (4.4.2) in Theorem 4.4 may be
stated as: The derivative $g=Df$ of the distribution $f$ is motivated by
$m\colon V\smallsetminus\\{0\\}\to\mathds{Q}$,
$m(\mathbf{x})=\log\mathbf{x}\boldsymbol{\omega}$.

##  5\. $1$-units

Having the distributions $(Z_{*,*})$ and $(Z_{*,*}^{\prime}(1))$ with values
in characteristic zero, we next define a similar distribution with values in
pro-$p$-groups.

### 5.1.

We make the following choices:

  * •

$\pi\in K$, a uniformizer at $\infty$;

  * •

for each $n\in\mathds{N}$, an $n$-th root $\pi^{1/n}\in C_{\infty}$ of $\pi$,
and such that for $m\mid n$, the rule $(\pi^{1/n})^{n/m}=\pi^{1/m}$ is
satisfied.

As with roots $S^{1/n}$ of $S$ in the last section, this requires Zorn’s lemma
in the general case, but as before, we will be working only on a finite level
$n$.

Then arbitrary powers $\pi^{x}$ ($x\in\mathds{Q}$) are defined, with the usual
rules holding. As the group $\log(C_{\infty}^{*})$ equals $\mathds{Q}$, we may
split

(5.1.1)
$\begin{split}C^{*}_{\infty}&\overset{\cong}{\longrightarrow}\pi^{\mathds{Q}}\times
U(C_{\infty}),\\\ z&\longmapsto(\pi^{{-}\log z/d_{\infty}},z\pi^{\log
z/d_{\infty}})\end{split}$

where $U(C_{\infty})=\\{z\in C_{\infty}\mid\lvert z\rvert=1\\}$ is the unit
sphere in $C_{\infty}$. For each $z\in U(C_{\infty})$, there exists a unique
root of unity $\operatorname{sgn}(z)$ such that $\operatorname{sgn}(z)^{-1}z$
is congruent to 1 modulo the valuation ideal of $C_{\infty}$, i.e., such that
$\lvert\operatorname{sgn}(z)^{-1}z-1\rvert<1$. This gives a surjective
homomorphism

(5.1.2) $\operatorname{sgn}\colon U(C_{\infty})\longrightarrow\mu(C_{\infty})$

to the group $\mu(C_{\infty})$ of roots of unity in $C_{\infty}$. Let
$U^{(1)}(C_{\infty})$ be its kernel, the group of $1$-units of $C_{\infty}$.
We extend $\operatorname{sgn}$ to $C_{\infty}^{*}$ by decreeing
$\operatorname{sgn}(\pi^{x})=1$ for each $x\in\mathds{Q}$. Together, we find a
decomposition

(5.1.3)
$\begin{split}C_{\infty}^{*}&\overset{\cong}{\longrightarrow}\mathds{Q}\times\mu(C_{\infty})\times
U^{(1)}(C_{\infty})\\\ z&\longmapsto({-}\log
z/d_{\infty},\operatorname{sgn}(z),\langle z\rangle),\end{split}$

where $\langle z\rangle\vcentcolon=z\cdot\pi^{\log
z/d_{\infty}}\operatorname{sgn}(z)^{-1}$ is the $1$-unit part of $z$. Note
that the component ${-}\log z/d_{\infty}$ is intrinsic, while
$\operatorname{sgn}(z)$ and $\langle z\rangle$ depend on the choice of $\pi$
and its roots. We also observe

#### 5.1.4.

The decomposition (5.1.3) imposes the discrete topology to the factors
$\mathds{Q}$ and $\mu(C_{\infty})$.

### 5.2.

In what follows, we fix an $A$-lattice $\Lambda\subset C_{\infty}$ of rank
$r$, and put for rational numbers $N<N^{\prime}$

(5.2.1)
$\Lambda_{N,N^{\prime}}=\Lambda_{N^{\prime}}\smallsetminus\Lambda_{N}.$

For some $a\in A$ of degree $d>0$, choose a section $s$ of the
$\mathds{F}$-linear map $\Lambda\to\Lambda/a\Lambda$, and let
$R\vcentcolon=s(\Lambda/a\Lambda)$. We assume that $N\gg_{\Lambda}0$ and also

(5.2.2) $N\geq C_{0}\vcentcolon=\max_{x\in R}\log x.$

Then

(5.2.3) $\begin{split}\Lambda_{N,N+d}\times
R&\overset{\cong}{\longrightarrow}\Lambda_{N+d,N+2d}\\\
(\lambda,x)&\longmapsto a\lambda+x\end{split}$

is bijective. Due to our assumptions,

(5.2.4) $\log(a\lambda+x)=\log
a+\log\lambda\quad\text{and}\quad\operatorname{sgn}(a\lambda+x)=\operatorname{sgn}(a)\operatorname{sgn}(\lambda).$

Let

(5.2.5) $H(X)\vcentcolon=\prod_{x\in
R}(X-x)=X^{q^{rd}}+h_{rd-1}X^{q^{rd-1}}+\cdots+h_{0}X$

be the $\mathds{F}$-linear polynomial corresponding to $R$. Its coefficients,
being composed of elementary symmetric functions in the $x$, satisfy

(5.2.6) $\log h_{i}\leq(q^{rd}-q^{i})C_{0}\quad(0\leq i<rd).$

For $\lambda\in\Lambda_{N,N+d}$ we find with easy (and omitted) estimates:

(5.2.7) $H(a\lambda)/(a\lambda)^{q^{rd}}=1+O(q^{{-}N})$

(that is, there exists $C_{1}>0$ such that the error term is $\leq
C_{1}q^{{-}N}$.) Therefore, by (5.2.4),

(5.2.8) $\prod_{x\in R}\langle
a\lambda+x\rangle=\operatorname{sgn}(a\lambda)^{{-}q^{rd}}\pi^{q^{rd}\log(a\lambda)/d_{\infty}}H(a\lambda)=\langle
a\lambda\rangle^{q^{rd}}(1+O(q^{{-}N})).$

Put

(5.2.9)
$U_{\Lambda,N}\vcentcolon=\sideset{}{{}^{\prime}}{\prod}_{\lambda\in\Lambda_{N}}\langle\lambda\rangle,\qquad
U_{\Lambda,N,N^{\prime}}\vcentcolon=\prod_{\lambda\in\Lambda_{N,N^{\prime}}}\langle\lambda\rangle,\quad\text{and}\quad
U_{u,\Lambda,N}\vcentcolon=\sideset{}{{}^{\prime}}{\prod}_{\begin{subarray}{c}\lambda\in
K\Lambda\\\ \lambda\equiv u\leavevmode\nobreak\ (\mathrm{mod}\Lambda)\\\
\log\lambda\leq N\end{subarray}}\langle\lambda\rangle.$

Taking the product of (5.2.8) over $\lambda\in\Lambda_{N,N+d}$ yields

(5.2.10) $U_{\Lambda,N+d,N+2d}=\langle
a\rangle^{q^{rd}\\#(\Lambda_{N,N+d})}(U_{\Lambda,N,N+d})^{q^{rd}}(1+O(q^{{-}N})).$

Now, as $\langle a\rangle$ and $U_{\Lambda,N}$ are $1$-units, we see that

(5.2.11) $\lim_{k\to\infty}U_{\Lambda,N+kd,N+(k+1)d}=1$

and thus $\lim_{k\to\infty}U_{\Lambda,N+kd}$ exists. One easily shows that
neither the convergence nor the value of the limit depends on the shape $N+kd$
of the sequence approaching $\infty$. Thus finally

(5.2.12) $U_{\Lambda}\vcentcolon=\lim_{N\to\infty}U_{\Lambda,N}\in
U^{(1)}(C_{\infty})$

is well-defined. With the same reasoning, we get for each $u\in K\Lambda$ the
existence of the limit

(5.2.13) $U_{u,\Lambda}\vcentcolon=\lim_{N\to\infty}U_{u,\Lambda,N}$

as an element of $U^{(1)}(C_{\infty})$.

######  Proposition 5.3:

  1. $\mathrm{(i)}$

For each $u\in K\Lambda$, the limit
$U_{u,\Lambda}=\lim_{N\to\infty}U_{u,\Lambda,N}$ exists.

  2. $\mathrm{(ii)}$

$U_{u,\Lambda}$ depends only on the class of $u$ modulo $\Lambda$.

  3. $\mathrm{(iii)}$

For a lattice pair $\Lambda\subset\Lambda^{\prime}\subset K\Lambda$ and $v\in
K\Lambda$, the distribution relation

$\prod_{\begin{subarray}{c}u\in K\Lambda\\\ u\equiv v\leavevmode\nobreak\
(\mathrm{mod}\Lambda^{\prime})\end{subarray}}U_{u,\Lambda}=U_{v,\Lambda^{\prime}}\quad\text{holds.}$

  4. $\mathrm{(iv)}$

If $c\in C_{\infty}^{*}$ then

$U_{cu,c\Lambda}=\begin{cases}U_{u,\Lambda}&(u\notin\Lambda)\\\ \langle
c\rangle^{-1}U_{u,\Lambda}&(u\in\Lambda).\end{cases}$

###### Proof.

(i) has been shown, and (ii) and (iii) are then easy consequences. As to (iv):
We have $U_{cu,c\Lambda}=\lim_{N\to\infty}\langle
c\rangle^{i(N)}U_{u,\Lambda}$ with $i(N)=\\#(\Lambda_{N})$ in case
$u\notin\Lambda$ and $i(N)=\\#(\Lambda_{N}\smallsetminus\\{0\\})$ in case
$u\in\Lambda$. Now $\langle c\rangle^{\\#(\Lambda_{N})}$ converges to 1, as
$\\#(\Lambda_{N})$ is a power of $q$ growing with $N$. ∎

The proposition assures that for given $\boldsymbol{\omega}\in\Psi$, the rule
$f\colon(\mathbf{u},Y)\mapsto
U_{\mathbf{u}\boldsymbol{\omega},Y_{\boldsymbol{\omega}}}$ is a distribution
on $\mathfrak{Y}$ with values in $U^{(1)}(C_{\infty})$. It is motivated by
$m\colon V\smallsetminus\\{0\\}\to U^{(1)}(C_{\infty})$,
$m(\mathbf{x})=\langle\mathbf{x}\boldsymbol{\omega}\rangle$.

##  6\. Roots of unity

We would like to dispose of a similar distribution with values in
$\mu(C_{\infty})$ motivated by $\operatorname{sgn}(\lambda)$, where
$0\neq\lambda\in\Lambda=Y_{\boldsymbol{\omega}}$. This fails however, due to
the lacking of an analogue of (5.2.11) for the sign function
$\operatorname{sgn}$.

### 6.1.

For each subset $S$ of $C_{\infty}$, define

(6.1.1) $\operatorname{sgn}(S)\vcentcolon=\text{the subgroup of
$\mu(C_{\infty})$ generated by $\operatorname{sgn}(x)$, where $0\neq x\in
S$}.$

If $S$ is finite, then put

(6.1.2) $\varepsilon(S)\vcentcolon=\sideset{}{{}^{\prime}}{\prod}_{x\in
S}\operatorname{sgn}(x).$

Fix an $A$-lattice $\Lambda\subset C_{\infty}$ of rank $r$. We first observe:

######  Proposition 6.2:

$\operatorname{sgn}(\Lambda)$ is finite.

###### Proof.

Let $a\in A$ of degree $d>0$ and $N\gg 0$ such that (5.2.3) is fulfilled.
Proceeding as in 5.2, we find that $\operatorname{sgn}(\Delta_{N+2d})$ is
generated by $\operatorname{sgn}(\Lambda_{N+d})$ and
$\operatorname{sgn}(A)=\mu_{w}$, the group of $w$-th root of unity,
$w=q_{\infty}-1$. Now use induction. ∎

### 6.3.

In what follows, we assume that $N\in\mathds{Q}$ is large enough with respect
to the lattices $\Lambda\subset\Lambda^{\prime}$, the choices of $a\in A$ of
degree $d>0$, and of the system $R$ of representatives for $\Lambda/a\Lambda$,
and w.r.t. $\log u$, so that, e.g., (5.2.3) and (5.2.4) apply.

######  Theorem 6.4:

  1. $\mathrm{(i)}$

The root of unity

(6.4.1)
$\varepsilon^{\Lambda}\vcentcolon=\varepsilon(\Lambda_{N})^{q_{\infty}^{r}}/\varepsilon(\Lambda_{N+d_{\infty}})$

is independent of $N\gg 0$ and therefore an invariant of $\Lambda$. For a
multiple $d$ of $d_{\infty}$,

(6.4.2)
$\varepsilon(\Lambda_{N})^{q^{rd}}/\varepsilon(\Lambda_{N+d})=(\varepsilon^{\Lambda})^{(q^{rd}-1)/(q_{\infty}^{r}-1)}$

holds. Further, for $0\neq c\in K$,

(6.4.3) $\varepsilon^{c\Lambda}=\varepsilon^{\Lambda}.$

  2. $\mathrm{(ii)}$

Define for $u\in K\Lambda\smallsetminus\Lambda$

(6.4.4)
$\varepsilon_{u}^{\Lambda}\vcentcolon=\varepsilon_{u,N}^{\Lambda}\vcentcolon=\prod_{\begin{subarray}{c}\lambda\in
K\Lambda\\\ \lambda\equiv u\leavevmode\nobreak\ (\mathrm{mod}\Lambda)\\\
\log\lambda\leq
N\end{subarray}}\operatorname{sgn}(\lambda)/\varepsilon(\Lambda_{N}).$

This is independent of $N\gg 0$, and an invariant of the class of
$u\pmod{\Lambda}$ and $\Lambda$. For $0\neq c\in K$,

(6.4.5)
$\varepsilon_{cu}^{c\Lambda}=\operatorname{sgn}(c)\varepsilon_{u}^{\Lambda}.$

  3. $\mathrm{(iii)}$

Let $\Lambda\subset\Lambda^{\prime}\subset K\Lambda$ with another lattice
$\Lambda^{\prime}$ of rank $r$, and define

(6.4.6)
$\varepsilon^{\Lambda^{\prime}|\Lambda}\vcentcolon=\Big{(}\sideset{}{{}^{\prime}}{\prod}_{u\in\Lambda^{\prime}/\Lambda}\varepsilon_{u}^{\Lambda}\Big{)}^{-1}=\varepsilon(\Lambda_{N})^{[\Lambda^{\prime}:\Lambda]}(\varepsilon(\Lambda_{N}^{\prime}))^{-1},$

where $N$ is large enough. For $v\in K\Lambda\smallsetminus\Lambda^{\prime}$,
we have

(6.4.7) $\prod_{\begin{subarray}{c}u\in K\Lambda/\Lambda\\\ u\equiv
v\leavevmode\nobreak\
(\mathrm{mod}\Lambda^{\prime})\end{subarray}}\varepsilon_{u}^{\Lambda}\cdot\varepsilon^{\Lambda^{\prime}|\Lambda}=\varepsilon_{v}^{\Lambda^{\prime}}.$

  4. $\mathrm{(iv)}$

Let $w_{r}\vcentcolon=q_{\infty}^{r}-1$. Then

(6.4.8)
$(\varepsilon^{\Lambda^{\prime}|\Lambda})^{w_{r}}=(\varepsilon^{\Lambda})^{[\Lambda^{\prime}:\Lambda]}/\varepsilon^{\Lambda^{\prime}}.$

###### Proof.

  1. (i)

Choose $a\in A$ of degree $d>0$. From (5.2.3) and (5.2.4) we find

(6.4.9)
$\varepsilon(\Lambda_{N+d,N+2d})=\varepsilon(\Lambda_{N,N+d})^{q^{rd}}.$

This implies that

$\varepsilon^{(d)}\vcentcolon=\varepsilon(\Lambda_{N})^{q^{rd}}/\varepsilon(\Lambda_{N+d})$

is independent of $N\gg 0$. Calculation shows that for $d_{i}=\deg a_{i}$
($i=1,2$),

$\varepsilon^{(d_{1}+d_{2})}=\varepsilon^{(d_{2})}(\varepsilon^{(d_{1})})^{q^{rd_{2}}}=\varepsilon^{(d_{1})}(\varepsilon^{(d_{2})})^{q^{rd_{1}}},$

and so

(6.4.10)
$(\varepsilon^{(d_{1})})^{(q^{rd_{2}}-1)}=(\varepsilon^{(d_{2})})^{q^{rd_{1}}-1}.$

Now we choose $a_{1},a_{2}$ such that $\gcd(d_{1},d_{2})=d_{\infty}$ and apply
2.8 to get
$\varepsilon^{(d_{1})}=(\varepsilon^{(d_{\infty})})^{(q^{rd_{1}}-1)/(q_{\infty}^{r}-1)}$
where $\varepsilon^{(d_{\infty})}=\varepsilon^{\Lambda}$ is an invariant of
$\Lambda$. This shows (6.4.1) and (6.4.2), and (6.4.3) is obvious, as
$\operatorname{sgn}(c)$ is a $w$-th root of unity ($w=q_{\infty}-1$) and
$q_{\infty}^{r}-1\equiv 0\pmod{w}$.

  2. (ii)

Replacing the bound $N$ by $N+d_{\infty}$, the newly appearing elements of the
index sets of the numerator and of the denominator correspond to each other by
$\lambda\leftrightarrow\lambda-u$. Since $N>\log u$, we have
$\operatorname{sgn}(\lambda)=\operatorname{sgn}(\lambda-u)$, that is
$\varepsilon_{u,N+d_{\infty}}^{\Lambda}=\varepsilon_{u,N}^{\Lambda}$. This
gives the independence on $N$; the independence of the representative
$u\pmod{\Lambda}$ is obvious, and (6.4.5) comes out as the number of factors
in the numerator of $\varepsilon_{u,N}^{\Lambda}$ is one larger than the
number of factors in the denominator $\varepsilon(\Lambda_{N})$.

  3. (iii)

By definition of $\varepsilon_{u}^{\Lambda}$,

$\sideset{}{{}^{\prime}}{\prod}_{u\in\Lambda^{\prime}/\Lambda}\varepsilon_{u}^{\Lambda}=\varepsilon(\Lambda_{N}^{\prime}\smallsetminus\Lambda_{N})\varepsilon(\Lambda_{N})^{1-[\Lambda^{\prime}:\Lambda]}$

if $N>\max\log u$, where $u$ runs through a system of representatives for
$\Lambda^{\prime}/\Lambda\smallsetminus\\{0\\}$, and the right hand side
equals
$\varepsilon(\Lambda_{N}^{\prime})\cdot\varepsilon(\Lambda_{N})^{{-}[\Lambda^{\prime}:\Lambda]}$.
This gives the equality stated in (6.4.6), while (6.4.7) results from a
straightforward calculation left to the reader.

  4. (iv)

Inserting (6.4.6), the wanted identity is equivalent with
($\ell\vcentcolon=[\Lambda^{\prime}:\Lambda]$):

$\frac{\varepsilon(\Lambda_{N})^{\ell
w_{r}}}{\varepsilon(\Lambda^{\prime}_{N})^{w_{r}}}=\frac{\varepsilon(\Lambda_{N})^{q_{\infty}^{r}\cdot\ell}\cdot\varepsilon(\Lambda^{\prime}_{N+d_{\infty}})}{\varepsilon(\Lambda_{N+d_{\infty}})^{\ell}\cdot\varepsilon(\Lambda_{N}^{\prime})^{q_{\infty}^{r}}}$

for $N\gg 0$, and by cancelling, with

$\frac{\varepsilon(\Lambda_{N})^{\ell}}{\varepsilon(\Lambda_{N}^{\prime})}=\frac{\varepsilon(\Lambda_{N+d_{\infty}})^{\ell}}{\varepsilon(\Lambda^{\prime}_{N+d_{\infty}})}.$

But the latter are two different expressions for the same quantity
$\varepsilon^{\Lambda^{\prime}|\Lambda}$.

∎

######  Remarks 6.5:

  1. (i)

(6.4.7) together with (ii) of the theorem states that, for fixed
$\boldsymbol{\omega}\in\Psi$, the map
$g\colon(\mathbf{u},Y)\mapsto\varepsilon_{\mathbf{u}\boldsymbol{\omega}}^{Y_{\boldsymbol{\omega}}}$
is a derived distribution with values in $\mu(C_{\infty})$, and is motivated
by $m\colon V\smallsetminus\\{0\\}\to\mu(C_{\infty})$, where
$m(\mathbf{x})=\operatorname{sgn}(\mathbf{x}\boldsymbol{\omega})$. And (6.4.8)
asserts that its $w_{r}$-th power
$G\colon(\mathbf{u},Y)\mapsto(\varepsilon_{\mathbf{u}\boldsymbol{\omega}}^{Y_{\boldsymbol{\omega}}})^{w_{r}}$
is the derivative of the distribution $F$, where $F(0,Y)=\varepsilon^{Y}$ and
$F(\mathbf{u},Y)=G(\mathbf{u},Y)F(0,Y)$ if $\mathbf{u}\notin Y$.

  2. (ii)

One is tempted to ask if already $g$ itself could be completed to a
distribution like $G=g^{w_{r}}$, or if possibly a proper divisor $n$ of
$w_{r}$ does the job. This is actually the case for $n=w_{r}/(q-1)$, as the
discriminant $\Delta$ defined in Corollary 7.6, regarded as a modular form for
$\Gamma_{Y}=\operatorname{GL}(Y)$, has a $(q-1)$-th root $h_{Y}$ as a
holomorphic function on the Drinfeld space
$\Omega=C_{\infty}^{*}\backslash\Psi$ ([5], [19]). These $h_{Y}$ are well-
defined up to $(q-1)$-th roots of unity, and may be chosen in a manner
consistent for all $Y$. We will not pursue that topic here.

##  7\. Product expansions for discriminants and division points

### 7.1.

We fix a lattice pair $Y^{\prime}|Y$ in $V=K^{r}$ and let
$\Delta^{Y^{\prime}|Y}$ be the associated discriminant function as in (1.6.2),
that is

(7.1.1)
$\Delta^{Y^{\prime}|Y}(\boldsymbol{\omega})=\Big{(}\sideset{}{{}^{\prime}}{\prod}_{\mathbf{u}\in
Y^{\prime}/Y}d_{\mathbf{u}}^{Y}(\boldsymbol{\omega})\Big{)}^{-1}.$

We also fix $\boldsymbol{\omega}\in\Psi$ and put
$\Lambda\vcentcolon=Y_{\boldsymbol{\omega}}$,
$\Lambda^{\prime}=Y_{\boldsymbol{\omega}}^{\prime}$, and
$\Delta^{\Lambda^{\prime}|\Lambda}=\Delta^{Y^{\prime}|Y}(\boldsymbol{\omega})$.
The following results will be intrinsic for $\Lambda$ and $\Lambda^{\prime}$
and independent from the presentation $\Lambda=Y_{\boldsymbol{\omega}}$, etc.,
which is only used to embed our current situation into the context of Sections
1 and 2.

From (7.1.1) and the definition of
$d_{\mathbf{u}}^{Y}(\boldsymbol{\omega})=e^{Y_{\boldsymbol{\omega}}}(\mathbf{u}\boldsymbol{\omega})$,
we get

(7.1.2)
$(\Delta^{Y^{\prime}|Y})^{-1}=\sideset{}{{}^{\prime}}{\prod}_{u\in\Lambda^{\prime}/\Lambda}e^{\Lambda}(u).$

Therefore, we first treat $e^{\Lambda}(u)$.

### 7.2.

We have

$e^{\Lambda}(u)=\lim_{N\to\infty}e^{\Lambda_{N}}(u)$

with

(7.2.1)
$e^{\Lambda_{N}}(u)=u\sideset{}{{}^{\prime}}{\prod}_{\lambda\in\Lambda_{N}}(1-u/\lambda)=u\sideset{}{{}^{\prime}}{\prod}_{\lambda\in\Lambda_{N}}\left(\frac{\lambda+u}{\lambda}\right)=\prod_{\begin{subarray}{c}\lambda\in
K\Lambda\\\ \lambda\equiv u\leavevmode\nobreak\ (\mathrm{mod}\Lambda)\\\
\log\lambda\leq
N\end{subarray}}\lambda/\sideset{}{{}^{\prime}}{\prod}_{\lambda\in\Lambda_{N}}\lambda,$

supposing that $N\gg 0$. Now we decompose the factors according to (5.1.3).
This yields

(7.2.2)
$e^{\Lambda_{N}}(u)=\pi^{{-}k}\varepsilon_{u,N}^{\Lambda}U_{u,\Lambda,N}/U_{\Lambda,N}^{-1},$

where for $N\geq\log u$,

$k=d_{\infty}^{-1}\Big{(}\sum_{\begin{subarray}{c}\lambda\in K\Lambda\\\
\lambda\equiv u\leavevmode\nobreak\ (\mathrm{mod}\Lambda)\\\ \log\lambda\leq
N\end{subarray}}\log\lambda-\sideset{}{{}^{\prime}}{\sum}_{\lambda\in\Lambda_{N}}\log\lambda\Big{)}.$

For $N\to\infty$, $k$ and $\varepsilon_{u,N}^{\Lambda}$ become stationary,
that is, for $N\gg 0$,

(7.2.3)
$k=d_{\infty}^{{-}1}[Z_{u,\Lambda}^{\prime}(1)-Z_{\Lambda}^{\prime}(1)]$

by Theorem 4.4(iii) and

(7.2.4) $\varepsilon_{u,N}^{\Lambda}=\varepsilon_{u}^{\Lambda}$

as defined in Theorem 6.4, while $U_{u,\Lambda,N}\to U_{u,\Lambda}$ and
$U_{\Lambda,N}\to U_{\Lambda}$.

### 7.3.

Now we use the distribution properties of $Z_{*,*}$, $U_{*,*}$ and
$\varepsilon_{*}^{*}$ to get

(7.3.1)
$\sideset{}{{}^{\prime}}{\sum}_{u\in\Lambda^{\prime}/\Lambda}(Z_{u,\Lambda}^{\prime}(1)-Z_{\Lambda}^{\prime}(1))=Z_{\Lambda^{\prime}}^{\prime}(1)-[\Lambda^{\prime}:\Lambda]Z_{\Lambda}^{\prime}(1),$
(7.3.2)
$\sideset{}{{}^{\prime}}{\prod}_{u\in\Lambda^{\prime}/\Lambda}(U_{u,\Lambda}U_{\Lambda}^{-1})=U_{\Lambda^{\prime}}\cdot
U_{\Lambda}^{{-}[\Lambda^{\prime}:\Lambda]}$

and

(7.3.3)
$\sideset{}{{}^{\prime}}{\prod}_{u\in\Lambda^{\prime}/\Lambda}\varepsilon_{u}^{\Lambda}=(\varepsilon^{\Lambda^{\prime}|\Lambda})^{-1}.$

This finally gives our principal result (part (iii) of the Main Theorem).

######  Theorem 7.4:

Let $\Lambda^{\prime}|\Lambda$ be a lattice pair of rank $r$ in $C_{\infty}$
and $u\in K\Lambda\smallsetminus\Lambda$.

  1. $\mathrm{(i)}$

The value of the division point $e^{\Lambda}(u)$ of the Drinfeld module
$\phi^{\Lambda}$ associated with $\Lambda$ is

(7.4.1)
$e^{\Lambda}(u)=\pi^{{-}k}\varepsilon_{u}^{\Lambda}U_{u,\Lambda}U_{\Lambda}^{-1}$

with $k=d_{\infty}^{-1}(Z_{u,\Lambda}^{\prime}(1)-Z_{\Lambda}^{\prime}(1))$.

  2. $\mathrm{(ii)}$

The value $\Delta(\Lambda^{\prime}|\Lambda)$ of the discriminant (see 1.6) is
given by

(7.4.2)
$\Delta(\Lambda^{\prime}|\Lambda)=\pi^{k}\varepsilon^{\Lambda^{\prime}|\Lambda}U_{\Lambda}^{[\Lambda^{\prime}:\Lambda]}U_{\Lambda^{\prime}}^{-1}$

with
$k=d_{\infty}^{-1}(Z_{\Lambda}^{\prime}(1)-[\Lambda^{\prime}:\Lambda]Z_{\Lambda}^{\prime}(1))$.

######  Corollary 7.5:

Let $a\in A$ have degree $d>0$. Then the usual discriminant
$\Delta_{a}^{Y}(\boldsymbol{\omega})$ of the Drinfeld module
$\phi^{Y_{\boldsymbol{\omega}}}$ is

(7.5.1)
$\Delta_{a}(\Lambda)=\Delta_{a}^{Y}(\boldsymbol{\omega})=\pi^{k}\operatorname{sgn}(a)(\varepsilon^{\Lambda})^{(q^{rd}-1)/w_{r}}U_{\Lambda}^{q^{rd}-1}$

with $k=d_{\infty}^{-1}(1-q^{rd})Z_{\Lambda}^{\prime}(1)$ and
$w_{r}=q_{\infty}^{r}-1$.

###### Proof.

This is the case of Theorem 7.4 where $Y^{\prime}=a^{-1}Y$. Here
$\Delta_{a}^{Y}(\boldsymbol{\omega})=a\Delta^{Y^{\prime}|Y}(\boldsymbol{\omega})=a\Delta(\Lambda^{\prime}|\Lambda)$,
where $\Delta(\Lambda^{\prime}|\Lambda)$ has the following components:

  * •

$\pi^{k}$,
$k=d_{\infty}^{-1}(Z_{a^{-1}\Lambda}^{\prime}(1)-q^{rd}Z_{\Lambda}^{\prime}(1))$

  * •

$\varepsilon^{a^{-1}\Lambda|\Lambda}$

  * •

$U_{\Lambda}^{q^{rd}}/U_{a^{-1}\Lambda}$,

and $a$ splits as $a=\pi^{{-}\log a/d_{\infty}}\operatorname{sgn}(a)\langle
a\rangle$.

Now $Z_{a^{-1}\Lambda}^{\prime}(1)=Z_{\Lambda}^{\prime}(1)+\log a$ by (4.3.2),
the term $\log a$ cancels, and $k$ in (7.5.1) is as stated.

Next,
$\varepsilon^{a^{-1}\Lambda|\Lambda}=\varepsilon(\Lambda_{N})^{q^{rd}}\varepsilon^{{-}1}((a^{{-}1}\Lambda)_{N})$
for $N\gg 0$ by (6.4.6). Further, $(a^{-1}\Lambda)_{N}=a^{-1}(\Lambda_{N+d})$,
so

$\varepsilon((a^{-1}\Lambda)_{N})=\operatorname{sgn}(a)^{1-\\#(\Lambda_{N+d})}\varepsilon(\Lambda_{N+d})=\varepsilon(\Lambda_{N+d}),$

as $\\#(\Lambda_{N+d})-1$ is divisible by $w=q_{\infty}-1$, and

$\varepsilon^{a^{-1}\Lambda|\Lambda}=\varepsilon(\Lambda_{N})^{q^{rd}}/\varepsilon(\Lambda_{N+d})=(\varepsilon^{\Lambda})^{(q^{rd}-1)/w_{r}}$

by (6.4.2). This gives the unit root part of (7.5.1). Finally,
$U_{\Lambda}^{q^{rd}}/U_{a^{-1}\Lambda}=\langle
a\rangle^{-1}U_{\Lambda}^{q^{rd}-1}$ by 5.3(iv). ∎

######  Corollary/Definition 7.6:

There is a canonical discriminant $\Delta(\Lambda)$ with the property

(7.6.1)
$\Delta_{a}(\Lambda)=\operatorname{sgn}(a)\Delta(\Lambda)^{(q^{rd}-1)/w_{r}}$

for $0\neq a\in A$ of degree $d$. It also satisfies the product formula

(7.6.2) $\Delta(\Lambda)=\pi^{k}\varepsilon^{\Lambda}U_{\Lambda}^{w_{r}}$

with $k={-}d_{\infty}^{{-}1}w_{r}Z_{\Lambda}^{\prime}(1)$, and

(7.6.3) the weight of $\Delta$ as a modular form is $w_{r}=q_{\infty}^{r}-1$.

(This is item(i) of the Main Theorem.)

###### Proof.

Let $a,a^{\prime}$ be elements of $A$ of respective degrees $d,d^{\prime}$ and
with $\gcd(d,d^{\prime})=d_{\infty}$. By (2.8.1) applied to
$q^{\prime}=q^{r}$, we also have
$\gcd(q^{rd}-1,q^{rd^{\prime}}-1)=q_{\infty}^{r}-1=w_{r}$. Now we apply
(2.8.2) to the modular forms
$\tilde{\Delta}_{a}\vcentcolon=\operatorname{sgn}(a)^{-1}\Delta_{a}$,
$\tilde{\Delta}_{a^{\prime}}\vcentcolon=\operatorname{sgn}(a^{\prime})^{-1}\Delta_{a^{\prime}}$,
of weights $q^{rd}-1$, $q^{rd^{\prime}}-1$, respectively, and, writing
$d_{\infty}=xd+x^{\prime}d^{\prime}$ ($x,x^{\prime}\in\mathds{Z}$), we define

(7.6.4)
$\Delta\vcentcolon=\tilde{\Delta}_{a}^{x}\tilde{\Delta}_{a^{\prime}}^{x^{\prime}}.$

Then (7.6.2) follows from (7.5.1), which also shows the independence of
$\Delta$ from the choices of $a$, $a^{\prime}$, $x$, and $x^{\prime}$, as well
as (7.6.1). Finally, (7.6.4) gives the weight $w_{r}$ for $\Delta$, i.e.,
(7.6.3). ∎

######  Remark 7.7:

In [19] we had defined $\Delta$ of weight $w_{r}$, but it was well-defined
only up to roots of unity. The same problem occurs in [5] Proposition 16.4,
where $\Delta$ was also defined up to roots of unity. The present $\Delta$ is
in so far canonical as it depends only on the sign function
“$\operatorname{sgn}$” restricted to $K$ (or, what amounts to the same, the
choice of the uniformizer $\pi$ of $K_{\infty}$ modulo
$\mathfrak{p}_{\infty}^{2}$, where $\mathfrak{p}_{\infty}$ is the valuation
ideal in $K_{\infty}$), but not on the other choices made.

######  Corollary 7.8:

The absolute values of the discriminants are as follows:

  1. $\mathrm{(i)}$

$\lvert\Delta(\Lambda^{\prime}|\Lambda)\rvert=q^{k}$,
$k=[\Lambda^{\prime}:\Lambda]Z_{\Lambda}^{\prime}(1)-Z_{\Lambda^{\prime}}^{\prime}(1)$

  2. $\mathrm{(ii)}$

$\lvert\Delta_{a}(\Lambda)\rvert=q^{k}$, $k=(q^{rd}-1)Z_{\Lambda}^{\prime}(1)$

  3. $\mathrm{(iii)}$

$\lvert\Delta(\Lambda)\rvert=q^{k}$, $k=w_{r}Z_{\Lambda}^{\prime}(1)$

### 7.9.

So far, we had
$\Delta(\Lambda^{\prime}|\Lambda)=\Delta^{Y^{\prime}|Y}(\boldsymbol{\omega})$,
$\Delta_{a}(\Lambda)=\Delta_{a}^{Y}(\boldsymbol{\omega})$,
$\Delta(\Lambda)=\Delta^{Y}(\boldsymbol{\omega})$ with a fixed
$\boldsymbol{\omega}\in\Psi$. Now we allow $\boldsymbol{\omega}$ to vary over
$\Psi$. For $H$ one of the factors $\mathds{Q}$, $\mu(C_{\infty})$,
$U^{(1)}(C_{\infty})$ of $C_{\infty}^{*}$ (see (5.1.3)), we let
$\operatorname{Maps}(\Psi,H)$ be the group of $H$-valued functions on $\Psi$.
The following hold since we know the corresponding properties argumentwise:

#### 7.9.1.

The map

$\displaystyle\mathfrak{Y}^{*}$
$\displaystyle\longrightarrow\operatorname{Maps}(\Psi,\mathds{Q})$
$\displaystyle(\mathbf{u},Y)$ $\displaystyle\longmapsto\log
d_{\mathbf{u}}^{Y}(\boldsymbol{\omega})=Z_{\mathbf{u}\boldsymbol{\omega},Y_{\boldsymbol{\omega}}}^{\prime}(1)-Z_{Y_{\boldsymbol{\omega}}}^{\prime}(1)$
is a derived distribution, motivated by $\displaystyle V\smallsetminus\\{0\\}$
$\displaystyle\longrightarrow\operatorname{Maps}(\Psi,\mathds{Q}).$
$\displaystyle x$ $\displaystyle\longmapsto\log\mathbf{x}\boldsymbol{\omega}$
It is the derivative of the distribution $\displaystyle\mathfrak{Y}$
$\displaystyle\longrightarrow\operatorname{Maps}(\Psi,\mathds{Q})$
$\displaystyle(\mathbf{u},Y)$ $\displaystyle\longmapsto
Z_{\mathbf{u}\boldsymbol{\omega},Y_{\boldsymbol{\omega}}}^{\prime}(1).$

#### 7.9.2.

Second,

$\displaystyle\mathfrak{Y}$
$\displaystyle\longrightarrow\operatorname{Maps}(\Psi,U^{(1)}(C_{\infty}))$
$\displaystyle(\mathbf{u},Y)$ $\displaystyle\longmapsto U_{\mathbf{u},Y}$

is a distribution, and is motivated by
$\mathbf{x}\mapsto\langle\mathbf{x}\boldsymbol{\omega}\rangle$.

#### 7.9.3.

Third,

$\displaystyle\mathfrak{Y}^{*}$
$\displaystyle\longrightarrow\operatorname{Maps}(\Psi,\mu(C_{\infty}))$
$\displaystyle(\mathbf{u},Y)$
$\displaystyle\longmapsto\varepsilon_{\mathbf{u}\boldsymbol{\omega}}^{Y_{\boldsymbol{\omega}}}$

is a derived distribution motivated by
$\mathbf{x}\mapsto\operatorname{sgn}(\mathbf{x}\boldsymbol{\omega})$. Its
$w_{r}$-th power
$(\mathbf{u},Y)\mapsto(\varepsilon_{\mathbf{u}\boldsymbol{\omega}}^{Y_{\boldsymbol{\omega}}})^{w_{r}}$
is the derivative of a distribution.

We put these together to the next result (item (ii) of the Main Theorem).

######  Corollary 7.10:

Let $\mathcal{O}(\Psi)^{*}$ be the group of invertible holomorphic functions
on the Drinfeld space $\Psi=\Psi^{r}$. Then the map

$\displaystyle F\colon\mathfrak{Y}$
$\displaystyle\longrightarrow\mathcal{O}(\Psi)^{*}$
$\displaystyle(\mathbf{u},Y)$
$\displaystyle\longmapsto\begin{cases}(d_{\mathbf{u}}^{Y})^{w_{r}}\Delta^{Y}&(\mathbf{u}\notin
Y)\\\ \Delta^{Y}&(\mathbf{u}\in Y)\end{cases}$

is a distribution with derivative
$G\colon(\mathbf{u},Y)\mapsto(d_{\mathbf{u}}^{Y})^{w_{r}}$. $G$ is motivated
by $m\colon V\smallsetminus\\{0\\}\to\mathcal{O}(\Psi)^{*}$,
$\mathbf{x}\mapsto(\mathbf{x}\boldsymbol{\omega})^{w_{r}}$.

###  7.11. A heuristic consideration

We would like to regard the product formula (7.6.2) for $\Delta$ (and
similarly, the formula (7.4.1) for $e^{\Lambda}(u)$) in Euler’s style as a
formula

(7.11.1)
$?\Delta(\Lambda)=\sideset{}{{}^{\prime}}{\prod}_{\lambda\in\Lambda}\lambda^{w_{r}}?$

and accordingly for $e^{\Lambda}(u)$. A first attempt to endow it with some
reason was to replace it by

(7.11.2)
$?\Delta(\Lambda)=\lim_{N\to\infty}\sideset{}{{}^{\prime}}{\prod}_{\lambda\in\Lambda_{N}}\lambda^{w_{r}}?,$

and to split the latter into components according to (5.1.3). This works
perfectly for the $1$-unit part, in view of Proposition 5.3; that is, the
$1$-unit part of (7.11.2) is true. It works less smoothly for the absolute
value part, since
$\lim_{N\to\infty}\sideset{}{{}^{\prime}}{\sum}_{\lambda\in\Lambda_{N}}\log\lambda$
doesn’t exist. But at least,

$\lim_{N\to\infty}\Big{(}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\lambda\in
K\Lambda\\\ \lambda\equiv u\leavevmode\nobreak\ (\mathrm{mod}\Lambda)\\\
\log\lambda\leq
N\end{subarray}}\log\lambda-\sideset{}{{}^{\prime}}{\sum}_{\lambda\in\Lambda_{N}}\log\lambda\Big{)}$

exists and equals
$d_{\infty}^{-1}(Z_{u,\Lambda}^{\prime}(1)-Z_{\Lambda}^{\prime}(1))$ by
(4.4.2). Then
$\lim_{N\to\infty}\sideset{}{{}^{\prime}}{\sum}_{\lambda\in\Lambda_{N}}\log\lambda$
is assigned the value $Z_{\Lambda}^{\prime}(1)$ through the distribution
property of $Z_{*,*}^{\prime}(1)$. Note that this evaluation resembles the
procedure of analytic continuation. Finally, for the unit root part, we have
to slightly alter

$\sideset{}{{}^{\prime}}{\prod}_{\lambda\in\Lambda_{N}}\operatorname{sgn}(\lambda)^{w_{r}}=\Big{(}\sideset{}{{}^{\prime}}{\prod}_{\lambda\in\Lambda_{N}}\operatorname{sgn}(\lambda)^{q_{\infty}^{r}}\Big{)}/\sideset{}{{}^{\prime}}{\prod}_{\lambda\in\Lambda_{N}}\operatorname{sgn}(\lambda)$

to

$\Big{(}\sideset{}{{}^{\prime}}{\prod}_{\lambda\in\Lambda_{N}}\operatorname{sgn}(\lambda)^{q_{\infty}^{r}}\Big{)}/\sideset{}{{}^{\prime}}{\prod}_{\lambda\in\Lambda_{N+d_{\infty}}}\operatorname{sgn}(\lambda)\qquad(N\gg
0)$

in order to get the correct value $\varepsilon^{\Lambda}$ for the unit root
part of formula (7.6.2). Together, it seems justified to regard (7.6.2) as a
regularized product like (7.11.1) over $\Lambda$.

##  8\. The case $A=\mathds{F}[T]$

As an example for the preceding, we now restrict to the most important case
where the curve $\mathcal{C}$ in 1.2 is the projective line and $\infty$ the
usual place at infinity, and so the Drinfeld ring $A$ is a polynomial ring
$\mathds{F}[T]$ as in 1.2.2.

Here, the corresponding distributions for $r=1$ (related to the Carlitz module
and its division points) have already been studied 1974 by Hayes [22] and 1980
by Galovich and Rosen [8]. In the case $r=2$ there is a large amount of work
about the corresponding modular forms, starting 1980 with [20] and [9], [11],
[10], [13], and since then continued by many authors. The higher rank case
$r>2$ has become a topic of serious research only about 2017 with a series of
preprints [2], [3], [4] by Basson, Breuer and Pink (which now are to appear in
[5]) and the ongoing series “On Drinfeld modular forms of higher rank I, II,
…” by the present author [15], [18], [19]. Actually it was the desire to find
a common framework for the distributive aspects of these (and other) papers
that led to the present work.

### 8.1.

Assume $r=1$. As each lattice $Y$ in $V=K=\mathds{F}(T)$ is free of rank $1$
of shape $Ay$, we may replace the distribution domain $\mathfrak{Y}$ by
$\mathfrak{Y}=K/A$ and the distribution property (2.1.2) by

(8.1.1) $\sum_{\begin{subarray}{c}u\in K/A\\\ u\equiv v\leavevmode\nobreak\
(\mathrm{mod}\leavevmode\nobreak\
n^{-1}A)\end{subarray}}f(u)=f(v)\qquad\text{for }0\neq n\in A.$

This is the point of view of [8]. We take $\pi=T^{-1}$ as uniformizer at
infinity, $\operatorname{sgn}(a)=$ leading coefficient of $a\in A$ as a
polynomial in $T$. The $Z$-function is

(8.1.2) $Z_{A}(S)=(q-1)\sum_{i\geq 0}(qS)^{i}=(q-1)/(1-qS)^{-1}.$

For $0\neq u\in K/A$, represented by $a/n$ with $(a,n)=1$, $a,n\in A$ with $n$
monic and $0\leq d\vcentcolon=\deg a<\deg n$, we have

(8.1.3) $Z_{u,A}(S)=S^{d-\deg n}+Z_{A}(S),$

and so

(8.1.4) $Z_{A}^{\prime}(1)=q/(q-1),\quad Z_{u,A}^{\prime}(1)=q/(q-1)+d-\deg
n.$

Inserting these into the formulas of the last section, we get expressions for
the sizes of $d_{u}^{A}(1)=e_{A}(u)$ and $\Delta^{A}(1)=\Delta(A)$. Let
$\overline{\pi}$ be the period of the Carlitz module (well-defined up to a
$(q-1)$-th root of unity), so that $A_{\overline{\pi}}=A\overline{\pi}$ is its
period lattice.333$\overline{\pi}$ should not be confused with $\pi=$
uniformizer at $\infty=T^{-1}$ (here). By the general formalism

(8.1.5) $d_{u}^{A}(\overline{\pi})=\overline{\pi}d_{u}^{A}(1)$

is the corresponding division point of the Carlitz module and

(8.1.6) $\Delta^{A}(\overline{\pi})=\overline{\pi}^{1-q}\Delta^{A}(1)=1$

by definition of the Carlitz module. We find from (7.5.1)

(8.1.7)
$\overline{\pi}^{q-1}=\Delta^{A}(1)=\pi^{k}\varepsilon^{A}U_{A}^{q-1},$

where $k=(1-q)Z_{A}^{\prime}(1)={-}q$, $\varepsilon^{A}={-}1$ and
$U_{A}=\lim_{N\to\infty}\sideset{}{{}^{\prime}}{\prod}_{\begin{subarray}{c}a\in
A\\\ \deg a<N\end{subarray}}\langle a\rangle$, $\langle
a\rangle=\operatorname{sgn}(a)^{{-}1}T^{{-}\deg a}d$, which agrees with the
formulas in [12] IV 4.10 or [13] 4.11. Similarly, for the division points,

(8.1.8) $\begin{split}\log d_{u}^{A}(\overline{\pi})&=\log\overline{\pi}+\log
d_{u}^{A}(1)\\\
&=\frac{q}{q-1}+(Z_{u,A}^{\prime}(1)-Z_{A}^{\prime}(1))=Z_{u,A}^{\prime}(1)=\frac{q}{q-1}+d-\deg
n,\end{split}$

in accordance with the values given, e.g., in [8] or [12] IV 4.13.

### 8.2.

Now we consider the case where $r=2$. We restrict to demonstrate how
$\lvert\Delta(\boldsymbol{\omega})\rvert$ and $\lvert
d_{\mathbf{u}}^{Y}(\boldsymbol{\omega})\rvert$ or rather their logarithms for
$Y=A^{2}$ may be computed from our general results. For this we assume that
$\boldsymbol{\omega}\in\Psi^{2}$ actually belongs to
$\Omega^{2}=\\{(\omega_{1},\omega_{2})\in\Psi^{2}\mid\omega_{2}=1\\}$, i.e.,
that $\boldsymbol{\omega}=(\omega,1)$ with $\omega\in C_{\infty}\smallsetminus
K_{\infty}$. This is the usual framework of Drinfeld modular forms of rank 2
for the group $\Gamma=\operatorname{GL}(2,A)$ as described, e.g., in [13].
Moreover, we assume that $\omega$ belongs to the fundamental domain
$\mathbf{F}$ for $\Gamma$,

(8.2.1) $\mathbf{F}=\\{\omega\in C_{\infty}\smallsetminus
K_{\infty}\mid\lvert\omega\rvert=\lvert\omega\rvert_{i}\geq 1\\},$

where $\lvert\omega_{i}\rvert_{i}\vcentcolon=\inf_{x\in
K_{\infty}}\lvert\omega-x\rvert$. Now the logarithms of the invertible
functions $\Delta(\omega)$ and $d_{\mathbf{u}}^{Y}(\omega)$ depend only on
$\log\omega$ and interpolate linearly from integer values of $\log\omega$.
(This has been known for long time by results of van der Put, and has been
generalized to higher ranks in [18] Theorems 2.4 and 2.6.) Therefore we may
assume that

(8.2.2) $\omega\in\mathbf{F}_{\ell}\vcentcolon=\\{\omega\in
C_{\infty}\smallsetminus
K_{\infty}\mid\lvert\omega\rvert=\lvert\omega\rvert_{i}=q^{\ell}\\}$

with $\ell\in\mathds{N}_{0}=\\{0,1,2,\dots\\}$. Let $\Lambda$ be the lattice
$Y_{\boldsymbol{\omega}}=A\omega+A\subset C_{\infty}$. Due to (8.2.2),

(8.2.3)
$Z_{\Lambda}(S)=(q-1)\frac{1-(qS)^{\ell}}{1-qS}+(q^{2}-1)\frac{(qS)^{\ell}}{1-q^{2}S}.$

Let $0\neq\mathbf{u}=n^{-1}(a_{1},a_{2})\in n^{-1}Y$, $a_{1},a_{2},n\in A$,
$d_{1}=\deg a_{1}$, $d_{2}=\deg a_{2}$, $d_{1},d_{2}<\deg n$,
$u\vcentcolon=\mathbf{u}\boldsymbol{\omega}=n^{-1}(a_{1}\omega+a_{2})$. Then

(8.2.4) $\log u=\begin{cases}\ell+d_{1}-\deg n,&\text{if }d_{2}\leq
d_{1}+\ell,\\\ d_{2}-\deg n,&\text{if }d_{2}>d_{1}+\ell\end{cases}$

(note that always $\log u<\ell)$ and therefore

(8.2.5) $Z_{u,\Lambda}(S)=\begin{cases}Z_{\Lambda}(S)+(qS)^{\log
u}-(q-1)\sum_{0\leq i<\log u}(qS)^{i},&\text{if }\log u\geq 0,\\\
Z_{\Lambda}(S)+S^{\log u},&\text{if }\log u<0.\end{cases}$

We find

(8.2.6) $Z_{\Lambda}^{\prime}(1)=\frac{q^{2}+q-q^{\ell+1}}{q^{2}-1}$

and

$Z_{u,\Lambda}^{\prime}(1)-Z_{\Lambda}^{\prime}(1)=\frac{q}{q-1}(q^{\log
u}-1)=\begin{cases}\frac{q}{q-1}(q^{\ell+d_{1}-\deg n}-1),&\text{if }\log
u\geq 0,\\\ \log u,&\text{if }\log u<0.\end{cases}$

Finally, from Theorem 7.4 and its corollaries,

(8.2.7) $\displaystyle\log\Delta(\omega)$ $\displaystyle=q^{2}+q-q^{\ell+1}$
and (8.2.8) $\displaystyle\log d_{\mathbf{u}}^{Y}(\omega)$
$\displaystyle=\begin{cases}\frac{q}{q-1}(q^{\ell+d_{1}-\deg n}-1),&\text{if
}\log u\geq 0,\\\ \log u,&\text{if }\log u<0.\end{cases}$

for $\omega\in\mathbf{F}_{\ell}$, $u=n^{.1}(a_{1}\omega+a_{2})$.

The first of these, (8.2.7), has been found in [14] Theorem 2.13, using a
difficult argument involving the van der Put transform of $\Delta$ and its
Fourier coefficients. To the author’s best knowledge, (8.2.8), although known
to him for quite some time, is so far nowhere published.

If $a_{1}=0$, then always $\log u=d_{2}-\deg n<0$, and $\log
d_{\mathbf{u}}^{Y}(\omega)=d_{2}-\deg n$ is constant along $\mathbf{F}_{0}$,
$\mathbf{F}_{1}$, $\mathbf{F}_{2}$, …If $a_{1}\neq 0$ then for $\ell\geq
d_{2}-d_{1}$, the first case of (8.2.4) prevails, and if moreover
$\ell\geq\deg n-d_{1}$, then $\log d_{\mathbf{u}}^{Y}(\omega)$ is given by the
first formula of (8.2.8), and grows very fast with $\ell$.

### 8.3.

We conclude this set of examples with the case $r=3$, where we restrict to
give formulas for $\log\Delta(\boldsymbol{\omega})$. The situation, although
considerably more complex, is analogous with the just considered case $r=2$.
Again we assume that $\boldsymbol{\omega}=(\omega_{1},\omega_{2},\omega_{3})$
lies in the fundamental domain $\mathbf{F}=\mathbf{F}^{3}$ of
$\Gamma=\operatorname{GL}(3,A)$. That is

(8.3.1) $\omega_{1},\omega_{2}\text{ and }\omega_{3}=1\text{ are orthogonal
(see \ref{Subsubsection.Being.uqp-d-is-stable}) and
}\lvert\omega_{1}\rvert\geq\lvert\omega_{2}\rvert\geq 1.$

Moreover, still because of [18] Theorems 2.4 and 2.6, we assume that
$\boldsymbol{\omega}$ lies above a vertex of the Bruhat-Tits building, which
here means that

$a\vcentcolon=\log\omega_{1}\quad\text{and}\quad b\vcentcolon=\log\omega_{2}$

are integers. For such $\boldsymbol{\omega}$, let
$\Lambda=Y_{\boldsymbol{\omega}}=A\omega_{1}\oplus A\omega_{2}+A\subset
C_{\infty}$. We distinguish the cases

(8.3.2) $(1)\leavevmode\nobreak\ a=b=0;\qquad(2)\leavevmode\nobreak\
a>b=0;\qquad(3)\leavevmode\nobreak\ a=b>0;\qquad(4)\leavevmode\nobreak\
a>b>0.$

Case (1) $a=b=0$ Here the $Z$-function is
$Z_{\Lambda}(S)=(q^{3}-1)/(1-q^{3}S)$, which gives

(8.3.3) $\log\Delta(\boldsymbol{\omega})=q^{3}.$

This is well-known and could be seen by bare eye.

Case (2) $a>b=0$ Here for $N\in\mathds{N}_{0}$,

$\dim_{\mathds{F}}\Lambda_{N}=2(N+1)\text{ if }N<a\quad\text{and}\quad
3(N+1)-a\text{ if }N\geq a.$

This implies

$Z_{\Lambda}(S)=(q^{2}-1)\frac{(q^{2}S)^{a}-1}{q^{2}S-1}+(q^{3}-1)\frac{(q^{2}S)^{a}}{1-q^{3}S}$

and we find

(8.3.4)
$\log\Delta(\boldsymbol{\omega})=(q^{3}-1)Z_{\Lambda}^{\prime}(1)=q^{2a+3}-(q^{3}-1)q^{2}\frac{q^{2a}-1}{q^{2}-1}.$

Case (3) $a=b>0$ We have for $N\in\mathds{N}_{0}$

$\dim_{\mathds{F}}\Lambda_{N}=N+1\text{ if }N<a\quad\text{and}\quad
3(N+1)-2a\text{ if }N\geq a,$

and so

$Z_{\Lambda}(S)=(q-1)\frac{(qS)^{a}-1}{qS-1}+(q^{3}-1)\frac{(qS)^{a}}{1-q^{3}S},$

which gives

(8.3.5)
$\log\Delta(\boldsymbol{\omega})=(q^{3}-1)Z_{\Lambda}^{\prime}(1)=q^{3}-(q^{a}-1)(q^{2}+q).$

Case (4) $a>b>0$ The dimension of $\Lambda_{N}$ is given by
$\dim\Lambda_{N}=N+1$, $2(N+1)-b$, $3(N+1)-a-b$ if $N<b$, $b\leq N<a$, $N\geq
a$, respectively. We obtain

$Z_{\Lambda}(S)=(q-1)\frac{(qS)^{b}-1}{qS-1}+(q^{2}-1)(qS)^{b}\frac{(qS)^{a-b}-1}{q^{2}S-1}+(q^{3}-1)q^{a-b}\frac{(qS)^{a}}{1-q^{3}S}$

and thus

(8.3.6) $\log\Delta(\boldsymbol{\omega})=(q^{3}-1)Z_{\Lambda}^{\prime}(1)$

with the above $Z_{\Lambda}(S)$ (we omit to write it out).

### 8.4.

A few of the values of $\log\Delta(\boldsymbol{\omega})$ for
$\boldsymbol{\omega}\in\mathbf{F}=\mathbf{F}^{3}$ had already been calculated,
with great pain, in [15] (combine Figure 3 with Theorem 4.13 loc. cit.) in
form of its increments on the Bruhat-Tits building. Luckily, these agree with
the values obtained above!

It is an exercise in intelligent notation to write down a general formula for
$\log\Delta(\boldsymbol{\omega})$, where
$\boldsymbol{\omega}\in\mathbf{F}^{r}$ lies above a vertex in the fundamental
domain of the Bruhat-Tits building $\mathcal{BT}^{r}$, or to write a computer
program. Similarly, it isn’t but a matter of patience to work out
$Z_{u,\Lambda}(S)$ for $u=\mathbf{u}\boldsymbol{\omega}$ and $\mathbf{u}\in
K^{r}/A^{r}$, where $r\geq 2$ is general, and thereby to get closed
expressions for $\log d_{\mathbf{u}}^{A^{r}}(\boldsymbol{\omega})$ in the
style of (8.2.8).

## References

  * [1] Dirk Basson  “A product formula for the higher rank Drinfeld discriminant function”  In _J. Number Theory_ 178, 2017, pp. 190–200
  * [2] Dirk Basson, Florian Breuer and Richard Pink  “Drinfeld modular forms of arbitrary rank, Part I: Analytic Theory”, 2018  arXiv:1805.12335 [math.NT]
  * [3] Dirk Basson, Florian Breuer and Richard Pink  “Drinfeld modular forms of arbitrary rank, Part II: Comparison with Algebraic Theory”, 2018  arXiv:1805.12337 [math.NT]
  * [4] Dirk Basson, Florian Breuer and Richard Pink  “Drinfeld modular forms of arbitrary rank, Part III: Examples”, 2018  arXiv:1805.12339 [math.NT]
  * [5] Dirk Basson, Florian Breuer and Richard Pink  “Drinfeld modular forms of arbitrary rank”  In _Memoirs of the AMS_ , to appear
  * [6] Leonard Carlitz  “On certain functions connected with polynomials in a Galois field”  In _Duke Math. J._ 1, 1935, pp. 137–168
  * [7] V.. Drinfel’d  “Elliptic modules”  In _Mat. Sb. (N.S.)_ 94(136), 1974, pp. 594–627656
  * [8] Steven Galovich and Michael Rosen  “Distributions on rational function fields”  In _Math. Ann._ 256, 1981, pp. 549–560
  * [9] Ernst-Ulrich Gekeler  “Drinfeld-Moduln und modulare Formen über rationalen Funktionenkörpern” 119, Bonner Mathematische Schriften [Bonn Mathematical Publications]  Universität Bonn, Mathematisches Institut, Bonn, 1980
  * [10] Ernst-Ulrich Gekeler  “Modulare Einheiten für Funktionenkörper”  In _J. Reine Angew. Math._ 348, 1984, pp. 94–115
  * [11] Ernst-Ulrich Gekeler  “A product expansion for the discriminant function of Drinfel’d modules of rank two”  In _J. Number Theory_ 21, 1985, pp. 135–140
  * [12] Ernst-Ulrich Gekeler  “Drinfel’d modular curves” 1231, Lecture Notes in Mathematics  Springer-Verlag, Berlin, 1986
  * [13] Ernst-Ulrich Gekeler  “On the coefficients of Drinfel’d modular forms”  In _Invent. Math._ 93, 1988, pp. 667–700
  * [14] Ernst-Ulrich Gekeler  “On the Drinfeld discriminant function”  In _Compositio Math._ 106, 1997, pp. 181–202
  * [15] Ernst-Ulrich Gekeler  “On Drinfeld modular forms of higher rank”  In _J. Théor. Nombres Bordeaux_ 29, 2017, pp. 875–902
  * [16] Ernst-Ulrich Gekeler  “Towers of ${\rm GL}(r)$-type of modular curves”  In _J. Reine Angew. Math._ 754, 2019, pp. 87–141
  * [17] Ernst-Ulrich Gekeler  “Invertible functions on nonarchimedean symmetric spaces”  In _Algebra Number Theory_ 14, 2020, pp. 2481–2504
  * [18] Ernst-Ulrich Gekeler  “On Drinfeld modular forms of higher rank II”  In _J. Number Theory_ 232, 2022, pp. 4–32
  * [19] Ernst-Ulrich Gekeler  “On Drinfeld modular forms of higher rank VII: Expansions at the boundary”, 2023  arXiv:2311.02131 [math.NT]
  * [20] David Goss  “$\pi$-adic Eisenstein series for function fields”  In _Compositio Math._ 41, 1980, pp. 3–38
  * [21] David Goss  “Basic structures of function field arithmetic” 35, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]  Springer-Verlag, Berlin, 1996
  * [22] David R. Hayes  “Explicit class field theory for rational function fields”  In _Trans. Amer. Math. Soc._ 189, 1974, pp. 77–91
  * [23] David R. Hayes  “Stickelberger elements in function fields”  In _Compositio Math._ 55, 1985, pp. 209–239
  * [24] Daniel S. Kubert and Serge Lang  “Modular units” 244, Grundlehren der Mathematischen Wissenschaften  Springer-Verlag, New York-Berlin, 1981
  * [25] Gilles Robert  “Concernant la relation de distribution satisfaite par la fonction $\phi$ associée à un réseau complexe”  In _Invent. Math._ 100, 1990, pp. 231–257
  * [26] Gilles Robert  “La racine 12-ième canonique de $\Delta(L)^{[\underline{L}\colon L]}/\Delta(\underline{L})$”  In _Séminaire de Théorie des Nombres, Paris, 1989–90_ 102, Progr. Math.  Birkhäuser Boston, Boston, MA, 1992, pp. 209–232
  * [27] Michael Rosen  “Number theory in function fields” 210, Graduate Texts in Mathematics  Springer-Verlag, New York, 2002
  * [28] Goro Shimura  “Introduction to the arithmetic theory of automorphic functions” Publications of the Mathematical Society of Japan, No. 11  Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, NJ, 1971