# Elliptic Units Above Fields With Exactly One Complex Place

Pierre Morain

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# Elliptic Units Above Fields With Exactly One Complex Place

Pierre Morain

► To cite this version:

Pierre Morain. Elliptic Units Above Fields With Exactly One Complex Place. 2024. hal-04607848v2

HAL Id: hal-04607848

https://hal.science/hal-04607848v2

Preprint submitted on 4 Mar 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Elliptic units above fields with exactly one complex place

Pierre L. L. Morain<fnref n="1" />

**Abstract:**

In this work we explore the construction of abelian extensions of fields with exactly one complex place using multivariate analytic functions in the spirit of Hilbert’s 12th problem. To this end we study the special values of the multiple elliptic Gamma functions introduced in the early 2000s by Nishizawa following the work of Felder and Varchenko on the elliptic Gamma function of Ruijsenaars. We construct geometric variants of these functions enjoying transformation properties under an action of $SL_d(\mathbb{Z})$ for $d \ge 2$. The evaluation of these functions at points of a degree $d$ field $\mathbb{K}$ with exactly one complex place following the scheme of a recent article by Bergeron, Charollois and García [1] seems to produce algebraic numbers. More precisely, we conjecture that such infinite products yield algebraic units in abelian extensions of $\mathbb{K}$ related to conjectural Stark units and we provide numerical evidence to support this conjecture for cubic, quartic and quintic fields.

**Acknowledgments:** This work is part of an on-going PhD work and the author would like to thank his advisors Pierre Charollois and Antonin Guilloux for their guidance and their helpful comments on this work.

<footnote n="1">¹Sorbonne Université, IMJ-PRG, CNRS, Ouragan Team, INRIA, Paris, France. This PhD work is funded by the École polytechnique, Palaiseau, France.</footnote>
Elliptic units above fields with exactly one complex place

Pierre L. L. Morain<fnref n="1" />

**Abstract:**

In this work we explore the construction of abelian extensions of number
fields with exactly one complex place using multivariate analytic functions
in the spirit of Hilbert’s 12th problem. To this end we study the special val-
ues of the multiple elliptic Gamma functions introduced in the early 2000s
by Nishizawa following the work of Felder and Varchenko on Ruijsenaars’
eliptic Gamma function. We construct geometric variants of these functions
enjoying transformation properties under an action of $SL_d(\mathbb{Z})$ for $d \ge 2$. The
evaluation of these functions at points of a degree $d$ field $\mathbb{K}$ with exactly one
complex place following the scheme of a recent article by Bergeron, Charollois
and García [1] seems to produce algebraic numbers. More precisely, we con-
jecture that such infinite products yield algebraic units in abelian extensions
of $\mathbb{K}$ related to conjectural Stark units and we provide numerical evidence to
support this conjecture for cubic, quartic and quintic fields.

**Acknowledgments:** This work is part of an on-going PhD work and the au-
thor would like to thank his advisors Pierre Charollois and Antonin Guilloux
for their guidance and their helpful comments on this work.

<footnote n="1">¹Sorbonne Université and Université Paris Cité, CNRS, IMJ-PRG, F-75005 Paris, France. This PhD work is funded by the École polytechnique, Palaiseau, France.</footnote>
# Contents

<table><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>2</td><td>Multiple elliptic Gamma functions and their geometric variants</td><td>4</td></tr><tr><td>2.1</td><td>The geometric elliptic Gamma functions</td><td>4</td></tr><tr><td>2.2</td><td>Multiple elliptic Gamma functions</td><td>7</td></tr><tr><td>2.3</td><td>Modified $G_r$ functions and smoothings</td><td>16</td></tr><tr><td>3</td><td>An algebraicity conjecture on special values of $G_r$ functions</td><td>18</td></tr><tr><td>3.1</td><td>Arithmetic $G_r$ functions</td><td>18</td></tr><tr><td>3.2</td><td>Formulation of the conjecture</td><td>21</td></tr><tr><td>4</td><td>Computing the arithmetic $G_r$ functions</td><td>24</td></tr><tr><td>4.1</td><td>Geometric setup</td><td>24</td></tr><tr><td>4.2</td><td>Computing the $G_r$ functions</td><td>29</td></tr><tr><td>4.3</td><td>The choice of vectors $(h_\rho)_{\rho \in \mathfrak{S}_r}$</td><td>31</td></tr><tr><td>5</td><td>Numerical evidence to support the conjecture</td><td>38</td></tr><tr><td>5.1</td><td>Cubic examples</td><td>39</td></tr><tr><td>5.2</td><td>Quartic examples</td><td>42</td></tr><tr><td>5.3</td><td>One Quintic example</td><td>46</td></tr></table>

# 1 Introduction

In explicit class field theory the main task is to provide a way to compute the abelian extensions of a number field $K$. The celebrated theorem of Kronecker and Weber shows that the exponential function plays a key role in the description of the abelian extensions of $\mathbb{Q}$. Since then, the theory of Complex Multiplication has provided explicit constructions to build abelian extensions of imaginary quadratic fields. There are many candidates for the function which plays the role of the exponential function here, including Klein's $j$-invariant, Weber's functions or Jacobi's $\theta$ functions, which are all related to modular forms for $\text{SL}_2(\mathbb{Z})$. Elliptic units are obtained by evaluating ratios of $\theta$ functions at points of an imaginary quadratic field. For instance, define

$$ \theta(z, \tau) = \prod_{m \ge 0} (1 - \exp(2i\pi((m+1)\tau - z)))(1 - \exp(2i\pi(m\tau + z))) $$
# Contents

<table><tbody><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>2</td><td>Multiple elliptic Gamma functions and their geometric variants</td><td>4</td></tr><tr><td>2.1</td><td>The geometric elliptic Gamma functions</td><td>5</td></tr><tr><td>2.2</td><td>Multiple elliptic Gamma functions</td><td>7</td></tr><tr><td>3</td><td>An algebraicity conjecture on special values of $G_r$ functions</td><td>17</td></tr><tr><td>3.1</td><td>Arithmetic $G_r$ functions</td><td>17</td></tr><tr><td>3.2</td><td>Formulation of the conjecture</td><td>19</td></tr><tr><td>4</td><td>Computing the arithmetic $G_r$ functions</td><td>24</td></tr><tr><td>4.1</td><td>Geometric setup</td><td>25</td></tr><tr><td>4.2</td><td>Computing the $G_r$ functions</td><td>30</td></tr><tr><td>4.3</td><td>The choice of vectors $(h_\rho)_{\rho \in \mathfrak{S}_r}$</td><td>32</td></tr><tr><td>4.4</td><td>The main algorithm</td><td>43</td></tr><tr><td>5</td><td>Numerical evidence to support the conjecture</td><td>44</td></tr><tr><td>5.1</td><td>Cubic examples</td><td>45</td></tr><tr><td>5.2</td><td>Quartic examples</td><td>51</td></tr><tr><td>5.3</td><td>One Quintic example</td><td>56</td></tr><tr><td>5.4</td><td>Quartic examples with $\kappa < \#\mathcal{Z}_f^1$</td><td>58</td></tr><tr><td>A</td><td>Appendix: Pure cubic table for $2 \le m \le 50$</td><td>62</td></tr></tbody></table>

## 1 Introduction

In explicit class field theory the main task is to provide a way to compute the abelian extensions of a number field $\mathbb{K}$. The celebrated theorem of Kronecker and Weber shows that the exponential function plays a key role in the description of the abelian extensions of $\mathbb{Q}$. Since then, the theory of Complex Multiplication has provided explicit constructions for abelian extensions of imaginary quadratic fields. There are many candidates for the function which plays the role of the exponential function here, including Klein's $j$-invariant, Weber's functions or Jacobi's $\theta$ functions, which are all related to modular forms for $\text{SL}_2(\mathbb{Z})$. Elliptic units are obtained by evaluating ratios of $\theta$ functions at points of an imaginary quadratic field. For instance, define

$$ \theta(z, \tau) = \prod_{m \ge 0} (1 - \exp(2i\pi((m+1)\tau - z)))(1 - \exp(2i\pi(m\tau + z))) $$
for $z \in \mathbb{C}$ and $\tau \in \mathbb{H} = \{w \in \mathbb{C}, \Im(w) > 0\}$. Then we may compute

$$ \theta\left(\frac{7}{91}, \frac{10 + e^{2i\pi/3}}{91}\right)^7 \theta\left(\frac{7}{13}, \frac{10 + e^{2i\pi/3}}{13}\right)^{-1} \approx 0.0196475... - i \cdot 0.6399892... $$

which is a unit outside an ideal above 13 in the field $\mathbb{Q}[x]/(x^4+3x^3+32x^2+13)$, an extension of $\mathbb{Q}(e^{2i\pi/3})$ ramified only above a prime ideal of norm 13. For a more complete presentation on the subject, we refer to [17].

Finding general analytic functions which play the role of the exponential function for other number fields constitutes Hilbert's 12th problem. The situation becomes more complicated outside of the two cases above due to the presence of an infinite set of units in $\mathbb{K}$. Other efforts have been made to compute elements in class fields using analytic functions, especially in the case of real quadratic fields. In [20], Stark analyses regulators in abelian extensions and studies how they can be factored. This led him to refine Dirichlet's class number formula and to conjecture the existence of special units in class fields which should yield the exponential of the values of $L$-functions at $s = 0$ as a combination of their valuations at infinite places. To relate this to Hilbert's 12th problem, we are interested in finding a way to compute these units directly, instead of computing absolute values. It was later shown that there are two simpler cases in the Stark conjectures (see for instance [6]). The first case deals with totally real fields and it was solved using $p$-adic methods by Dasgupta and Kakde in [7]. In this article we focus on the second case where $\mathbb{K}$ is an "almost totally real field" (ATR), meaning that $\mathbb{K}$ has exactly one complex place. Note that imaginary quadratic fields are ATR in this sense, so that we may hope to find an analogue to the construction of elliptic units above ATR fields. To this end, Bergeron, Charollos and García [1] used the elliptic Gamma function from mathematical physics of Ruijsenaars [18] studied by Felder and Varchenko [10] to build conjectural elliptic units above complex cubic fields and related their construction to the conjectural Stark units. For instance, if $z$ is the root of the polynomial $x^3 - 2$ in the upper half-plane $\mathbb{H}$ we may compute with high precision (more than 1000 digits) the quotient

$$ \frac{\Gamma\left(\frac{-1}{3}, \frac{-z^2-z-3}{15}, \frac{-z+8}{15}\right)^5}{\Gamma\left(\frac{-5}{3}, \frac{-z^2-z-3}{3}, \frac{-z+8}{3}\right)} \approx -0.3362599... - i \cdot -0.3832104... $$

(see section 2.1.1 for the definition of the elliptic Gamma function). It is close to a root of the polynomial $x^6 + 3x^5 + 6x^4 + 5x^3 + 6x^2 + 3x + 1$ which is the minimal polynomial over $\mathbb{Q}$ of a unit in an abelian extension of $\mathbb{Q}(z)$. We treat this example in section 5.1.2 below.
for $z \in \mathbb{C}$ and $\tau \in \mathbb{H} = \{w \in \mathbb{C}, \Im(w) > 0\}$. Then we may compute

$$ \theta\left(\frac{7}{91}, \frac{10 + e^{2i\pi/3}}{91}\right)^7 \theta\left(\frac{7}{13}, \frac{10 + e^{2i\pi/3}}{13}\right)^{-1} \approx 0.0196475\dots - i \cdot 0.6399892\dots $$

which is a unit outside an ideal above 13 in the field $\mathbb{Q}[x]/(x^4+3x^3+32x^2+13)$, an abelian extension of $\mathbb{Q}(e^{2i\pi/3})$ ramified only above a prime ideal of norm 13. For a more complete presentation on the subject, we refer to [20].

Finding general analytic functions which play the role of the exponential function for other number fields constitutes Hilbert's 12th problem. The situation becomes more complicated outside of the two cases above due to the presence of an infinite set of units in $\mathbb{K}$. Other efforts have been made to compute elements in class fields using analytic functions, especially in the case of real quadratic fields. In [23], Stark analyses regulators in abelian extensions and studies how they can be factored. This led him to refine Dirichlet's class number formula and to conjecture the existence of special units in class fields which should yield the exponential of the values of $L$-functions at $s=0$ as a combination of their valuations at infinite places. To relate this to Hilbert's 12th problem, we are interested in finding a way to compute these units directly, instead of computing absolute values. It was later shown that there are two simpler cases in the Stark conjectures (see for instance [7]). The first case deals with totally real number fields and it was solved using $p$-adic methods by Dasgupta and Kakde in [8]. The second case deals with number fields with exactly one complex place, the so-called "almost totally real fields" (ATR for short).

In this article we are concerned with the case of ATR number fields of degree $d \ge 3$ for which Hilbert's 12th problem and Stark's conjectures are still open (the case $d=2$ of imaginary quadratic fields is dealt with using CM theory). Previous work on the abelian Stark conjecture for complex cubic fields was carried out by Ren and Sczech in [19] where a conjectural explicit formula is given for the Stark unit. Recently, Bergeron, Charollois and García [1] used Ruijsenaars' elliptic Gamma function coming from mathematical physics (see [21]), to build conjectural elliptic units above complex cubic fields and related their construction to the conjectural Stark units. For instance, if $z = e^{2i\pi/3}2^{1/3}$ is the root of the polynomial $x^3 - 2$ in the upper half-plane $\mathbb{H}$ we may compute with high precision (more than 1000 digits) the quotient

$$ \frac{\Gamma\left(\frac{-1}{3}, \frac{z^2+z+3}{15}, \frac{z-8}{15}\right)^{-5}}{\Gamma\left(\frac{-5}{3}, \frac{z^2+z+3}{3}, \frac{z-8}{3}\right)^{-1}} \approx -1.2937005\dots + i \cdot 1.4743341\dots $$

(see section 2.1.1 for the definition of the elliptic Gamma function). It is close to a root of the polynomial $x^6 + 3x^5 + 6x^4 + 5x^3 + 6x^2 + 3x + 1$ which is
This work proposes a generalisation beyond complex cubic fields of the construction carried out in [1] using multiple elliptic Gamma functions (denoted $G_r$ for $r \ge 0$) which were introduced by Nishizawa in [13] and studied by Narukawa in [12] (see section 2.2). In order to achieve this, we first construct geometric variants of the $G_r$ functions of Nishizawa, upgrading the construction of the geometric elliptic Gamma function in [9] and we show that these functions enjoy nice properties under the action of $SL_{r+2}(\mathbb{Z})$. In section 3, we show how to evaluate these functions at points in an ATR field $\mathbb{K}$ of degree $d = r + 2$ to obtain conjectural algebraic units in abelian extensions of $\mathbb{K}$. This is expressed in conjecture 10 which generalises the conjecture in [1] to higher degree ATR fields. In section 4, under some assumptions, we show how to choose the evaluation points for the geometric $G_r$ functions to efficiently obtain conjectural algebraic units. We also describe the algorithmic approach to obtaining numerical evidence to support conjecture 10, and in section 5, we provide many examples for ATR cubic, quartic and quintic fields.

# 2 Multiple elliptic Gamma functions and their geometric variants

In this section, we recall the properties of the elliptic Gamma function and its geometric variants. Then we will present the multiple elliptic Gamma functions of Nishizawa and construct geometric variants of these functions, upgrading the construction of the geometric elliptic Gamma functions in [9].

## 2.1 The geometric elliptic Gamma functions

### 2.1.1 The elliptic Gamma function

The elliptic Gamma function was first introduced by Ruijsenaars [18] and was studied later at length by Felder and Varchenko [10] who made the connexion with the above $\theta$ function. For $z \in \mathbb{C}$ and $\tau, \sigma \in \mathbb{H}$ put

$$ \Gamma(z, \tau, \sigma) = \prod_{m,n \ge 0} \left( \frac{1 - \exp(2i\pi((m+1)\tau + (n+1)\sigma - z))}{1 - \exp(2i\pi(m\tau + n\sigma + z))} \right) $$

This function converges absolutely for $z \in \mathbb{C}, z \notin \mathbb{Z} + \mathbb{Z}_{\le 0}\tau + \mathbb{Z}_{\le 0}\sigma$ and it is meromorphic in this domain. It is worth noting that this function has a nice
the minimal polynomial over $\mathbb{Q}$ of a unit in an abelian extension of $\mathbb{Q}(z)$. We
treat this example in section 5.1.2 below. The elliptic Gamma function had
been studied by Felder and Varchenko [12] and shown to satisfy modular
identities for the group $\text{SL}_3(\mathbb{Z})$ involving $\theta$ functions. The construction of
conjectural elliptic units in [1] is inspired by the construction of elliptic units
above imaginary quadratic fields using $\theta$ functions, and the constructions
carried out in this article will follow the same principles for arbitrary ATR
number fields. Previous work was also carried out in the case of even degree
$2d'$ ATR number fields containing a maximal real subfield $\mathbb{F}$ of degree $d'$ in [3]
using Hilbert modular forms for the modular group $\text{SL}_2(\mathcal{O}_{\mathbb{F}})$. It is interesting
to note that the existence of such a real subfield will often be an obstacle to
the approach developed in this article.

This work proposes a generalisation beyond complex cubic fields of the
construction carried out in [1] using multiple elliptic Gamma functions (de-
noted $G_r$ for $r \ge 0$) which were introduced by Nishizawa in [16] and studied
by Narukawa in [14] (see section 2.2). In order to achieve this, we first
construct geometric variants of the $G_r$ functions of Nishizawa in section 2,
upgrading the construction of the geometric elliptic Gamma function in [11]
and we show that these functions enjoy nice properties under the action of
$\text{SL}_{r+2}(\mathbb{Z})$. In section 3, we show how to evaluate these functions at points
in an ATR number field $\mathbb{K}$ of degree $d = r + 2$ to obtain conjectural alge-
braic units in abelian extensions of $\mathbb{K}$. This is expressed in Conjecture 11
which generalises the conjecture in [1] to higher degree ATR number fields.
In section 4, under some assumptions, we show how to choose the evalua-
tion points for the geometric $G_r$ functions to efficiently obtain conjectural
algebraic units. We also describe the algorithmic approach to obtaining nu-
merical evidence to support Conjecture 11, and in section 5, we provide many
examples for ATR cubic, quartic and quintic fields.

## 2 Multiple elliptic Gamma functions and their geometric variants

In this section, we recall the properties of the elliptic Gamma function and
its geometric variants. Then we will present the multiple elliptic Gamma
functions of Nishizawa and construct geometric variants of these functions,
upgrading the construction of the geometric elliptic Gamma functions in [11].
expression as the exponential of an infinite sum involving sinuses:

$$ \Gamma(z, \tau, \sigma) = \exp \left( \sum_{j \ge 1} \frac{1}{2ij} \frac{\sin(\pi j (2z - \tau - \sigma))}{\sin(\pi j \tau) \sin(\pi j \sigma)} \right) \quad (1) $$

The range of parameters $\tau, \sigma$ can thus be extended to $\tau, \sigma \in \mathbb{C} - \mathbb{R}$ by putting:

$$ \Gamma(z, -\tau, \sigma) = \frac{1}{\Gamma(z + \tau, \tau, \sigma)}, \quad \Gamma(z, \tau, -\sigma) = \frac{1}{\Gamma(z + \sigma, \tau, \sigma)} $$

In [[10], Theorems 3.1 and 4.1], Felder and Varchenko proved the following properties for the elliptic Gamma function related to the $\theta$ function of Complex Multiplication. If $\tau, \sigma \in \mathbb{C} - \mathbb{R}$ then

$$ \begin{aligned} \Gamma(z + 1, \tau, \sigma) &= \Gamma(z, \tau + 1, \sigma) = \Gamma(z, \tau, \sigma + 1) = \Gamma(z, \tau, \sigma) \\ \Gamma(z + \tau, \tau, \sigma) &= \theta(z, \sigma)\Gamma(z, \tau, \sigma) \\ \Gamma(z + \tau + \sigma, \tau, \sigma) &= \Gamma(-z, \tau, \sigma)^{-1} \end{aligned} $$

Finally, if $\sigma/\tau \notin \mathbb{R}$ then

$$ \Gamma(z, \tau, \sigma)^{-1} \Gamma\left(\frac{z}{\tau}, \frac{-1}{\tau}, \frac{\sigma}{\tau}\right) \Gamma\left(\frac{z-\tau}{\sigma}, \frac{-\tau}{\sigma}, \frac{-1}{\sigma}\right)^{-1} = \exp(i\pi P_1(z, \tau, \sigma)) \quad (2) $$

where

$$ P_1(z, \tau, \sigma) = \frac{z^3}{3\tau\sigma} - \frac{\tau + \sigma - 1}{2\tau\sigma}z^2 + \frac{\tau^2 + \sigma^2 + 3\tau\sigma - 3\tau - 3\sigma + 1}{6\tau\sigma}z + \frac{1}{12}(\tau + \sigma - 1)\left(\frac{1}{\tau} + \frac{1}{\sigma} - 1\right) $$

These properties are transformation properties for the elliptic Gamma function under the action of $SL_3(\mathbb{Z})$. This will be more apparent for the geometric elliptic Gamma functions which we introduce now.

### 2.1.2 Geometric variants of the elliptic Gamma function

In [9], Felder, Henriques, Rossi and Zhu have built elliptic Gamma functions associated to pairs of primitive vectors in a free $\mathbb{Z}$-module of rank 3. We briefly review the construction which we will adapt to Nishizawa's $G_r$ functions. Consider two linearly independent primitive vectors $a, b$ in $\Lambda$ a free $\mathbb{Z}$-module of rank 3. Fix a basis $B$ (and therefore an orientation) of $\Lambda$ and write $\det_B(a, b, \cdot) = s\gamma$ for a unique integer $s > 0$ and a unique linear form
## 2.1 The geometric elliptic Gamma functions

### 2.1.1 The elliptic Gamma function

The elliptic Gamma function was first introduced by Ruijsenaars [21] and was studied later at length by Felder and Varchenko [12] who made the connexion with the above $\theta$ function. For $z \in \mathbb{C}$ and $\tau, \sigma \in \mathbb{H}$ put

$$ \Gamma(z, \tau, \sigma) = \prod_{m,n \ge 0} \left( \frac{1 - \exp(2i\pi((m+1)\tau + (n+1)\sigma - z))}{1 - \exp(2i\pi(m\tau + n\sigma + z))} \right) $$

This function converges absolutely for $z \in \mathbb{C}, z \notin \mathbb{Z} + \mathbb{Z}_{\le 0}\tau + \mathbb{Z}_{\le 0}\sigma$ and it is meromorphic in this domain. It is worth noting that this function has a nice expression as the exponential of an infinite sum involving sinuses when $|\Im(2z - \tau - \sigma)| < |\Im(\tau)| + |\Im(\sigma)|$:

$$ \Gamma(z, \tau, \sigma) = \exp \left( \sum_{j \ge 1} \frac{1}{2ij} \frac{\sin(\pi j (2z - \tau - \sigma))}{\sin(\pi j \tau) \sin(\pi j \sigma)} \right) \quad (1) $$

The range of parameters $\tau, \sigma$ can thus be extended to $\tau, \sigma \in \mathbb{C} - \mathbb{R}$ by putting:

$$ \Gamma(z, -\tau, \sigma) = \frac{1}{\Gamma(z + \tau, \tau, \sigma)}, \quad \Gamma(z, \tau, -\sigma) = \frac{1}{\Gamma(z + \sigma, \tau, \sigma)} $$

In [[12], Theorems 3.1 and 4.1], Felder and Varchenko proved the following properties for the elliptic Gamma function related to the $\theta$ function of Complex Multiplication. If $\tau, \sigma \in \mathbb{C} - \mathbb{R}$ then

$$ \begin{aligned} \Gamma(z + 1, \tau, \sigma) &= \Gamma(z, \tau + 1, \sigma) = \Gamma(z, \tau, \sigma + 1) = \Gamma(z, \tau, \sigma) \\ \Gamma(z + \tau, \tau, \sigma) &= \theta(z, \sigma)\Gamma(z, \tau, \sigma) \\ \Gamma(z + \tau + \sigma, \tau, \sigma) &= \Gamma(-z, \tau, \sigma)^{-1} \end{aligned} $$

Finally, if $\sigma/\tau \notin \mathbb{R}$ then

$$ \Gamma(z, \tau, \sigma)^{-1} \Gamma\left(\frac{z}{\tau}, \frac{-1}{\tau}, \frac{\sigma}{\tau}\right) \Gamma\left(\frac{z-\tau}{\sigma}, -\frac{\tau}{\sigma}, -\frac{1}{\sigma}\right)^{-1} = \exp(i\pi P_1(z, \tau, \sigma)) \quad (2) $$

where

$$ P_1(z, \tau, \sigma) = \frac{z^3}{3\tau\sigma} - \frac{\tau + \sigma - 1}{2\tau\sigma} z^2 + \frac{\tau^2 + \sigma^2 + 3\tau\sigma - 3\tau - 3\sigma + 1}{6\tau\sigma} z \\ + \frac{1}{12}(\tau + \sigma - 1) \left( \frac{1}{\tau} + \frac{1}{\sigma} - 1 \right) $$

These properties are transformation properties for the elliptic Gamma function under the action of $\mathrm{SL}_3(\mathbb{Z})$. This will be more apparent for the geometric elliptic Gamma functions which we introduce now.
$\gamma \in L = \operatorname{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{Z})$. The basis $B$ identifies $\Lambda \simeq \mathbb{Z}^3$ which fixes an action of $\operatorname{SL}_3(\mathbb{Z})$ on $\Lambda$. Now write

$$C(a, b) = \{\delta \in L, \delta(a) > 0, \delta(b) \le 0\}$$

The geometric elliptic $\Gamma_{a,b}$ function associated to the pair $(a, b)$ is defined as follows:

$$\Gamma_{a,b}(w, x, \Lambda) = \frac{\prod_{\delta \in C(a,b)/\mathbb{Z}\gamma} (1 - \exp(-2i\pi(\delta(x) - w)/\gamma(x)))}{\prod_{\delta \in C(b,a)/\mathbb{Z}\gamma} (1 - \exp(2i\pi(\delta(x) - w)/\gamma(x)))}$$

where $w \in \mathbb{C}$ and $x \in \operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{C})$ and where we understand $\delta(x) = x(\delta)$ by biduality. The study of the convergence properties of this function was carried out in [9] and is better understood with an alternative definition of this function. Consider $\alpha, \beta \in L$ such that $\beta(a) = \alpha(b) = 0$, $\alpha(a) > 0$, and $\beta(b) > 0$. We call such a pair a positive dual family to $(a, b)$. We define

$$F(\alpha, \beta) = \{\delta \in L, 0 \le \delta(a) < \alpha(a), 0 \le \delta(b) < \beta(b)\}$$

Then in [9] the authors prove that the following alternative definition holds:

$$\Gamma_{a,b}(w, x, \Lambda) = \prod_{\delta \in F(\alpha, \beta)/\mathbb{Z}\gamma} \Gamma\left(\frac{w + \delta(x)}{\gamma(x)}, \frac{\alpha(x)}{\gamma(x)}, \frac{\beta(x)}{\gamma(x)}\right) \quad (3)$$

The right-hand side is a finite product of elliptic Gamma functions and is independent of the choice for $\alpha, \beta$. As a function of $x$ it converges outside the hyperplane $\gamma(x) = 0$ whenever $\alpha(x)/\gamma(x)$ and $\beta(x)/\gamma(x)$ are not real. As a function of $w$, it converges outside the discrete set of poles of every elliptic Gamma function involved. Note that the conditions $\alpha(x)/\gamma(x), \beta(x)/\gamma(x) \in \mathbb{R}$ describe a $\mathbb{R}$-hyperplane of the $\mathbb{R}$-vector space $\operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{C})$, so that the geometric elliptic Gamma function is well-defined on a dense open set of $\mathbb{C} \times \operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{C})$ endowed with the finite dimensional $\mathbb{C}$-vector space topology.

**Remark:** Formula (3) could have been taken as the definition of the geometric elliptic Gamma function associated to the pair $(a, b)$. This will be done below (see Proposition 4) when considering $G_r$ functions associated to $r+1$ vectors in a free $\mathbb{Z}$-module of rank $r+2$. It will remain to check that this definition is independent of the choice for $\alpha, \beta$, and this can be done by direct computation in reverting the proof of formula (3) carried out in [9].

On the computational side, the geometric function $\Gamma_{a,b}$ is here computed as a product of $\#|F(\alpha, \beta)/\mathbb{Z}\gamma|$ ordinary elliptic Gamma functions. A standard choice gives $\alpha(a) = \beta(b) = s = \#|F/\mathbb{Z}\gamma|$. This choice is given by considering a third vector $c$ such that $\det_B(a, b, c) = s$ and writing the transpose of the comatrix of $(a, b, c)$ as $(\alpha, \beta, s\gamma)^T$. The properties of the elliptic
### 2.1.2 Geometric variants of the elliptic Gamma function

In [11], Felder, Henriques, Rossi and Zhu have built elliptic Gamma functions associated to pairs of primitive vectors in a free $\mathbb{Z}$-module of rank 3. We briefly review the construction which we will adapt to Nishizawa's $G_r$ functions. Consider two linearly independent primitive vectors $a, b$ in $\Lambda$ a free $\mathbb{Z}$-module of rank 3. Fix a basis $B$ (and therefore an orientation) of $\Lambda$ and write $\det_B(a, b, \cdot) = s\gamma$ for a unique integer $s > 0$ and a unique linear form $\gamma \in L = \operatorname{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{Z})$. The basis $B$ identifies $\Lambda \simeq \mathbb{Z}^3$ which fixes an action of $\operatorname{SL}_3(\mathbb{Z})$ on $\Lambda$. Now write

$$C(a, b) = \{\delta \in L, \delta(a) > 0, \delta(b) \le 0\}$$

The geometric elliptic $\Gamma_{a,b}$ function associated to the pair $(a, b)$ is defined as follows:

$$\Gamma_{a,b}(w, x, \Lambda) = \frac{\prod_{\delta \in C(a,b)/\mathbb{Z}\gamma} (1 - \exp(-2i\pi(\delta(x) - w)/\gamma(x)))}{\prod_{\delta \in C(b,a)/\mathbb{Z}\gamma} (1 - \exp(2i\pi(\delta(x) - w)/\gamma(x)))}$$

where $w \in \mathbb{C}$ and $x \in \operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{C})$ and where we understand $\delta(x) = x(\delta)$ by biduality. The study of the convergence properties of this function was carried out in [11] and is better understood with an alternative definition of this function. Consider $\alpha, \beta \in L$ such that $\beta(a) = \alpha(b) = 0$, $\alpha(a) > 0$, and $\beta(b) > 0$. We call such a pair a positive dual family to $(a, b)$. We define

$$F(\alpha, \beta) = \{\delta \in L, 0 \le \delta(a) < \alpha(a), 0 \le \delta(b) < \beta(b)\}$$

Then in [11] the authors prove that the following alternative definition holds:

$$\Gamma_{a,b}(w, x, \Lambda) = \prod_{\delta \in F(\alpha, \beta)/\mathbb{Z}\gamma} \Gamma\left(\frac{w + \delta(x)}{\gamma(x)}, \frac{\alpha(x)}{\gamma(x)}, \frac{\beta(x)}{\gamma(x)}\right) \quad (3)$$

The right-hand side is a finite product of elliptic Gamma functions and is independent of the choice for $\alpha, \beta$. As a function of $x$ it converges outside the hyperplane $\gamma(x) = 0$ whenever $\alpha(x)/\gamma(x)$ and $\beta(x)/\gamma(x)$ are not real. As a function of $w$, it converges outside the discrete set of poles of every elliptic Gamma function involved. Note that the conditions $\alpha(x)/\gamma(x), \beta(x)/\gamma(x) \in \mathbb{R}$ describe a $\mathbb{R}$-hyperplane of the $\mathbb{R}$-vector space $\operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{C})$, so that the geometric elliptic Gamma function is well-defined on a dense open set of $\mathbb{C} \times \operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{C})$ endowed with the finite dimensional $\mathbb{C}$-vector space topology.

**Remark:** Formula (3) could have been taken as the definition of the geometric elliptic Gamma function associated to the pair $(a, b)$. This will be done below (see Proposition 4) when considering $G_r$ functions associated to
Gamma function translate into nice properties for the geometric $\Gamma_{a,b}$ functions under the action of $SL_3(\mathbb{Z})$ as shown in [9]. Namely, in the domains where every term converges (which are dense open sets), we have the following inversion relation:

$$ \Gamma_{a,b}(w, x, \Lambda)\Gamma_{b,a}(w, x, \Lambda) = 1 $$

We also have for $a, b, c$ linearly independent:

$$ \Gamma_{a,b}(w, x, \Lambda)\Gamma_{b,c}(w, x, \Lambda)\Gamma_{c,a}(w, x, \Lambda) = \exp(i\pi P_{a,b,c}(w, x, \Lambda)) \quad (4) $$

where $P_{a,b,c}$ is a polynomial in $w$ of degree 3. Furthermore, we have the equivariance relations: for all $g \in SL_3(\mathbb{Z})$,

$$ \Gamma_{ga,gb}(w, gx, \Lambda) = \Gamma_{a,b}(w, x, \Lambda) $$

$$ P_{ga,gb,gc}(w, gx, \Lambda) = P_{a,b,c}(w, x, \Lambda) $$

The polynomial $P_{a,b,c}$ will be presented below alongside other Bernoulli polynomials serving the same purpose for higher degree (see definition 5 in section 2.2.3). It is worth noting that the 3-term relation (4) can be understood as a 1-cocycle relation for $SL_3(\mathbb{Z})$. Consider the cocycle $\psi_a(g) = \Gamma_{a,ga}$ for a primitive vector $a \in \Lambda$ and $g \in SL_3(\mathbb{Z})$. Then we may rewrite formula (4) for fixed $w, x$ as

$$ \Gamma_{a,g_1a}\Gamma_{g_1a,g_1g_2a}\Gamma_{g_1g_2a,a} = \exp(i\pi P_{a,g_1a,g_1g_2a}) $$

which yields the coboundary relation:

$$ d\psi_a(g_1, g_2) := \psi_a(g_1)(g_1 \cdot \psi_a(g_2))\psi_a(g_1g_2)^{-1} = \exp(i\pi P_{a,g_1a,g_1g_2a}) \quad (5) $$

## 2.2 Multiple elliptic Gamma functions

Here we recall the definition and properties of Nishizawa's $G_r$ functions which generalise the $\theta$ and elliptic Gamma functions above, and we construct their geometric variants $G_{r,a}$ upgrading the construction in [9] to higher degrees.

### 2.2.1 Nishizawa's $G_r$ functions

In [13], Nishizawa introduced the hierarchy of $G_r$ functions that generalise both the $\theta$ and the elliptic Gamma functions. Later, Narukawa proved the modularity property for the $G_r$ functions by relating them to multiple sine
$r + 1$ vectors in a free $\mathbb{Z}$-module of rank $r + 2$. It will remain to check that this definition is independent of the choice for $\alpha, \beta$, and this can be done by direct computation in reverting the proof of formula (3) carried out in [11].

On the computational side, the geometric function $\Gamma_{a,b}$ is here computed as a product of $\#|F(\alpha, \beta)/\mathbb{Z}\gamma|$ ordinary elliptic Gamma functions. A standard choice gives $\alpha(a) = \beta(b) = s = \#|F/\mathbb{Z}\gamma|$. This choice is given by considering a third vector $c$ such that $\det_B(a, b, c) = s$ and writing the transpose of the comatrix of $(a, b, c)$ as $(\alpha, \beta, s\gamma)^T$. The properties of the elliptic Gamma function translate into nice properties for the geometric $\Gamma_{a,b}$ functions under the action of $\text{SL}_3(\mathbb{Z})$ as shown in [11]. Namely, in the domains where every term converges (which are dense open sets), we have the following inversion relation:

$$ \Gamma_{a,b}(w, x, \Lambda)\Gamma_{b,a}(w, x, \Lambda) = 1 $$

We also have for $a, b, c$ linearly independent:

$$ \Gamma_{a,b}(w, x, \Lambda)\Gamma_{b,c}(w, x, \Lambda)\Gamma_{c,a}(w, x, \Lambda) = \exp(i\pi P_{a,b,c}(w, x, \Lambda)) \quad (4) $$

where $P_{a,b,c}$ is a polynomial in $w$ of degree 3. Furthermore, we have the equivariance relations: for all $g \in \text{SL}_3(\mathbb{Z})$,

$$ \Gamma_{ga,gb}(w, gx, \Lambda) = \Gamma_{a,b}(w, x, \Lambda) $$

$$ P_{ga,gb,gc}(w, gx, \Lambda) = P_{a,b,c}(w, x, \Lambda) $$

The polynomial $P_{a,b,c}$ will be presented below alongside other Bernoulli polynomials serving the same purpose for higher degree (see definition 5 in section 2.2.3). It is worth noting that the 3-term relation (4) can be understood as a 1-cocycle relation for $\text{SL}_3(\mathbb{Z})$. Consider the cocycle $\psi_a(g) = \Gamma_{a,ga}$ for a primitive vector $a \in \Lambda$ and $g \in \text{SL}_3(\mathbb{Z})$. Then we may rewrite formula (4) for fixed $w, x$ as

$$ \Gamma_{a,g_1a}\Gamma_{g_1a,g_1g_2a}\Gamma_{g_1g_2a,a} = \exp(i\pi P_{a,g_1a,g_1g_2a}) $$

which yields the coboundary relation:

$$ d\psi_a(g_1, g_2) := \psi_a(g_1)(g_1 \cdot \psi_a(g_2))\psi_a(g_1g_2)^{-1} = \exp(i\pi P_{a,g_1a,g_1g_2a}) \quad (5) $$

## 2.2 Multiple elliptic Gamma functions

Here we recall the definition and properties of Nishizawa's $G_r$ functions which generalise the $\theta$ and elliptic Gamma functions above, and we construct their geometric variants $G_{r,a}$ upgrading the construction in [11] to higher degrees.
functions. To simplify the notations, we will write as in [13], with a clear separation for base and exponential variables:

$$ \underline{\tau} = (\tau_j)_{0 \le j \le r} \in \mathbb{C}^{r+1} ; \quad \underline{q} = (q_j)_{0 \le j \le r} = (\exp(2i\pi\tau_j))_{0 \le j \le r} $$

$$ |\underline{\tau}| = \sum_{j=0}^{r} \tau_j ; \quad |\underline{q}| = \prod_{j=0}^{r} q_j $$

$$ \underline{\tau}^{-}(j) = (\tau_{n})_{0 \le n \neq j \le r} ; \quad \underline{q}^{-}(j) = (q_{n})_{0 \le n \neq j \le r} $$

$$ \underline{\tau}[j] = (\tau_0, \dots, \tau_{j-1}, -\tau_j, \tau_{j+1}, \dots, \tau_r) ; \quad \underline{q}[j] = (q_0, \dots, q_{j-1}, q_j^{-1}, q_{j+1}, \dots, q_r) $$

Moreover, for a $(r+1)$-uple $\underline{m} = (m_0, \dots, m_r) \in \mathbb{Z}^{r+1}$, we write:

$$ \underline{m} + \underline{1} = (m_j + 1)_{0 \le j \le r}, \quad \underline{m} \cdot \underline{\tau} = (m_j \tau_j)_{0 \le j \le r}, \quad \underline{q}^{\underline{m}} = (q_j^{m_j})_{0 \le j \le r} $$

$$ \prod_{\underline{m} \ge 0} = \prod_{m_0 \ge 0} \cdots \prod_{m_r \ge 0} $$

Let $r \ge 0$ and $\underline{\tau} = (\tau_0, \dots, \tau_r) \in \mathbb{H}^{r+1}$. For $z \in \mathbb{C}$, $(z \notin \mathbb{Z} + \sum_{j=0}^r \mathbb{Z}_{\le 0} \tau_j)$ if $r$ is odd), put $x = \exp(2i\pi z)$ and define as in [13]:

$$ G_r(z, \underline{\tau}) = \prod_{\underline{m} \ge 0} (1 - |\underline{q}^{\underline{m}+1}| x^{-1}) (1 - |\underline{q}^{\underline{m}}| x)^{(-1)^r} $$

Note that $G_0$ and $G_1$ are respectively the $\theta$ function and the elliptic Gamma function. It is once again worth noting that the $G_r$ functions have nice expressions as the exponential of infinite sums involving sinuses (see [[13], Proposition 3.6]), namely:

$$ G_r(z, \underline{\tau}) = \begin{cases} \exp \left( \sum_{j \ge 1} \frac{1}{(2i)^r j} \frac{\sin(\pi j (2z - |\underline{\tau}|))}{\prod_{k=0}^r \sin(\pi j \tau_k)} \right) & \text{if } r \text{ is odd} \\ \exp \left( \sum_{j \ge 1} \frac{2}{(2i)^{r+1} j} \frac{\cos(\pi j (2z - |\underline{\tau}|))}{\prod_{k=0}^r \sin(\pi j \tau_k)} \right) & \text{if } r \text{ is even} \end{cases} \quad (6) $$

This expression allows us to extend the range of parameters to $\underline{\tau} \in (\mathbb{C} - \mathbb{R})^{r+1}$ by putting:

$$ G_r(z, \underline{\tau}[j]) = G_r(z + \tau_j, \underline{\tau})^{-1} \quad (7) $$

The $G_r$ functions satisfy modular properties similar to that of the elliptic Gamma function, as proven by Nishizawa in [13] and later by Narukawa in [12]. First, the $G_r$ functions are 1-periodic in each of their arguments and they satisfy:

$$ \begin{aligned} G_r(z + |\underline{\tau}|, \underline{\tau}) &= G_r(-z, \underline{\tau})^{(-1)^r} \\ G_r(-z, -\underline{\tau}) &= G_r(z, \underline{\tau})^{-1} \end{aligned} \quad (8) $$
### 2.2.1 Nishizawa's $G_r$ functions

In [16], Nishizawa introduced the hierarchy of $G_r$ functions that generalise both the $\theta$ and the elliptic Gamma functions. Later, Narukawa proved the modularity property for the $G_r$ functions by relating them to multiple sine functions. To simplify the notations, we will write as in [16], with a clear separation for base and exponential variables:

$$ \underline{\tau} = (\tau_j)_{0 \le j \le r} \in \mathbb{C}^{r+1} ; \quad \underline{q} = (q_j)_{0 \le j \le r} = (\exp(2i\pi\tau_j))_{0 \le j \le r} $$

$$ |\underline{\tau}| = \sum_{j=0}^{r} \tau_j ; \quad |\underline{q}| = \prod_{j=0}^{r} q_j $$

$$ \underline{\tau}^{-}(j) = (\tau_n)_{0 \le n \neq j \le r} ; \quad \underline{q}^{-}(j) = (q_n)_{0 \le n \neq j \le r} $$

$$ \underline{\tau}[j] = (\tau_0, \dots, \tau_{j-1}, -\tau_j, \tau_{j+1}, \dots, \tau_r) ; \quad \underline{q}[j] = (q_0, \dots, q_{j-1}, q_j^{-1}, q_{j+1}, \dots, q_r) $$

Moreover, for a $(r+1)$-uple $\underline{m} = (m_0, \dots, m_r) \in \mathbb{Z}^{r+1}$, we write:

$$ \underline{m} + \underline{1} = (m_j + 1)_{0 \le j \le r}, \quad \underline{m} \cdot \underline{\tau} = (m_j \tau_j)_{0 \le j \le r}, \quad \underline{q}^{\underline{m}} = (q_j^{m_j})_{0 \le j \le r} $$

$$ \prod_{\underline{m} \ge 0} = \prod_{m_0 \ge 0} \cdots \prod_{m_r \ge 0} $$

Let $r \ge 0$ and $\underline{\tau} = (\tau_0, \dots, \tau_r) \in \mathbb{H}^{r+1}$. For $z \in \mathbb{C}$, $(z \notin \mathbb{Z} + \sum_{j=0}^r \mathbb{Z}_{\le 0} \tau_j)$ if $r$ is odd), put $x = \exp(2i\pi z)$ and define as in [16]:

$$ G_r(z, \underline{\tau}) = \prod_{\underline{m} \ge 0} (1 - |\underline{q}^{\underline{m}+1}| x^{-1}) (1 - |\underline{q}^{\underline{m}}| x)^{(-1)^r} $$

Note that $G_0$ and $G_1$ are respectively the $\theta$ function and the elliptic Gamma function. It is once again worth noting that the $G_r$ functions have nice expressions as the exponential of infinite sums involving sinuses (see [[16], Proposition 3.6]), namely:

$$ G_r(z, \underline{\tau}) = \begin{cases} \exp \left( \sum_{j \ge 1} \frac{1}{(2i)^r j} \frac{\sin(\pi j (2z - |\underline{\tau}|))}{\prod_{k=0}^r \sin(\pi j \tau_k)} \right) & \text{if } r \text{ is odd} \\ \exp \left( \sum_{j \ge 1} \frac{2}{(2i)^{r+1} j} \frac{\cos(\pi j (2z - |\underline{\tau}|))}{\prod_{k=0}^r \sin(\pi j \tau_k)} \right) & \text{if } r \text{ is even} \end{cases} \quad (6) $$

This expression is valid provided that $\underline{\tau} \in (\mathbb{C} - \mathbb{R})^{r+1}$ and $|\mathfrak{F}(2z - |\underline{\tau}|)| < \sum_{j=0}^r |\mathfrak{F}(\tau_j)|$ and allows us to extend the range of parameters to $\underline{\tau} \in (\mathbb{C} - \mathbb{R})^{r+1}$ by putting:

$$ G_r(z, \underline{\tau}[j]) = G_r(z + \tau_j, \underline{\tau})^{-1} \quad (7) $$

The $G_r$ functions satisfy modular properties similar to that of the elliptic Gamma function, as proven by Nishizawa in [16] and later by Narukawa
Furthermore, if $r \ge 1$, the $G_r$ functions are almost periodic in $z$ with periods $\tau_j$ for $0 \le j \le r$ with a correction factor involving a lower degree function:

$$G_r(z + \tau_j, \underline{\tau}) = G_{r-1}(z, \underline{\tau}^{-}(j))G_r(z, \underline{\tau})$$

A consequence of these properties which we will use is the following formula for complex conjugation:

$$G_r(\bar{z}, \bar{\tau}_0, \dots, \bar{\tau}_r) = \overline{G_r(z, \tau_0, \dots, \tau_r)}^{(-1)^r} \quad (9)$$

### 2.2.2 Bernoulli polynomials and Narukawa's theorem

The last missing property is crucial for arithmetic purposes and was later proved by Narukawa. To state Narukawa's theorem, we first need to introduce the multiple Bernoulli polynomials. Let $\underline{\omega} = (\omega_1, \dots, \omega_d) \in (\mathbb{C} - \{0\})^d$ with $\omega_j \ne 0$ for all $1 \le j \le d$. We define the multiple Bernoulli polynomials $B_{d,n}^*(z, \underline{\omega})$ with the following generating function:

$$e^{zt} \prod_{j=1}^{d} \frac{\omega_j t}{e^{\omega_j t} - 1} = \sum_{n \ge 0} B_{d,n}^*(z, \underline{\omega}) \frac{t^n}{n!} = \left( \sum_{m \ge 0} \frac{z^m t^m}{m!} \right) \prod_{1 \le j \le d} \left( \sum_{k_j \ge 0} B_{k_j} \omega_j^{k_j} \frac{t^{k_j}}{k_j!} \right)$$

where the $B_{k_j}$ are the usual Bernoulli numbers. The classic Bernoulli polynomials are obtained with the choice $d = 1$, $\omega_1 = 1$, and the Bernoulli numbers are obtained for $z = 0$. These polynomials obey many relations which we sum up below and which can easily be obtained from the properties of the generating function. For $\underline{\omega} \in (\mathbb{C} - \{0\})^d$, $B_{d,n}^*(z, \underline{\omega})$ is a degree $n$ homogeneous polynomial in $d+1$ variables, which is symmetric in the $d$ variables of $\underline{\omega}$. Moreover, the coefficients of $B_{d,n}^*(z, \underline{\omega})$ are rational numbers. In [12], Narukawa used the rescaled version $B_{d,n}(z, \underline{\omega}) = (\prod_{j=1}^d \omega_j^{-1}) B_{d,n}^*(z, \underline{\omega})$ and proved the following properties:

$$B_{d,n}(\alpha z, \alpha \underline{\omega}) = \alpha^{n-d} B_{d,n}(z, \underline{\omega}), \quad \forall \alpha \in \mathbb{C} - \{0\}$$

$$B_{d,n}(z + |\underline{\omega}|, \underline{\omega}) = (-1)^n B_{d,n}(-z, \underline{\omega})$$

$$B_{d,n}(z + \omega_j, \underline{\omega}) = B_{d,n}(z, \underline{\omega}) + n B_{d-1,n-1}(z, \underline{\omega}^{-}(j))$$

$$B_{d,n}(z + \omega_j, \underline{\omega}) = -B_{d,n}(z, \underline{\omega}[j])$$

$$\frac{\partial}{\partial z} B_{d,n}(z, \underline{\omega}) = n B_{d,n-1}(z, \underline{\omega})$$

Narukawa's modularity theorem (see [[12], Theorem 7]) involves the polynomials $B_{d,d}^*$ or rather the rescaled versions $B_{d,d}$. For instance, we have

$$(\omega_1 \omega_2) B_{2,2}(z, \omega_1, \omega_2) = B_{2,2}^*(z, \omega_1, \omega_2) = z^2 - z(\omega_1 + \omega_2) + \frac{\omega_1^2 + \omega_2^2 + 3\omega_1\omega_2}{6}$$
in [14]. First, the $G_r$ functions are 1-periodic in each of their arguments and they satisfy:

$$ 
\begin{aligned}
G_r(z + |\underline{\tau}|, \underline{\tau}) &= G_r(-z, \underline{\tau})^{(-1)^r} \\
G_r(-z, -\underline{\tau}) &= G_r(z, \underline{\tau})^{-1}
\end{aligned} 
\qquad (8) 
$$

Furthermore, if $r \ge 1$, the $G_r$ functions are almost periodic in $z$ with periods $\tau_j$ for $0 \le j \le r$ with a correction factor involving a lower degree function:

$$ G_r(z + \tau_j, \underline{\tau}) = G_{r-1}(z, \underline{\tau}^{-}(j))G_r(z, \underline{\tau}) $$

A consequence of these properties which we will use is the following formula for complex conjugation:

$$ G_r(\bar{z}, \bar{\tau}_0, \dots, \bar{\tau}_r) = \overline{G_r(z, \tau_0, \dots, \tau_r)}^{(-1)^r} \qquad (9) $$

### 2.2.2 Bernoulli polynomials and Narukawa's theorem

The last missing property is crucial for arithmetic purposes and was later proved by Narukawa. To state Narukawa's theorem, we first need to introduce the multiple Bernoulli polynomials. Consider an integer $d \ge 1$. Let $\underline{\omega} = (\omega_1, \dots, \omega_d) \in (\mathbb{C} - \{0\})^d$ with $\omega_j \ne 0$ for all $1 \le j \le d$. We define the multiple Bernoulli polynomials $B_{d,n}^*(z, \underline{\omega})$ with the following generating function:

$$ e^{zt} \prod_{j=1}^{d} \frac{\omega_j t}{e^{\omega_j t} - 1} = \sum_{n \ge 0} B_{d,n}^*(z, \underline{\omega}) \frac{t^n}{n!} = \left( \sum_{m \ge 0} \frac{z^m t^m}{m!} \right) \prod_{1 \le j \le d} \left( \sum_{k_j \ge 0} B_{k_j, \omega_j} \omega_j^{k_j} \frac{t^{k_j}}{k_j!} \right) $$

where the $B_{k_j}$ are the usual Bernoulli numbers. The classic Bernoulli polynomials are obtained with the choice $d=1$, $\omega_1=1$, and the Bernoulli numbers are obtained for $z=0$. These polynomials obey many relations which we sum up below and which can easily be obtained from the properties of the generating function. For $\underline{\omega} \in (\mathbb{C} - \{0\})^d$, $B_{d,n}^*(z, \underline{\omega})$ is a degree $n$ homogeneous polynomial in $d+1$ variables, which is symmetric in the $d$ variables of $\underline{\omega}$. Moreover, the coefficients of $B_{d,n}^*(z, \underline{\omega})$ are rational numbers. In [14], Narukawa used the rescaled version $B_{d,n}(z, \underline{\omega}) = (\prod_{j=1}^d \omega_j^{-1}) B_{d,n}^*(z, \underline{\omega})$ and proved the following properties:

$$ 
\begin{aligned}
B_{d,n}(\alpha z, \alpha \underline{\omega}) &= \alpha^{n-d} B_{d,n}(z, \underline{\omega}), \quad \forall \alpha \in \mathbb{C} - \{0\} \\
B_{d,n}(z + |\underline{\omega}|, \underline{\omega}) &= (-1)^n B_{d,n}(-z, \underline{\omega}) \\
B_{d,n}(z + \omega_j, \underline{\omega}) &= B_{d,n}(z, \underline{\omega}) + n B_{d-1,n-1}(z, \underline{\omega}^{-}(j)) \\
B_{d,n}(z + \omega_j, \underline{\omega}) &= -B_{d,n}(z, \underline{\omega}[j]) \\
\frac{\partial}{\partial z} B_{d,n}(z, \underline{\omega}) &= n B_{d,n-1}(z, \underline{\omega})
\end{aligned}
$$
Then, Narukawa's theorem can be stated as follows. Let $r \ge 0$. Take $\omega \in (\mathbb{C} - \{0\})^{r+2}$ and suppose that $\omega_j/\omega_n \in \mathbb{C} - \mathbb{R}$ for all $j \ne n$. Then for $z \in \mathbb{C}$ outside the discrete set of poles of the left-hand side

$$ \prod_{j=0}^{r+1} G_r \left( \frac{z}{\omega_j}, \left( \frac{\omega_n}{\omega_j} \right)_{n \ne j} \right) = \exp \left( \frac{-2i\pi}{(r+2)!} B_{r+2, r+2}(z, \omega) \right) \quad (10) $$

### 2.2.3 Geometric $G_{r,a}$ functions

In this section we upgrade the construction of section 2.1.2 and define geometric variants of the $G_r$ functions. We also prove (Theorem 6) the modular property of the geometric variants and their equivariance property under the action of $SL_{r+2}(\mathbb{Z})$. In the literature one may find other variants of these functions, such as the $G_r$ functions associated to cones by Winding in [23]. Although our construction inspired by [9] differs from Winding's construction, it would be interesting to understand the links between the two variants. As mentioned, we will use the alternative definition of 2.1.2 to start and prove that this definition is indeed valid. Throughout this section, we consider a free $\mathbb{Z}$-module $\Lambda$ of rank $r+2$ with an orientation form given by the determinant in a fixed basis. Let us start by defining positive dual families which will be used in the rest of this work.

**Definition 1:** Let $(a_0, \dots, a_m)$ be a family of $m+1$ independent primitive vectors in a lattice $\Lambda$. We call $(\alpha_0, \dots, \alpha_m) \in L = \text{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{Z})$ a positive dual family to $\underline{a} = (a_0, \dots, a_m)$ if for all $0 \le j \le m$ the following holds:

$$ \alpha_j(a_j) > 0, \quad \alpha_j(a_k) = 0, \forall k \ne j $$

The important cases in this work will be those where $m=r$ and $m=r+1$ in the lattice $\Lambda$ of rank $r+2$. The following lemmas show that in these cases two positive dual families to the same family $\underline{a}$ are closely related.

**Lemma 2:** Let $\underline{a} = (a_0, \dots, a_r)$ be a family of $r+1$ linearly independent primitive vectors in the oriented lattice $\Lambda$ of rank $r+2$. Take two positive dual families $(\alpha_0, \dots, \alpha_r), (\alpha'_0, \dots, \alpha'_r)$ to $\underline{a}$ in $L = \text{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{Z})$. For $0 \le j \le r$ write $s_j = \alpha_j(a_j)$ and $s'_j = \alpha'_j(a_j)$. Then, there are rational numbers $r_j$ such that $\alpha_j = (s_j/s'_j)\alpha'_j + r_j\det(a_0, \dots, a_r, \cdot)$ for all $0 \le j \le r$.

**Proof:**

Consider $\gamma_j = s'_j\alpha_j - s_j\alpha'_j$. Then for all $0 \le k \le r$, we get $\gamma_j(a_k) = 0$. This means that either $\gamma_j = 0$ and then $r_j = 0$ or the $\mathbb{Q}$-linear forms $\gamma_j$ and
Narukawa's modularity theorem (see [[14], Theorem 7]) involves the polynomials $B_{d,d}^*$ or rather the rescaled versions $B_{d,d}$. For instance, we have

$$ (\omega_1\omega_2)B_{2,2}(z, \omega_1, \omega_2) = B_{2,2}^*(z, \omega_1, \omega_2) = z^2 - z(\omega_1 + \omega_2) + \frac{\omega_1^2 + \omega_2^2 + 3\omega_1\omega_2}{6} $$

Then, Narukawa's theorem can be stated as follows. Let $r \ge 0$. Take $\underline{\omega} \in (\mathbb{C} - \{0\})^{r+2}$ and suppose that $\omega_j/\omega_n \in \mathbb{C} - \mathbb{R}$ for all $j \ne n$. Then for $z \in \mathbb{C}$ outside the discrete set of poles of the left-hand side

$$ \prod_{j=0}^{r+1} G_r \left( \frac{z}{\omega_j}, \left( \frac{\omega_n}{\omega_j} \right)_{n \ne j} \right) = \exp \left( \frac{-2i\pi}{(r+2)!} B_{r+2,r+2}(z, \underline{\omega}) \right) \quad (10) $$

### 2.2.3 Geometric $G_{r,\underline{a}}$ functions

In this section we upgrade the construction of section 2.1.2 and define geometric variants of the $G_r$ functions. We also prove (Theorem 6) the modular property of the geometric variants and their equivariance property under the action of $SL_{r+2}(\mathbb{Z})$. In the literature one may find other variants of these functions, such as the $G_r$ functions associated to cones by Winding in [26]. Although our construction inspired by [11] differs from Winding's construction, it would be interesting to understand the links between the two variants. As mentioned, we will use the alternative definition of 2.1.2 to start and prove that this definition is indeed valid. Throughout this section, we consider a free $\mathbb{Z}$-module $\Lambda$ of rank $r+2$ with an orientation form given by the determinant in a fixed basis. Let us start by defining positive dual families which will be used in the rest of this work.

**Definition 1:** Let $(a_0, \dots, a_m)$ be a family of $m+1$ independent primitive vectors in a lattice $\Lambda$. We call $(\alpha_0, \dots, \alpha_m) \in L = \text{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{Z})$ a positive dual family to $\underline{a} = (a_0, \dots, a_m)$ if for all $0 \le j \le m$ the following holds:

$$ \alpha_j(a_j) > 0, \quad \alpha_j(a_k) = 0, \forall k \ne j $$

The important cases in this work will be those where $m = r$ and $m = r+1$ in the lattice $\Lambda$ of rank $r+2$. The following lemmas show that in these cases two positive dual families to the same family $\underline{a}$ are closely related.

**Lemma 2:** Let $\underline{a} = (a_0, \dots, a_r)$ be a family of $r+1$ linearly independent primitive vectors in the oriented lattice $\Lambda$ of rank $r+2$. Take two positive dual families $(\alpha_0, \dots, \alpha_r)$, $(\alpha'_0, \dots, \alpha'_r)$ to $\underline{a}$ in $L = \text{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{Z})$. For $0 \le j \le r$
$\det(a_0, \dots, a_r, \cdot)$ defined on the $\mathbb{Q}$-vector space $\Lambda \otimes \mathbb{Q}$ share the same kernel, and therefore, they must be linearly dependent.

□

**Lemma 3:** A family $\underline{a} = (a_0, \dots, a_{r+1})$ of $r+2$ linearly independent primitive vectors in a lattice $\Lambda$ of rank $r+2$ has exactly one positive dual family $\underline{\alpha}$ containing only primitive vectors. We call this family the primitive positive dual family to $\underline{a}$.

**Proof:**

Consider two such families $\underline{\alpha}, \underline{\alpha}'$ and write again $s_j = \alpha_j(a_j), s'_j = \alpha'_j(a_j)$. Consider once again $\gamma_j = s'_j\alpha_j - s_j\alpha'_j$ such that for all $0 \le k \le r+1$, $\gamma_j(a_k) = 0$. The family $(a_0, \dots, a_{r+1})$ is a basis of the $\mathbb{Q}$-vector space $\Lambda \otimes \mathbb{Q}$, so $\gamma_j = 0$ for all $0 \le j \le r+1$. This means that $\alpha_j = s_j\alpha'_j/s'_j$ and we write $s_jn_j + s'_jn'_j = d_j$ where $d_j = \gcd(s_j, s'_j)$. Then $n_j\alpha_j + n'_j\alpha'_j = d_j\alpha'_j/s'_j \in L$. Because $\alpha'_j$ is a primitive vector in $L$ by assumption, we must have $d_j = s'_j$, and then because $\alpha_j = s_j\alpha'_j/d_j$ is primitive in $L$ we must also have $d_j = s_j$. The conditions $\alpha_j(a_j) = s_j, \alpha_j(a_k) = 0$ for $k \ne j$ describe exactly one linear form on the $\mathbb{Q}$-vector space $\Lambda \otimes \mathbb{Q}$. To compute $\underline{\alpha}$ we compute the comatrix of $\underline{a}$ in a suitable basis of $\Lambda$ and we rescale the linear forms to obtain primitive elements in $L$.

□

Now we may define the geometric variants of the $G_r$ functions using lemma 2:

**Proposition 4:** Let $\underline{a} = (a_0, \dots, a_r)$ be a family of $r+1$ linearly independent primitive vectors in the oriented lattice $\Lambda$ of rank $r+2$. There is a unique $s \in \mathbb{Z}_{>0}$ and a unique primitive element $\gamma \in L = \operatorname{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{Z})$ such that $\forall c \in \Lambda$, $\det(\underline{a}, c) = s\gamma(c)$. For any choice of positive dual family $(\alpha_0, \dots, \alpha_r)$ to $\underline{a}$ in $L$ the function

$$G_{\underline{r}, \underline{a}}^{\underline{\alpha}}(w, x, \Lambda) = \prod_{\delta \in F(\underline{\alpha})/\mathbb{Z}\gamma} G_r \left( \frac{w + \delta(x)}{\gamma(x)}, \frac{1}{\gamma(x)}\underline{\alpha}(x) \right) \quad (11)$$

$$F(\underline{\alpha}) = \{\delta \in L, \forall 0 \le j \le r, 0 \le \delta(a_j) < \alpha_j(a_j)\}$$

is well defined for $(w, x)$ in a dense open set of the $\mathbb{C}$-vector space $\mathbb{C} \times \operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{C}) \simeq \mathbb{C} \times \mathbb{C}^{r+2}$ endowed with the finite dimensional $\mathbb{C}$-vector space topology. Furthermore, it is independent of the choice for $\underline{\alpha}$.

**Proof:**

The set $\operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{C})$ is a $\mathbb{R}$-vector space of dimension $2r+4$. The condition $\gamma(x) = 0$ describes a subspace of dimension $2r+2$ and the condition $\alpha_j(x)/\gamma(x) \in \mathbb{R}$ describes a subspace of dimension $2r+3$. The function $G_{\underline{r}, \underline{a}}^{\underline{\alpha}}(w, x, \Lambda)$ is therefore well defined for $x$ outside a finite union of
write $s_j = \alpha_j(a_j)$ and $s'_j = \alpha'_j(a_j)$. Then, there are rational numbers $r_j$ such that $\alpha_j = (s_j/s'_j)\alpha'_j + r_j\det(a_0, \dots, a_r, \cdot)$ for all $0 \le j \le r$.

**Proof:**

Consider $\gamma_j = s'_j\alpha_j - s_j\alpha'_j$. Then for all $0 \le k \le r$, we get $\gamma_j(a_k) = 0$. This means that either $\gamma_j = 0$ and then $r_j = 0$ or the $\mathbb{Q}$-linear forms $\gamma_j$ and $\det(a_0, \dots, a_r, \cdot)$ defined on the $\mathbb{Q}$-vector space $\Lambda \otimes \mathbb{Q}$ share the same kernel, and therefore, they must be linearly dependent. $\square$

**Lemma 3:** A family $\underline{a} = (a_0, \dots, a_{r+1})$ of $r+2$ linearly independent primitive vectors in a lattice $\Lambda$ of rank $r+2$ has exactly one positive dual family $\underline{\alpha}$ containing only primitive vectors. We call this family the primitive positive dual family to $\underline{a}$.

**Proof:**

Consider two such families $\underline{\alpha}$, $\underline{\alpha}'$ and write again $s_j = \alpha_j(a_j)$, $s'_j = \alpha'_j(a_j)$. Consider once again $\gamma_j = s'_j\alpha_j - s_j\alpha'_j$ such that for all $0 \le k \le r+1$, $\gamma_j(a_k) = 0$. The family $(a_0, \dots, a_{r+1})$ is a basis of the $\mathbb{Q}$-vector space $\Lambda \otimes \mathbb{Q}$, so $\gamma_j = 0$ for all $0 \le j \le r+1$. This means that $\alpha_j = s_j\alpha'_j/s'_j$ and we write $s_jn_j + s'_jn'_j = d_j$ where $d_j = \gcd(s_j, s'_j)$. Then $n_j\alpha_j + n'_j\alpha'_j = d_j\alpha'_j/s'_j \in L$. Because $\alpha'_j$ is a primitive vector in $L$ by assumption, we must have $d_j = s'_j$, and then because $\alpha_j = s_j\alpha'_j/d_j$ is primitive in $L$ we must also have $d_j = s_j$. The conditions $\alpha_j(a_j) = s_j$, $\alpha_j(a_k) = 0$ for $k \ne j$ describe exactly one linear form on the $\mathbb{Q}$-vector space $\Lambda \otimes \mathbb{Q}$. To compute $\underline{\alpha}$ we compute the comatrix of $\underline{a}$ in a suitable basis of $\Lambda$ and we rescale the linear forms to obtain primitive elements in $L$. $\square$

Now we may define the geometric variants of the $G_r$ functions using lemma (2) and adapting formula 3 to higher rank lattices:

**Proposition 4:** Let $\underline{a} = (a_0, \dots, a_r)$ be a family of $r+1$ linearly independent primitive vectors in the oriented lattice $\Lambda$ of rank $r+2$. There is a unique $s \in \mathbb{Z}_{>0}$ and a unique primitive element $\gamma \in L = \text{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{Z})$ such that $\forall c \in \Lambda$, $\det(\underline{a}, c) = s\gamma(c)$. For any choice of positive dual family $(\alpha_0, \dots, \alpha_r)$ to $\underline{a}$ in $L$ the function

$$G_{r, \underline{a}}^{\underline{\alpha}}(w, x, \Lambda) = \prod_{\delta \in F(\underline{\alpha})/\mathbb{Z}\gamma} G_r \left( \frac{w + \delta(x)}{\gamma(x)}, \frac{1}{\gamma(x)}\underline{\alpha}(x) \right) \quad (11)$$

$$F(\underline{\alpha}) = \{\delta \in L, \forall 0 \le j \le r, 0 \le \delta(a_j) < \alpha_j(a_j)\}$$

is well defined for $(w, x)$ in a dense open set of the $\mathbb{C}$-vector space $\mathbb{C} \times \text{Hom}_{\mathbb{Z}}(L, \mathbb{C}) \simeq \mathbb{C} \times \mathbb{C}^{r+2}$ endowed with the finite dimensional $\mathbb{C}$-vector space topology. Furthermore, it is independent of the choice for $\underline{\alpha}$.
R-hyperplanes in $\operatorname{Hom}_Z(L, \mathbb{C})$, i.e. in a dense open set. The right-hand side in (11) is a finite product of meromorphic functions in $w$ and it has a discrete set of poles.

Let us now prove that this definition is indeed independent of the choice for the positive dual family $\underline{\alpha}$. Take $x \in \operatorname{Hom}_Z(L, \mathbb{C})$ such that $\gamma(x) \neq 0$ and $\alpha_j(x)/\gamma(x) \notin \mathbb{R}$ for all $0 \le j \le r$. Put for $0 \le j \le r$, $d_j = \pm 1$ such that $d_j\alpha_j(x)/\gamma(x) \in \mathbb{H}$. Put also $D = \sum_{j=0}^r (d_j - 1)/2$. Then using the inversion relation (7) we get:

$$
\begin{align*}
G_{r, \underline{\alpha}}^{\underline{\alpha}}(w, x, \Lambda)^{(-1)^D} &= \prod_{\delta \in F(\underline{\alpha})/\mathbb{Z}\gamma} G_r \left( \frac{w + \delta(x)}{\gamma(x)} + \sum_{j=0}^{r} \frac{d_j - 1}{2} \frac{\alpha_j(x)}{\gamma(x)}, \left( \frac{1}{\gamma(x)} d_j \alpha_j(x) \right) \Big|_{0 \le j \le r} \right) \\
G_{r, \underline{\alpha}}^{\underline{\alpha}}(w, x, \Lambda)^{(-1)^D} &= \prod_{\delta \in F(\underline{\alpha})/\mathbb{Z}\gamma} \prod_{\underline{m} \ge 0} \left[ \left( 1 - e^{2i\pi \left( \sum_{j=0}^r \frac{(d_j m_j + (1+d_j)/2)\alpha_j(x)}{\gamma(x)} - \frac{w+\delta(x)}{\gamma(x)} \right)} \right) \right. \\
&\qquad \left. \times \left( 1 - e^{2i\pi \left( \sum_{j=0}^r \frac{(d_j m_j + (1-d_j)/2)\alpha_j(x)}{\gamma(x)} + \frac{w+\delta(x)}{\gamma(x)} \right)} \right)^{(-1)^r} \right]
\end{align*}
$$

Now write $C^{\pm}(\underline{\alpha}, x)$ for the set of $\delta' \in L$ satisfying for all $0 \le j \le r$:

$$
\begin{cases}
\delta'(a_j) > 0, & \text{if } d_j = \pm 1 \\
\delta'(a_j) \le 0, & \text{if } d_j = \mp 1
\end{cases}
$$

Consider $\delta' \in C^+(\underline{\alpha}, x)$. This means that if $d_j = 1$ then $\delta'(a_j) > 0$ and we can perform Euclidian division by $\alpha_j(a_j)$ so that there exists a unique integer $m_j > 0$ such that $0 \le (m_j\alpha_j - \delta')(a_j) < \alpha_j(a_j)$. On the contrary, if $d_j = -1$ then $\delta'(a_j) \le 0$ and there exists a unique integer $m_j \ge 0$ such that $0 \le (-m_j\alpha_j - \delta')(a_j) < \alpha_j(a_j)$. Then for all $0 \le k \le r$:

$$
0 \le \left( \sum_{j=0}^{r} d_j m_j \alpha_j - \delta' \right) (a_k) < \alpha_k(a_k)
$$

This shows that $C^+(\underline{\alpha}, x)$ can be written as a disjoint union

$$
C^+(\underline{\alpha}, x) = \bigcup_{\delta \in F(\underline{\alpha})/\mathbb{Z}\gamma} \bigcup_{\underline{m} \ge 0} \left\{ -\delta + \sum_{j=0}^{r} (d_j m_j + (1+d_j)/2) \alpha_j \right\} + \mathbb{Z}\gamma
$$

and a similar statement holds for $C^-(\underline{\alpha}, x)$ so that $G_{r,\underline{\alpha}}^{\underline{\alpha}}(w, x, \Lambda)^{(-1)^D}$ is equal to

$$
\prod_{\delta' \in C^+(\underline{\alpha}, x)/\mathbb{Z}\gamma} \left(1 - e^{2i\pi \left(\frac{\delta'(x)-w}{\gamma(x)}\right)}\right) \prod_{\delta' \in C^-(\underline{\alpha}, x)/\mathbb{Z}\gamma} \left(1 - e^{-2i\pi \left(\frac{\delta'(x)-w}{\gamma(x)}\right)}\right)
$$
**Proof :**

The set $\operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{C})$ is a $\mathbb{R}$-vector space of dimension $2r + 4$. The condition $\gamma(x) = 0$ describes a subspace of dimension $2r + 2$ and the condition $\alpha_j(x)/\gamma(x) \in \mathbb{R}$ describes a subspace of dimension $2r + 3$. The function $G_{r, \underline{a}}^{\alpha}(w, x, \Lambda)$ is therefore well defined for $x$ outside a finite union of $\mathbb{R}$-hyperplanes in $\operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{C})$, i.e. in a dense open set. The right-hand side in (11) is a finite product of meromorphic functions in $w$ and it has a discrete set of poles.

Let us now prove that this definition is indeed independent of the choice for the positive dual family $\underline{a}$. Take $x \in \operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{C})$ such that $\gamma(x) \neq 0$ and $\alpha_j(x)/\gamma(x) \notin \mathbb{R}$ for all $0 \le j \le r$. Put for $0 \le j \le r$, $d_j = \pm 1$ such that $d_j\alpha_j(x)/\gamma(x) \in \mathbb{H}$. Put also $D = \sum_{j=0}^{r}(d_j - 1)/2$. Then using the inversion relation (7) we get:

$$G_{r, \underline{a}}^{\alpha}(w, x, \Lambda)^{(-1)D} = \prod_{\delta \in F(\underline{a})/\mathbb{Z}\gamma} G_r \left( \frac{w + \delta(x)}{\gamma(x)} + \sum_{j=0}^{r} \frac{d_j - 1}{2} \frac{\alpha_j(x)}{\gamma(x)}, \left( \frac{1}{\gamma(x)} d_j \alpha_j(x) \right)_{0 \le j \le r} \right)$$

$$G_{r, \underline{a}}^{\alpha}(w, x, \Lambda)^{(-1)D} = \prod_{\delta \in F(\underline{a})/\mathbb{Z}\gamma} \prod_{\underline{m} \ge 0} \left[ \left( 1 - e^{2i\pi \left( \sum_{j=0}^{r} \frac{(d_j m_j + (1+d_j)/2)\alpha_j(x)}{\gamma(x)} - \frac{w+\delta(x)}{\gamma(x)} \right)} \right) \right. \\ \left. \times \left( 1 - e^{2i\pi \left( \sum_{j=0}^{r} \frac{(d_j m_j + (1-d_j)/2)\alpha_j(x)}{\gamma(x)} + \frac{w+\delta(x)}{\gamma(x)} \right)} \right)^{(-1)^r} \right]$$

Now write $C^{\pm}(\underline{a}, x)$ for the set of $\delta' \in L$ satisfying for all $0 \le j \le r$:

$$\begin{cases} \delta'(a_j) > 0, & \text{if } d_j = \pm 1 \\ \delta'(a_j) \le 0, & \text{if } d_j = \mp 1 \end{cases}$$

Consider $\delta' \in C^{+}(\underline{a}, x)$. This means that if $d_j = 1$ then $\delta'(a_j) > 0$ and we can perform Euclidian division by $\alpha_j(a_j)$ so that there exists a unique integer $m_j > 0$ such that $0 \le (m_j\alpha_j - \delta')(a_j) < \alpha_j(a_j)$. On the contrary, if $d_j = -1$ then $\delta'(a_j) \le 0$ and there exists a unique integer $m_j \ge 0$ such that $0 \le (-m_j\alpha_j - \delta')(a_j) < \alpha_j(a_j)$. Then for all $0 \le k \le r$:

$$0 \le \left( \sum_{j=0}^{r} d_j m_j \alpha_j - \delta' \right) (a_k) < \alpha_k(a_k)$$
Then, we only need to show that the sets $C^{\pm}(\underline{\alpha}, x)$ are independent of the choice for $\underline{\alpha}$. Consider another positive dual family $\underline{\alpha}'$ to $\underline{\alpha}$. Write $\alpha_j(a_j) = s_j > 0$ and $\alpha'_j(a_j) = s'_j > 0$. Then from lemma 2 there is a rational number $r_j$ such that

$$ \alpha'_{j} = \frac{s'_{j}}{s_{j}}\alpha_{j} + r_{j}\gamma $$

For any $x \in \mathrm{Hom}_{\mathbb{Z}}(L, \mathbb{C})$ we get:

$$ d_j \frac{\alpha'_j(x)}{\gamma(x)} = d_j r_j + d_j \frac{s'_j}{s_j} \frac{\alpha_j(x)}{\gamma(x)} \in \mathbb{H} $$

This shows that the signs $d_j$ (and therefore also $D$) are independent of the choice for $\underline{\alpha}$. By construction we get $C^{\pm}(\underline{\alpha}, x) = C^{\pm}(\underline{\alpha}', x)$ and the definition of the geometric $G_{\underline{r},\underline{\alpha}}^{\underline{\alpha}}$ function is independent of the choice for $\underline{\alpha}$. $\square$

From now on we denote by $G_{\underline{r},\underline{\alpha}} := G_{\underline{r},\underline{\alpha}}^{\underline{\alpha}}$ the geometric $G_{\underline{r}}$ function associated to $\underline{\alpha}$ for any suitable choice of $\underline{\alpha}$. A choice that is interesting for us is defined by the comatrix or rather by the rescaled comatrix. Let $c$ be a vector such that $\gamma(c) = 1$. Then the elementary divisors of the concatenation $(\underline{\alpha}, c)$ can be written as $[A_r, A_{r-1}, \dots, A_1, 1, 1]$ with $A_i|A_{i+1}$ so that a possible choice for $\underline{\alpha}$ is given by:

$$ (\underline{\alpha} \ c) \begin{pmatrix} \underline{\alpha} \\ A_r \gamma \end{pmatrix} = A_r I_{r+2} \qquad (12) $$

To express the transformation properties of the function $G_{\underline{r},\underline{\alpha}}$ under the action of $\mathrm{SL}_{r+2}(\mathbb{Z})$ we need to introduce the family of geometric Bernoulli polynomials which encompass $P_{a,b,c}$. In what follows, we want to discuss an analogue of formula (4) for $G_{\underline{r},\underline{\alpha}}$, and we must consider families of $r+2$ vectors. Recall that we have defined the polynomials $B_{\underline{r}+2,\underline{r}+2}^*$ in section 2.2.2.

**Definition 5:** Let $w \in \mathbb{C}$ and $x \in \mathrm{Hom}_{\mathbb{Z}}(L, \mathbb{C}) \simeq \mathbb{C}^{r+2}$. Let $a_0, \dots, a_r, a_{r+1}$ be a family of $r+2$ linearly independent primitive vectors in the oriented lattice $\Lambda$. Let $\epsilon$ be the sign of $(-1)^{r+1}\det(a_0, \dots, a_{r+1})$ and put

$$ B_{\underline{r}+2, a_0, \dots, a_{r+1}}^*(w, x, \Lambda) = \frac{-2\epsilon}{(r+2)!} \sum_{\delta \in F} B_{\underline{r}+2, \underline{r}+2}^*(w + \delta(x), \alpha_0(x), \dots, \alpha_{r+1}(x)) $$

where $\alpha_0, \dots, \alpha_{r+1}$ is the unique primitive positive dual family to $(a_0, \dots, a_{r+1})$ in $L = \mathrm{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{Z})$ (see lemma 3) and

$$ F = F(\alpha_0, \dots, \alpha_{r+1}) = \{\delta \in L, 0 \le \delta(a_j) < \alpha_j(a_j), \forall 0 \le j \le r+1\} $$
This shows that $C^+(\underline{\alpha}, x)$ can be written as a disjoint union

$$C^+(\underline{\alpha}, x) = \bigcup_{\delta \in F(\underline{\alpha})/\mathbb{Z}\gamma} \bigcup_{\underline{m} \ge 0} \left\{ -\delta + \sum_{j=0}^{r} (d_j m_j + (1+d_j)/2) \alpha_j \right\} + \mathbb{Z}\gamma$$

and a similar statement holds for $C^-(\underline{\alpha}, x)$ so that $G_{r,\underline{\alpha}}^{\underline{\alpha}}(w, x, \Lambda)^{(-1)^D}$ is equal to

$$\prod_{\delta' \in C^+(\underline{\alpha}, x)/\mathbb{Z}\gamma} \left(1 - e^{2i\pi \left(\frac{\delta'(x)-w}{\gamma(x)}\right)}\right) \prod_{\delta' \in C^-(\underline{\alpha}, x)/\mathbb{Z}\gamma} \left(1 - e^{-2i\pi \left(\frac{\delta'(x)-w}{\gamma(x)}\right)}\right)^{(-1)^r}$$

Then, we only need to show that the sets $C^{\pm}(\underline{\alpha}, x)$ are independent of the choice for $\underline{\alpha}$. Consider another positive dual family $\underline{\alpha}'$ to $\underline{\alpha}$. Write $\alpha_j(a_j) = s_j > 0$ and $\alpha'_j(a_j) = s'_j > 0$. Then from lemma 2 there is a rational number $r_j$ such that

$$\alpha'_j = \frac{s'_j}{s_j} \alpha_j + r_j \gamma$$

For any $x \in \mathrm{Hom}_{\mathbb{Z}}(L, \mathbb{C})$ we get:

$$d_j \frac{\alpha'_j(x)}{\gamma(x)} = d_j r_j + d_j \frac{s'_j}{s_j} \frac{\alpha_j(x)}{\gamma(x)} \in \mathbb{H}$$

This shows that the signs $d_j$ (and therefore also $D$) are independent of the choice for $\underline{\alpha}$. By construction we get $C^{\pm}(\underline{\alpha}, x) = C^{\pm}(\underline{\alpha}', x)$ and the definition of the geometric $G_{r,\underline{\alpha}}^{\underline{\alpha}}$ function is independent of the choice for $\underline{\alpha}$. $\square$

From now on we denote by $G_{r,\underline{\alpha}} := G_{r,\underline{\alpha}}^{\underline{\alpha}}$ the geometric $G_r$ function associated to $\underline{\alpha}$ for any suitable choice of $\underline{\alpha}$. A choice that is interesting for us is defined by the comatrix or rather by the rescaled comatrix. Let $c$ be a vector such that $\gamma(c) = 1$. Then the elementary divisors of the concatenation $(\underline{\alpha}, c)$ can be written as $[A_r, A_{r-1}, \dots, A_1, 1, 1]$ with $A_i|A_{i+1}$ so that a possible choice for $\underline{\alpha}$ is given by:

$$(\underline{\alpha} \ c) \begin{pmatrix} \underline{\alpha} \\ A_r \gamma \end{pmatrix} = A_r I_{r+2} \qquad (12)$$

To express the transformation properties of the function $G_{r,\underline{\alpha}}$ under the action of $\mathrm{SL}_{r+2}(\mathbb{Z})$ we need to introduce the family of geometric Bernoulli polynomials which encompass $P_{a,b,c}$. In what follows, we want to discuss an analogue of formula (4) for $G_{r,\underline{\alpha}}$, and we must consider families of $r+2$ vectors. Recall that we have defined the polynomials $B_{r+2,r+2}^*$ in section 2.2.2.
is a finite set.

These geometric Bernoulli “polynomials” have the form

$$B_{r+2, a_0, \dots, a_{r+1}}^{*}(w, x, \Lambda) = \sum_{l+k_0+\dots+k_{r+1}=r+2} c_{l, k_0, \dots, k_{r+1}} w^l \prod_{j=0}^{r+1} \alpha_j(x)^{k_j}$$

where the sum is taken over non-negative integers and the coefficients $c_{l,k_0,\dots,k_{r+1}}$ are rational numbers which explicitly depend on the set $F = F(\alpha_0, \dots, \alpha_{r+1})$. The modularity property for the geometric $G_{r,a}$ functions will involve the rescaled versions of the geometric Bernoulli polynomials

$$B_{r+2, a_0, \dots, a_{r+1}}(w, x, \Lambda) = \left( \prod_{j=0}^{r+1} \alpha_j(x) \right)^{-1} B_{r+2, a_0, \dots, a_{r+1}}^{*}(w, x, \Lambda)$$

For $r = 1$ the polynomial $B_{3,a,b,c}$ may be identified with the polynomial $P_{a,b,c}$ appearing in formula (4). We can now generalise formula (4).

**Theorem 6:**

1. [Modular property] For a family $a_0, \dots, a_{r+1}$ of $r+2$ linearly independent primitive vectors in $\Lambda$:

$$\prod_{j=0}^{r+1} G_{r, (a_k)_{k \neq j}}(w, x, \Lambda)^{(-1)^j} = \exp(i\pi B_{r+2, a_0, \dots, a_{r+1}}(w, x, \Lambda)) \quad (13)$$

2. [Equivariance relations] For a family $a_0, \dots, a_r$ of $r+1$ linearly independent primitive vectors in $\Lambda$, for $w \in \mathbb{C}$ and $x \in \text{Hom}_{\mathbb{Z}}(L, \mathbb{C})$, for all $g \in \text{SL}_{r+2}(\mathbb{Z})$:

$$G_{r, ga}(w, gx, \Lambda) = G_{r, \underline{a}}(w, x, \Lambda) \\ B_{r+2, ga_0, \dots, ga_{r+1}}(w, gx, \Lambda) = B_{r+2, a_0, \dots, a_{r+1}}(w, x, \Lambda)$$

**Proof:**

1. The modular relation for the geometric $G_{r,a}$ functions is a consequence of the modular property (10) for the ordinary $G_r$ function. Let $\underline{a}$ be the primitive positive dual family to $\underline{a}$ in $L$ as given in lemma 3. Write $\epsilon$ for the sign of $\det_B(a_0, \dots, a_{r+1}).(-1)^{r+1}$. Then for all $0 \le j \le r+1$ we have

$$\det_B(a_0, \dots, a_{j-1}, a_{j+1}, \dots, a_{r+1}, \cdot) = s_j \cdot \epsilon \cdot (-1)^j \alpha_j$$
**Definition 5:** Let $w \in \mathbb{C}$ and $x \in \mathrm{Hom}_{\mathbb{Z}}(L, \mathbb{C}) \simeq \mathbb{C}^{r+2}$. Let $a_0, \dots, a_r, a_{r+1}$ be a family of $r+2$ linearly independent primitive vectors in the oriented lattice $\Lambda$. Let $\epsilon$ be the sign of $(-1)^{r+1}\det(a_0, \dots, a_{r+1})$ and put

$$B_{r+2,a_0,\dots,a_{r+1}}^*(w, x, \Lambda) = \frac{-2\epsilon}{(r+2)!} \sum_{\delta \in F} B_{r+2,r+2}^*(w + \delta(x), \alpha_0(x), \dots, \alpha_{r+1}(x))$$

where $\alpha_0, \dots, \alpha_{r+1}$ is the unique primitive positive dual family to $(a_0, \dots, a_{r+1})$ in $L = \mathrm{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{Z})$ (see lemma 3) and

$$F = F(\alpha_0, \dots, \alpha_{r+1}) = \{\delta \in L, 0 \le \delta(a_j) < \alpha_j(a_j), \forall 0 \le j \le r+1\}$$

is a finite set.

These geometric Bernoulli “polynomials” have the form

$$B_{r+2,a_0,\dots,a_{r+1}}^*(w, x, \Lambda) = \sum_{l+k_0+\dots+k_{r+1}=r+2} c_{l,k_0,\dots,k_{r+1}} w^l \prod_{j=0}^{r+1} \alpha_j(x)^{k_j}$$

where the sum is taken over non-negative integers and the coefficients $c_{l,k_0,\dots,k_{r+1}}$ are rational numbers which explicitly depend on the set $F = F(\alpha_0, \dots, \alpha_{r+1})$. The modularity property for the geometric $G_{r,\underline{a}}$ functions will involve the rescaled versions of the geometric Bernoulli polynomials

$$B_{r+2,a_0,\dots,a_{r+1}}(w, x, \Lambda) = \left( \prod_{j=0}^{r+1} \alpha_j(x) \right)^{-1} B_{r+2,a_0,\dots,a_{r+1}}^*(w, x, \Lambda)$$

For $r=1$ the polynomial $B_{3,a,b,c}$ may be identified with the polynomial $P_{a,b,c}$ appearing in formula (4). We can now generalise formula (4).

**Theorem 6:**

1. [Modular property] For a family $a_0, \dots, a_{r+1}$ of $r+2$ linearly independent primitive vectors in $\Lambda$:

$$\prod_{j=0}^{r+1} G_{r,(a_k)_{k \neq j}}(w, x, \Lambda)^{(-1)^j} = \exp(i\pi B_{r+2,a_0,\dots,a_{r+1}}(w, x, \Lambda)) \quad (13)$$

2. [Equivalence relations] For a family $a_0, \dots, a_r$ of $r+1$ linearly independent primitive vectors in $\Lambda$, for $w \in \mathbb{C}$ and $x \in \mathrm{Hom}_{\mathbb{Z}}(L, \mathbb{C})$, for all $g \in \mathrm{SL}_{r+2}(\mathbb{Z})$:

$$G_{r,\underline{g}}(w, gx, \Lambda) = G_{r,\underline{a}}(w, x, \Lambda) \\ B_{r+2,g a_0,\dots,g a_{r+1}}(w, gx, \Lambda) = B_{r+2,a_0,\dots,a_{r+1}}(w, x, \Lambda)$$
so that $\gamma_j = \epsilon \cdot (-1)^j \alpha_j$ and

$$G_{r,(a_k)_{k \neq j}}(w, x, \Lambda) = \prod_{\delta \in F_j / \mathbb{Z}\alpha_j} G_r \left( \frac{w + \delta(x)}{\alpha_j(x)}, \left( \frac{\alpha_k(x)}{\alpha_j(x)} \right)_{k \neq j} \right)^{(-1)^j \epsilon}$$

where

$$F_j = \{ \delta \in L, 0 \le \delta(a_k) < \alpha_k(a_k), \forall 0 \le k \neq j \le r+1 \}$$

Then

$$\prod_{j=0}^{r+1} G_{r,((a_k)_{k \neq j}}(w, x, \Lambda)^{(-1)^j} = \prod_{j=0}^{r+1} \prod_{\delta \in F_j / \mathbb{Z}\alpha_j} G_r \left( \frac{w + \delta(x)}{\alpha_j(x)}, \left( \frac{\alpha_k(x)}{\alpha_j(x)} \right)_{k \neq j} \right)^{\epsilon}$$

Put $\mathcal{F} = F(\alpha_0, \dots, \alpha_{r+1}) = \{\delta \in L, 0 \le \delta(a_j) < \alpha_j(a_j), \forall 0 \le j \le r+1\}$ so that we can write uniformly $\mathcal{F} \simeq F_j / \mathbb{Z}\alpha_j$ for all $0 \le j \le r+1$. Then, using Narukawa's theorem (10) for each $\delta$ in the finite set $\mathcal{F}$ yields:

$$\prod_{j=0}^{r+1} G_{r,((a_k)_{k \neq j}}(w, x, \Lambda)^{(-1)^j} = \prod_{\delta \in \mathcal{F}} \exp \left( \frac{-2i\pi\epsilon}{(r+2)!} B_{r+2,r+2}(w + \delta(x), \underline{\alpha}(x)) \right)$$

The identification of the right-hand side gives the conclusion:

$$\prod_{j=0}^{r+1} G_{r,((a_k)_{k \neq j}}(w, x, \Lambda)^{(-1)^j} = \exp(i\pi B_{r+2,a_0,\dots,a_{r+1}}(w, x, \Lambda))$$

2. Consider $g \in \text{SL}_{r+2}(\mathbb{Z})$. In the construction of $G_{r,\underline{a}}$ or $B_{r,a_0,\dots,a_{r+1}}$ replacing $\underline{a}$ with $ga$ replaces $\underline{\alpha}$ with $\underline{\alpha}g^{-1}$, $\gamma$ with $\gamma g^{-1}$ and $F(\underline{\alpha})$ with $F(\underline{\alpha}g^{-1})$ but $\epsilon$ is left unchanged. Therefore we may write:

$$G_{r,\underline{g}\underline{a}}(w, gx, \Lambda) = \prod_{\delta \in F(\underline{\alpha}g^{-1}) / \mathbb{Z}\gamma g^{-1}} G_r \left( \frac{w + \delta(gx)}{\gamma g^{-1}(gx)}, \frac{1}{\gamma g^{-1}(gx)} \underline{\alpha} g^{-1}(gx) \right)$$

Then, identifying $F(\underline{\alpha}g^{-1}) = F(\underline{\alpha})g^{-1}$ and putting $\delta' = \delta g$ gives

$$G_{r,\underline{g}\underline{a}}(w, gx, \Lambda) = \prod_{\delta' \in F(\underline{\alpha}) / \mathbb{Z}\gamma} G_r \left( \frac{w + \delta'(x)}{\gamma(x)}, \frac{1}{\gamma(x)} \underline{\alpha}(x) \right)$$

$$G_{r,\underline{g}\underline{a}}(w, gx, \Lambda) = G_{r,\underline{a}}(w, x, \Lambda)$$

As for $B_{r+2,ga_0,\dots,ga_{r+1}}$, write $\mathcal{F} = F(\alpha_0, \dots, \alpha_{r+1})$:

$$B_{r+2,ga_0,\dots,ga_{r+1}}(w, gx, \Lambda) = \frac{-2\epsilon}{(r+2)!} \sum_{\delta \in \mathcal{F}g^{-1}} B_{r+2,r+2}(w + \delta(gx), \underline{\alpha}(g^{-1}gx))$$
### Proof:

1. The modular relation for the geometric $G_{r,g}$ functions is a consequence of the modular property (10) for the ordinary $G_r$ function. Let $\underline{\alpha}$ be the primitive positive dual family to $\underline{g}$ in $L$ as given in lemma 3. Write $\epsilon$ for the sign of $\det_B(a_0, \dots, a_{r+1}).(-1)^{r+1}$. Then for all $0 \le j \le r+1$ we have

$$ \det_B(a_0, \dots, a_{j-1}, a_{j+1}, \dots, a_{r+1}, \cdot) = s_j \cdot \epsilon \cdot (-1)^j \alpha_j $$

so that $\gamma_j = \epsilon \cdot (-1)^j \alpha_j$ and

$$ G_{r, (a_k)_{k \neq j}}(w, x, \Lambda) = \prod_{\delta \in F_j / \mathbb{Z}\alpha_j} G_r \left( \frac{w + \delta(x)}{\alpha_j(x)}, \left( \frac{\alpha_k(x)}{\alpha_j(x)} \right)_{k \neq j} \right)^{(-1)^j \cdot \epsilon} $$

where

$$ F_j = \{ \delta \in L, 0 \le \delta(a_k) < \alpha_k(a_k), \forall 0 \le k \ne j \le r+1 \} $$

Then

$$ \prod_{j=0}^{r+1} G_{r, ((a_k)_{k \neq j}}(w, x, \Lambda)^{(-1)^j} = \prod_{j=0}^{r+1} \prod_{\delta \in F_j / \mathbb{Z}\alpha_j} G_r \left( \frac{w + \delta(x)}{\alpha_j(x)}, \left( \frac{\alpha_k(x)}{\alpha_j(x)} \right)_{k \neq j} \right)^{\epsilon} $$

Put $\mathcal{F} = F(\alpha_0, \dots, \alpha_{r+1}) = \{\delta \in L, 0 \le \delta(a_j) < \alpha_j(a_j), \forall 0 \le j \le r+1\}$ so that we can write uniformly $\mathcal{F} \simeq F_j / \mathbb{Z}\alpha_j$ for all $0 \le j \le r+1$. Then, using Narukawa's theorem (10) for each $\delta$ in the finite set $\mathcal{F}$ yields:

$$ \prod_{j=0}^{r+1} G_{r, ((a_k)_{k \neq j}}(w, x, \Lambda)^{(-1)^j} = \prod_{\delta \in \mathcal{F}} \exp \left( \frac{-2i\pi\epsilon}{(r+2)!} B_{r+2, r+2}(w + \delta(x), \underline{\alpha}(x)) \right) $$

The identification of the right-hand side gives the conclusion:

$$ \prod_{j=0}^{r+1} G_{r, ((a_k)_{k \neq j}}(w, x, \Lambda)^{(-1)^j} = \exp(i\pi B_{r+2, a_0, \dots, a_{r+1}}(w, x, \Lambda)) $$

2. Consider $g \in \text{SL}_{r+2}(\mathbb{Z})$. In the construction of $G_{r,g}$ or $B_{r,a_0,\dots,a_{r+1}}$ replacing $\underline{g}$ with $\underline{ga}$ replaces $\underline{\alpha}$ with $\underline{\alpha}g^{-1}$, $\gamma$ with $\gamma g^{-1}$ and $F(\underline{\alpha})$ with $F(\underline{\alpha}g^{-1})$ but $\epsilon$ is left unchanged. Therefore we may write:

$$ G_{r, ga}(w, gx, \Lambda) = \prod_{\delta \in F(\underline{\alpha}g^{-1}) / \mathbb{Z}\gamma g^{-1}} G_r \left( \frac{w + \delta(gx)}{\gamma g^{-1}(gx)}, \frac{1}{\gamma g^{-1}(gx)} \underline{\alpha} g^{-1}(gx) \right) $$

Then, identifying $F(\underline{\alpha}g^{-1}) = F(\underline{\alpha})g^{-1}$ and putting $\delta' = \delta g$ gives

$$ G_{r, ga}(w, gx, \Lambda) = \prod_{\delta' \in F(\underline{\alpha}) / \mathbb{Z}\gamma} G_r \left( \frac{w + \delta'(x)}{\gamma(x)}, \frac{1}{\gamma(x)} \underline{\alpha}(x) \right) $$
Put once again $\delta' = \delta g$, which gives:

$$B_{r+2,ga_0,...,ga_{r+1}}(w, gx, \Lambda) = \frac{-2\epsilon}{(r+2)!} \sum_{\delta' \in F(\alpha_0,...,\alpha_{r+1})} B_{r+2,r+2}(w + \delta(x), \underline{\alpha}(x))$$

Identify the right-hand side to conclude that

$$B_{r+2,ga_0,...,ga_{r+1}}(w, gx, \Lambda) = B_{r+2,a_0,...,a_{r+1}}(w, x, \Lambda)$$

□

We also add that the geometric $G_{r,a}$ functions behave nicely under permutation of vectors, namely for all permutation $\sigma \in \mathfrak{S}_{r+1}$, $G_{r,\sigma(a)} = G_{r,a}^{\epsilon(\sigma)}$ where $\epsilon(\sigma)$ is the sign of the permutation $\sigma$. As in the cubic case, the relation (13) can be restated and best understood as a multiplicative $r$-cocycle relation for $SL_{r+2}(\mathbb{Z})$. For $g_1, ..., g_l \in SL_{r+2}(\mathbb{Z})$ put

$$[g_1|\dots|g_l] = \left( g_1, \dots, \prod_{j=1}^{k} g_j, \dots, \prod_{j=1}^{l} g_j \right)$$

Then we may write $\psi_a(g_1, ..., g_r) = G_{r,a,g_1a,\dots,(\prod_{j=1}^r g_j)a} = G_{r,[1|g_1|\dots|g_r]a}$ for a primitive vector $a \in \Lambda$ and $g_1, ..., g_r \in SL_{r+2}(\mathbb{Z})$. The coboundary of $\psi_a$ is given by the modular property (13):

$$d\psi_a(g_1, ..., g_{r+1}) = \exp(i\pi B_{r+2,[1|g_1|\dots|g_{r+1}]a}) \quad (14)$$

where 1 stands for the identity matrix $I_{r+2}$. This property is essential to understand the type of evaluation of the $G_r$ functions we carry out in section 3.1. Indeed, in [19] Sczech uses Eisenstein ($d-1$)-cocycles for $SL_d(\mathbb{Z})$ to express the values of $L$-functions of totally real fields of degree $d$ at negative integers. In the case of a totally real field $\mathbb{K}$ of degree $d$, fixing a basis of $\mathcal{O}_{\mathbb{K}}$, the unit group $\mathcal{O}_{\mathbb{K}}^\times$ may be embedded inside $SL_d(\mathbb{Z})$ as an abelian group of rank $d-1$. In the case of ATR fields which we are concerned with in this article, we have here obtained a ($d-2$)-cocycle $\psi_a$ (up to the $\exp(i\pi B_{r+2,[1|g_1|\dots|g_{r+1}]a})$ term) for $SL_d(\mathbb{Z})$ which is compatible with the unit group of an ATR field of degree $d$ having rank $d-2$.

## 2.3 Modified $G_r$ functions and smoothings

Following an observation by Zudilin for $r = 2, 3$ (see for instance [14]), we modify the $G_r$ functions with an exponential prefactor to mimic the correction factors one may find attached to modular functions to ensure better
$$G_{r,g_a}(w, gx, \Lambda) = G_{r,a}(w, x, \Lambda)$$

As for $B_{r+2,ga_0,...,ga_{r+1}}$, write $\mathcal{F} = F(\alpha_0, ..., \alpha_{r+1})$:

$$B_{r+2,ga_0,...,ga_{r+1}}(w, gx, \Lambda) = \frac{-2\epsilon}{(r+2)!} \sum_{\delta \in \mathcal{F}g^{-1}} B_{r+2,r+2}(w + \delta(gx), \underline{\alpha}(g^{-1}gx))$$

Put once again $\delta' = \delta g$, which gives:

$$B_{r+2,ga_0,...,ga_{r+1}}(w, gx, \Lambda) = \frac{-2\epsilon}{(r+2)!} \sum_{\delta' \in F(\alpha_0,...,\alpha_{r+1})} B_{r+2,r+2}(w + \delta(x), \underline{\alpha}(x))$$

Identify the right-hand side to conclude that

$$B_{r+2,ga_0,...,ga_{r+1}}(w, gx, \Lambda) = B_{r+2,a_0,...,a_{r+1}}(w, x, \Lambda)$$

□

We also add that the geometric $G_{r,a}$ functions behave nicely under permutation of vectors, namely for all permutation $\sigma \in \mathfrak{S}_{r+1}$, $G_{r,\sigma(a)} = G_{r,a}^{\epsilon(\sigma)}$ where $\epsilon(\sigma)$ is the sign of the permutation $\sigma$. As in the cubic case, the relation (13) can be restated and best understood as a multiplicative $r$-cocycle relation for $SL_{r+2}(\mathbb{Z})$. For $g_1, ..., g_l \in SL_{r+2}(\mathbb{Z})$ put

$$[g_1 | \dots | g_l] = \left( g_1, \dots, \prod_{j=1}^{k} g_j, \dots, \prod_{j=1}^{l} g_j \right)$$

Then we may write $\psi_a(g_1, ..., g_r) = G_{r,a,g_1 a, \dots, (\prod_{j=1}^r g_j)a} = G_{r, [1|g_1|\dots|g_r]a}$ for a primitive vector $a \in \Lambda$ and $g_1, ..., g_r \in SL_{r+2}(\mathbb{Z})$. The coboundary of $\psi_a$ is given by the modular property (13):

$$d\psi_a(g_1, ..., g_{r+1}) = \exp(i\pi B_{r+2, [1|g_1|\dots|g_{r+1}]a}) \quad (14)$$

where 1 stands for the identity matrix $I_{r+2}$. This property is essential to understand the type of evaluation of the $G_r$ functions we carry out in section 3.1. Indeed, in [22] Schzech uses Eisenstein ($d-1$)-cocycles for $SL_d(\mathbb{Z})$ to express the values of $L$-functions of totally real fields of degree $d$ at negative integers. In the case of a totally real field $\mathbb{K}$ of degree $d$, fixing a basis of $\mathcal{O}_{\mathbb{K}}$, the unit group $\mathcal{O}_{\mathbb{K}}^\times$ may be embedded inside $SL_d(\mathbb{Z})$ as an abelian group of rank $d-1$. In the case of ATR fields which we are concerned with in this article, we have here obtained a ($d-2$)-cocycle $\psi_a$ (up to the $\exp(i\pi B_{r+2, [1|g_1|\dots|g_{r+1}]a})$ term) for $SL_d(\mathbb{Z})$ which is compatible with the unit group of an ATR field of degree $d$ having rank $d-2$.
arithmetic properties, as for the Dedekind $\eta$ function :

$$ \eta(\tau) = q^{1/24} \prod_{m \ge 1} (1 - q^m) $$

We will now derive the exponential prefactor from Narukawa’s modularity theorem:

$$ \prod_{j=0}^{r+1} G_r \left( \frac{z}{\omega_j}, \left( \frac{\omega_n}{\omega_j} \right)_{n \neq j} \right) = \exp \left( \frac{-2i\pi}{(r+2)!} B_{r+2,r+2}(z, \underline{\omega}) \right) $$

The rescaled Bernoulli polynomial may be written explicitly using the generating function:

$$ e^{zt} \prod_{j=0}^{r+1} \frac{\omega_j t}{e^{\omega_j t} - 1} = \left( \sum_{n \ge 0} \frac{(zt)^n}{n!} \right) \prod_{j=0}^{r+1} \left( \sum_{k_j \ge 0} \frac{(\omega_j t)^{k_j}}{(k_j)!} B_{k_j} \right) $$

where the $B_{k_j}$ are the usual Bernoulli numbers. This gives:

$$ \frac{B_{r+2,r+2}^*(z, \underline{\omega})}{(r+2)! \prod_{j=0}^{r+1} \omega_j} = \frac{B_{r+2,r+2}(z, \underline{\omega})}{(r+2)!} = \sum_{l+k_0+\cdots+k_{r+1}=r+2} \frac{z^l}{l!} \prod_{j=0}^{r+1} \frac{\omega_j^{k_j-1}}{k_j!} B_{k_j} $$

where the sum ranges over non-negative integers. Specialising to the case where $z = \sum_{j=0}^{r+1} a_j \omega_j$ with rational coefficients $a_j$ gives:

$$ B_{r+2,r+2} \left( \sum_{j=0}^{r+1} a_j \omega_j, \underline{\omega} \right) = (r+2)! \sum_{|\underline{l}|+|\underline{k}|=r+2} \prod_{j=0}^{r+1} \frac{(a_j \omega_j)^{l_j}}{l_j!} \frac{\omega_j^{k_j-1}}{k_j!} B_{k_j} $$

Therefore we define

$$ D_{r+2,m}(\underline{a}, \underline{\omega}) = \sum_{|\underline{l}|+|\underline{k}|=r+2}^{l_m=0, k_m=0} \frac{(r+2)!}{\#\{j, l_j = k_j = 0\}} \prod_{j=0}^{r+1} \frac{(a_j \omega_j)^{l_j}}{l_j!} \frac{\omega_j^{k_j-1}}{k_j!} B_{k_j} $$

Then $B_{r+2,r+2}$ naturally decomposes as

$$ B_{r+2,r+2} \left( \sum_{j=0}^{r+1} a_j \omega_j, \underline{\omega} \right) = (r+2)! \prod_{j=0}^{r+1} \left( a_j - \frac{1}{2} \right) + \sum_{m=0}^{r+1} D_{r+2,m}(\underline{a}, \underline{\omega}) $$

Thus we define

$$ H_r \left( \sum_{j=0}^{r+1} a_j \frac{\omega_j}{\omega_m}, \left( \frac{\omega_j}{\omega_m} \right)_{j \neq m} \right) = e^{\frac{2i\pi}{(r+2)!} D_{r+2,m}(\underline{a}, \underline{\omega})} \cdot G_r \left( \sum_{j=0}^{r+1} a_j \frac{\omega_j}{\omega_m}, \left( \frac{\omega_j}{\omega_m} \right)_{j \neq m} \right) $$
# 3 An algebraicity conjecture on special values of $G_r$ functions

## 3.1 Arithmetic $G_r$ functions

In their recent article [1], Bergeron, Charollois and García describe special values of the elliptic Gamma function evaluated at specific points in complex cubic fields which they relate to partial zeta values. They also conjecture that these special values are units lying in specific abelian extensions. We shall present a generalisation of this construction for fields of degree $d \ge 2$ with exactly one pair of complex embeddings yielding special values of $G_{d-2}$ functions.

Consider a degree $d \ge 2$ ATR number field $\mathbb{K}$. Fix a complex embedding $\sigma_{\mathbb{K}}$ of $\mathbb{K}$ and fix an ordering $\sigma_1, \dots, \sigma_{d-2}, \sigma_{d-1} = \sigma_{\mathbb{K}}, \sigma_d = \overline{\sigma_{d-1}}$ on the embeddings of $\mathbb{K}$. We fix a global orientation of $\mathbb{K}$ as follows. For any basis $B = (e_1, \dots, e_d)$ of the $\mathbb{Q}$-vector space $\mathbb{K}$, the determinant $i \cdot \det((\sigma_j(e_k))_{1 \le j, k \le d})$ is a non zero real number. We call a basis $B$ positive if the latter is positive. Consider an integral ideal $f \ne \mathcal{O}_{\mathbb{K}}$. We write $q\mathbb{Z} = \mathbb{Z} \cap f$. Fix an integral ideal $b$ coprime to $f$ and put $L = fb^{-1}$. Fix an integral ideal $a$ coprime to $fb$ such that $a^{-1}L/L$ is a cyclic group. We call such an ideal a *smoothing ideal* and we write $N = \mathcal{N}(a)$ for the norm of $a$. We consider specific vectors in $L$ which we will use to evaluate geometric $G_r$ functions.

**Definition 7:** A vector $h \in L = fb^{-1}$ is called admissible for the data $f, b, a$ if $h/q \equiv 1 \pmod L$ and $h/N$ generates the cyclic group $a^{-1}L/L$, where $q\mathbb{Z} = \mathbb{Z} \cap f$ and $N = \mathcal{N}(a)$ is the norm of the ideal $a$.

In this section we fix $h \in L$ an admissible vector and we fix a $\mathbb{Z}$-basis $B_L$ of the $\mathbb{Z}$-module $L$ which is a positive basis of $\mathbb{K}$. The unit group

$$ \mathcal{O}_{f}^{+ \times} = \{ \varepsilon \in \mathcal{O}_{\mathbb{K}}^{\times}, \varepsilon \equiv 1 \pmod{f}, \sigma(\varepsilon) > 0, \text{ for all real embedding } \sigma \text{ of } \mathbb{K} \} \quad (15) $$

is a free $\mathbb{Z}$-module of rank $r = d - 2$ acting on $L$, and we fix for now $u_1, \dots, u_r$ a set of fundamental units for $\mathcal{O}_f^{+ \times}$ such that the linear form $\det_{B_L}(h, u_1h, \dots, u_rh, \cdot)$ is non zero. Denote by $a$ the unique primitive element in $\operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{Z})$ satisfying $na = \det_{B_L}(h, u_1h, \dots, u_rh, \cdot)$ for a positive integer $n$.

**Definition 8:** Let $u_1, \dots, u_r$ be a set of fundamental units for $\mathcal{O}_f^{+ \times}$ as above. Let $h$ be an admissible vector in $fb^{-1}$ and $a$ be the linear form described above. We define for the complex embedding $\sigma_{\mathbb{K}}$ of $\mathbb{K}$ and the choice of sign $\pm$ the
This way, Narukawa's modularity theorem becomes a multiplicative cocycle relation for $H_r$ with values in $\exp(2i\pi\mathbb{Q})$:

$$ \prod_{m=0}^{r+1} H_r \left( \sum_{j=0}^{r+1} a_j \frac{\omega_j}{\omega_m}, \left( \frac{\omega_j}{\omega_m} \right)_{j \neq m} \right) = \exp \left( 2i\pi \prod_{j=0}^{r+1} \left( a_j - \frac{1}{2} \right) \right) $$

In what follows we will smooth out $G_r$ functions, and this will kill the factor $D_{r+2,m}$ because it is homogeneous of degree 1 in $\omega_0, \dots, \omega_{m-1}, \omega_{m+1}, \dots, \omega_{r+1}$:

$$ D_{r+2,m}(a, N\omega_0, \dots, N\omega_{m-1}, \omega_m, N\omega_{m+1}, \dots, \omega_{r+1} = N D_{r+2,m}(a, \omega) $$

This means that for any positive integer $N$, the level $N$ smoothings of $G_r$ and $H_r$ functions coincide:

$$ \frac{H_r \left( \sum_{j=0}^{r+1} a_j \frac{\omega_j}{\omega_m}, \left( \frac{\omega_j}{\omega_m} \right)_{j \neq m} \right)^N}{H_r \left( \sum_{j=0}^{r+1} a_j \frac{N\omega_j}{\omega_m}, \left( \frac{N\omega_j}{\omega_m} \right)_{j \neq m} \right)} = \frac{G_r \left( \sum_{j=0}^{r+1} a_j \frac{\omega_j}{\omega_m}, \left( \frac{\omega_j}{\omega_m} \right)_{j \neq m} \right)^N}{G_r \left( \sum_{j=0}^{r+1} a_j \frac{N\omega_j}{\omega_m}, \left( \frac{N\omega_j}{\omega_m} \right)_{j \neq m} \right)} $$

# 3 An algebraicity conjecture on special values of $G_r$ functions

## 3.1 Arithmetic $G_r$ functions

In their recent article [1], Bergeron, Charollos and García describe special values of the elliptic Gamma function in complex cubic fields which they relate to partial zeta values. They also conjecture that these special values are units lying in specific abelian extensions. We shall present a generalisation of this construction for fields of degree $d \ge 2$ with exactly one pair of complex embeddings yielding special values of $G_{d-2}$ functions.

Consider a degree $d \ge 2$ ATR field $\mathbb{K}$ and an integral ideal $f \neq \mathcal{O}_{\mathbb{K}}$. We write $q\mathbb{Z} = \mathbb{Z} \cap f$. Fix an integral ideal $b$ coprime to $f$ and put $L = fb^{-1}$. Choose $B_L$ a $\mathbb{Z}$-basis of the $\mathbb{Z}$-module $L$ and fix an integral ideal $a$ coprime to $fb$ such that $a^{-1}L/L$ is a cyclic group. We call such an ideal a *smoothing* ideal and we write $N = \mathcal{N}(a)$ for the norm of $a$. We consider specific vectors in $L$ which we will use to evaluate geometric $G_r$ functions.

**Definition 7:** A vector $h \in L = fb^{-1}$ is called admissible for the data $f, b, a$ if $h/q \equiv 1 \pmod L$ and $h/N$ generates the cyclic group $a^{-1}L/L$, where $q\mathbb{Z} = \mathbb{Z} \cap f$ and $N = \mathcal{N}(a)$ is the norm of the ideal $a$.
arithmetic $G_r$ function:

$$G_{r,f,b,a}^{+}(u_1, \dots, u_r; h, \sigma_{\mathbb{C}}) = \frac{G_{r,(a,au_1,\dots,au_r)}(\frac{h}{q}, \sigma_{\mathbb{C}}, \mathrm{Hom}_{\mathbb{Z}}(\mathfrak{f}\mathfrak{b}^{-1}, \mathbb{Z}))^N}{G_{r,(a,au_1,\dots,au_r)}(\frac{h}{q}, \sigma_{\mathbb{C}}, \mathrm{Hom}_{\mathbb{Z}}(\mathfrak{f}(\mathfrak{b}\mathfrak{a})^{-1}, \mathbb{Z}))}$$

$$G_{r,f,b,a}^{-}(u_1, \dots, u_r; h, \sigma_{\mathbb{C}}) = \frac{G_{r,(-a,-au_1,\dots,-au_r)}(\frac{h}{q}, \sigma_{\mathbb{C}}, \mathrm{Hom}_{\mathbb{Z}}(\mathfrak{f}\mathfrak{b}^{-1}, \mathbb{Z}))^N}{G_{r,(-a,-au_1,\dots,-au_r)}(\frac{h}{q}, \sigma_{\mathbb{C}}, \mathrm{Hom}_{\mathbb{Z}}(\mathfrak{f}(\mathfrak{b}\mathfrak{a})^{-1}, \mathbb{Z}))}$$

where in both cases the right-hand side is a quotient of two geometric $G_r$ functions as defined in Proposition 4.

**Remark** This is well-defined provided that a positive dual family $\alpha$ to $a, au_1, \dots, au_r$ satisfies $\sigma_{\mathbb{C}}(\alpha_j/h) \notin \mathbb{R}$ for all $0 \le j \le r$. This is always the case, except possibly when $\mathbb{K}$ is of even degree $2d'$ and contains a real subfield $\mathbb{F}$ of degree $\le d'$. We will briefly discuss this case in section 5.2.4.

One may use formula (9) to relate the two values $G_{r,f,b,a}^{\pm}(u_1, \dots, u_r; h, \sigma_{\mathbb{C}})$ and $G_{r,f,b,a}^{\pm}(u_1, \dots, u_r; h, \overline{\sigma_{\mathbb{C}}})$ for the two complex embeddings of $\mathbb{K}$ as:

$$G_{r,f,b,a}^{\pm}(u_1, \dots, u_r; h, \overline{\sigma_{\mathbb{C}}}) = \overline{G_{r,f,b,a}^{\pm}(u_1, \dots, u_r; h, \sigma_{\mathbb{C}})}^{(-1)^r}$$

This means that we only need to treat the case of one of the complex embeddings. Throughout the rest of this work, we will consider fields with a specific defining polynomial equation and we fix the complex embedding $\sigma_{\mathbb{C}}$ corresponding to the root of the polynomial lying in the upper half-plane. We shall then drop the $\sigma_{\mathbb{C}}$ in the writing. In the special cases $r = 0, r = 1$ we will identify

$$\theta_{f,b,a}^{\pm}(h) = G_{0,f,b,a}^{\pm}(\emptyset; h)$$

$$\Gamma_{f,b,a}^{\pm}(\varepsilon; h) = G_{1,f,b,a}^{\pm}(\varepsilon; h)$$

In [1] the authors use modular symbols (see the cocycle relation (5)) for $SL_3(\mathbb{Z})$ built from smoothed Eisenstein series to establish a link between this special value and the partial zeta values associated to the conductor $f$ and the ideal class $[b]$ of the integral ideal $b$ in the narrow ideal class group $Cl^+(f)$. Namely, the partial $\zeta$ function associated to this class is

$$\zeta_f([b], s) = \sum_{I \in [b]} \mathcal{N}(I)^{-s}$$

when $\Re(s) > 1$ and where the sum ranges over all integral ideals in the class $[b]$. This function can be extended into a meromorphic function over $\mathbb{C}$ with a pole at $s = 1$. In the specific case where $\mathbb{K}$ is a complex cubic field, this partial $\zeta$ function vanishes at the origin. In [1] the authors have proven that when
In this section we fix $h \in L$ an admissible vector. The unit group

$$ \mathcal{O}_{\mathfrak{f}}^{+, \times} = \{ \varepsilon \in \mathcal{O}_{\mathbb{K}}^{\times}, \varepsilon \equiv 1 \mod \mathfrak{f}, \sigma(\varepsilon) > 0, \text{ for all real embedding } \sigma \text{ of } \mathbb{K} \} $$

is a free $\mathbb{Z}$-module of rank $r = d - 2$ acting on $L$, and we fix for now $u_1, \dots, u_r$ a set of fundamental units for $\mathcal{O}_{\mathfrak{f}}^{+, \times}$ such that the linear form $\det_{B_L}(h, u_1h, \dots, u_rh, \cdot)$ is non degenerate. This means that it should be non-zero and that if $\mathbb{K}$ contains a totally real subfield an additional condition must be satisfied (see (*) in section 4.1). Denote by $a$ the unique primitive element in $\operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{Z})$ satisfying $na = \det_{B_L}(h, u_1h, \dots, u_rh, \cdot)$ for a positive integer $n$.

**Definition 8:** Let $u_1, \dots, u_r$ be a set of fundamental units for $\mathcal{O}_{\mathfrak{f}}^{+, \times}$ as above. Let $h$ be an admissible vector in $\mathfrak{f}\mathfrak{b}^{-1}$ and be the linear form described above. We define for the complex embedding $\sigma_{\mathbb{C}}$ of $\mathbb{K}$ and the choice of sign $\pm$ the arithmetic $G_r$ function:

$$ G_{r, \mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{+}(u_1, \dots, u_r; h, \sigma_{\mathbb{C}}) = \frac{G_{r, (a, au_1, \dots, au_r)}(\frac{h}{q}, \sigma_{\mathbb{C}}, \operatorname{Hom}_{\mathbb{Z}}(\mathfrak{f}\mathfrak{b}^{-1}, \mathbb{Z}))^{N}}{G_{r, (a, au_1, \dots, au_r)}(\frac{h}{q}, \sigma_{\mathbb{C}}, \operatorname{Hom}_{\mathbb{Z}}(\mathfrak{f}(\mathfrak{b}\mathfrak{a})^{-1}, \mathbb{Z}))} $$

$$ G_{r, \mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{-}(u_1, \dots, u_r; h, \sigma_{\mathbb{C}}) = \frac{G_{r, (-a, -au_1, \dots, -au_r)}(\frac{h}{q}, \sigma_{\mathbb{C}}, \operatorname{Hom}_{\mathbb{Z}}(\mathfrak{f}\mathfrak{b}^{-1}, \mathbb{Z}))^{N}}{G_{r, (-a, -au_1, \dots, -au_r)}(\frac{h}{q}, \sigma_{\mathbb{C}}, \operatorname{Hom}_{\mathbb{Z}}(\mathfrak{f}(\mathfrak{b}\mathfrak{a})^{-1}, \mathbb{Z}))} $$

where in both cases the right-hand side is a quotient of two geometric $G_r$ functions as defined in Proposition 4.

Note that this level $N$ smoothing is a product of level $N$ smoothings of ordinary $G_r$ functions which coincide with the level $N$ smoothings of $H_r$ functions as defined in section 2.3. On another note, we only have two complex embeddings $\sigma_{\mathbb{C}}, \overline{\sigma_{\mathbb{C}}}$ for the ATR field $\mathbb{K}$ and using formula (9) we get

$$ G_{r, \mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{\pm}(u_1, \dots, u_r; h, \overline{\sigma_{\mathbb{C}}}) = \overline{G_{r, \mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{\pm}(u_1, \dots, u_r; h, \sigma_{\mathbb{C}})}^{(-1)^r} $$

This means that we only need to treat the case of one of the complex embeddings. Throughout the rest of this work, we will consider fields with a specific defining polynomial equation and we fix the complex embedding $\sigma_{\mathbb{C}}$ corresponding to the root of the polynomial lying in the upper half-plane. We shall then drop the $\sigma_{\mathbb{C}}$ in the writing. In the special cases $r = 0, r = 1$ we will identify

$$ \theta_{\mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{\pm}(h) = G_{0, \mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{\pm}(\emptyset; h) $$

$$ \Gamma_{\mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{\pm}(\varepsilon; h) = G_{1, \mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{\pm}(\varepsilon; h) $$
$\mathcal{O}_{\mathfrak{f}}^{+, \times} = \varepsilon^{\mathbb{Z}}$ with $\sigma_{\mathbb{R}}(\varepsilon) < 1$ then the modulus of $\Gamma_{\mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{-}(\varepsilon, h)^{-1} = \Gamma_{\mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{+}(\varepsilon^{-1}, h)$ is independent of the choice for the admissible vector $h$ and the following Kronecker type limit formula holds:

$$ \mathcal{N}(\mathfrak{a})\zeta'_{\mathfrak{f}}([\mathfrak{b}], 0) - \zeta'_{\mathfrak{f}}([\mathfrak{ab}], 0) = \log |\Gamma_{\mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{-}(\varepsilon, h)^{-1}|^2 = \log |\Gamma_{\mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{+}(\varepsilon^{-1}, h)|^2 $$

They also conjectured that $\Gamma_{\mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{-}(\varepsilon, h)^{-1}$ is a unit in the associated class field $\mathbb{K}^{+}(\mathfrak{f})$ related to the conjectural Stark unit. We believe that a combination of special values of arithmetic $G_r$ functions for $r \ge 2$ will be linked to the partial $\zeta$ functions of an ATR field of degree $d = r + 2$ using modular symbols (see the cocycle relation (14)) for $\text{SL}_d(\mathbb{Z})$ related to some smoothed Eisenstein series. In view of this cohomological interpretation and following the work of Sczech in [22] on the Eisenstein cocycles for totally real fields, we consider the following combination of arithmetic $G_r$ functions:

**Definition 9:** Let $(\varepsilon_1, \dots, \varepsilon_r)$ be a set of fundamental units for $\mathcal{O}_{\mathfrak{f}}^{+, \times}$. Let $\underline{h} = (h_\rho)$ be a set of admissible vectors for $\mathfrak{f}, \mathfrak{b}, \mathfrak{a}$ where $\rho$ ranges over the permutations in $\mathfrak{S}_r$, the symmetric group on $r$ elements. For any choice of orientations for the $G_r$ functions $\underline{\mu} = (\mu_\rho), \underline{\nu} = (\nu_\rho)$ we define:

$$ I_{r, \mathfrak{f}, \mathfrak{b}, \mathfrak{a}}(\varepsilon_1, \dots, \varepsilon_r; \underline{h}, \underline{\mu}, \underline{\nu}) = \prod_{\rho \in \mathfrak{S}_r} G_{r, \mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{\mu_\rho}([\varepsilon_{\rho(1)}| \dots | \varepsilon_{\rho(r)}]; h_\rho)^{\nu_\rho} $$

where again

$$ [\varepsilon_1 | \dots | \varepsilon_r] = \left( \varepsilon_1, \dots, \prod_{j=1}^k \varepsilon_j, \dots, \prod_{j=1}^r \varepsilon_j \right) $$

and the orientations $\mu_\rho, \nu_\rho \in \{-1, +1\}$. This is well-defined if for all permutation $\rho$, the family $(1, \varepsilon_{\rho(1)}, \dots, \prod_{j=1}^r \varepsilon_{\rho(j)})$ is a family of linearly independent vectors in $\mathbb{K}$.

We will now present a conjecture on special values of these arithmetic $G_r$ functions for specific choices of admissible vectors $(h_\rho)_{\rho \in \mathfrak{S}_r}$ and orientations $(\mu_\rho)_{\rho \in \mathfrak{S}_r}, (\nu_\rho)_{\rho \in \mathfrak{S}_r}$ which we will describe in the next section.

## 3.2 Formulation of the conjecture

One aim of this article is to generalise the following conjecture of Bergeron, Charollois and García [1] on the special values of the elliptic Gamma function evaluated at points in complex cubic fields.

**Conjecture (Bergeron, Charollois and García):**

Suppose that $\mathbb{K}$ is a complex cubic field and $\mathfrak{f} \neq \mathcal{O}_{\mathbb{K}}$ is such that $\mathbb{K}^{+}(\mathfrak{f})$ is
In [1] the authors use modular symbols (see the cocycle relation (5)) for SL<sub>3</sub>(Z) built from smoothed Eisenstein series to establish a link between this special value and the partial zeta values associated to the conductor f and the ideal class [b] of the integral ideal b in the narrow ideal class group Cl<sup>+</sup>(f). Namely, the partial ζ function associated to this class is

$$
\zeta_f([b], s) = \sum_{I \in [b]} \mathcal{N}(I)^{-s}
$$

when ℜ(s) > 1 and where the sum ranges over all integral ideals in the class
[b]. This function can be extended into a meromorphic function over ℂ with
a pole at s = 1. In the specific case where ℚ is a complex cubic field, this
partial ζ function vanishes at the origin. In [1] the authors have proven that
the modulus of Γ<sup>±</sup><sub>f,b,a</sub>(ε; h) is independent of the choices for ε and h up to
signs and there is a sign δ = +1 or −1 such that

$$
\mathcal{N}(\mathfrak{a})\zeta'_{f}([\mathfrak{b}], 0) - \zeta'_{f}([\mathfrak{ab}], 0) = \delta \log |\Gamma_{f, \mathfrak{b}, \mathfrak{a}}^{\pm}(\varepsilon; h)|^2
$$

They also conjectured that Γ±<sub>f,b,a</sub>(ε; h) is a unit in the associated class field K+(f) related to the conjectural Stark unit. We believe that a combination of special values of arithmetic G<sub>r</sub> functions for r ≥ 2 will be linked to the partial ζ functions of an ATR field of degree d = r + 2 using modular symbols (see the cocycle relation (14)) for SL<sub>d</sub>(Z) related to some smoothed Eisenstein series. In view of this cohomological interpretation and following the work of Sczech in [19] on the Eisenstein cocycles for totally real fields, we consider the following combination of arithmetic G<sub>r</sub> functions:

**Definition 9:** Let $(\varepsilon_1, \dots, \varepsilon_r)$ be a set of fundamental units for $\mathcal{O}_f^{+, \times}$. Let $\underline{h} = (h_\rho)$ be a set of admissible vectors for $\mathfrak{f}, \mathfrak{b}, \mathfrak{a}$ where $\rho$ ranges over the permutations in $\mathfrak{S}_r$, the symmetric group on $r$ elements. For any choice of orientations for the $G_r$ functions $\underline{\mu} = (\mu_\rho), \underline{\nu} = (\nu_\rho)$ we define:

$$
I_{r, \mathfrak{f}, \mathfrak{b}, \mathfrak{a}}(\varepsilon_1, \dots, \varepsilon_r; \underline{h}, \underline{\mu}, \underline{\nu}) = \prod_{\rho \in \mathfrak{S}_r} G_{r, \mathfrak{f}, \mathfrak{b}, \mathfrak{a}}^{\mu_\rho} ([\varepsilon_{\rho(1)}] | \dots | [\varepsilon_{\rho(r)}]); h_\rho)^{\nu_\rho}
$$

where again

$$
[\varepsilon_1 | \dots | \varepsilon_r] = \left( \varepsilon_1, \dots, \prod_{j=1}^{k} \varepsilon_j, \dots, \prod_{j=1}^{r} \varepsilon_j \right)
$$

and the orientations $\mu_\rho, \nu_\rho \in \{-1, +1\}$. This is well-defined if for all permutation $\rho$, the family $(1, \varepsilon_{\rho(1)}, \dots, \prod_{j=1}^r \varepsilon_{\rho(j)})$ is a family of linearly independent vectors in $\mathbb{K}$.
totally complex. Take **b**, **a** as in section 3.1 and further ask that (*N*(**a**), 6) = 1. Define ε to be the generator of *O*<sub>f</sub><sup>+×</sup> satisfying σ<sub>ℝ</sub>(ε) < 1. Then for any admissible vector h ∈ L = **f****b**<sup>-1</sup>:

* the number $\Gamma_{\mathfrak{f},\mathfrak{b},\mathfrak{a}}^{-}(\varepsilon, h)^{-1} = \Gamma_{\mathfrak{f},\mathfrak{b},\mathfrak{a}}^{+}(\varepsilon^{-1}, h)$ is independent of the choice of $h$ and is the image in $\mathbb{C}$ of a unit $u_{L,\mathfrak{a}} \in \mathbb{K}^{+}(\mathfrak{f})$ under a complex embedding $\sigma'_{\mathbb{C}}$ extending $\sigma_{\mathbb{C}}$.

* any complex embedding of $\mathbb{K}^+(\mathfrak{f})$ above the real place of $\mathbb{K}$ sends $u_{L,\mathfrak{a}}$ to the unit circle.

* for $c \in I(\mathfrak{f})$, the reciprocity law is explicitly given by $u_{L,c^{-1},c}^{\sigma_c} = u_{L,c^{-1},a}$ where $\sigma_c$ is the corresponding element in $\text{Gal}(\mathbb{K}^+(\mathfrak{f})/\mathbb{K})$.

Consider the following example for the field $\mathbb{K} = \mathbb{Q}(z)$ where $z = e^{2i\pi/3}10^{1/3}$ is the root of the polynomial $x^3 - 10$ in the upper half-plane, $\mathfrak{f}^3 = (5)$, **b** = (1) and **a** is the unique degree one prime above 11 in $\mathbb{K}$. The fundamental unit is $\varepsilon = (2z^2 - z - 7)/3$. Then, for $h = -(35z^2 + 20z + 35)/3$ we compute

$$ \Gamma_{\mathfrak{f},\mathfrak{b},\mathfrak{a}}^{-}(\varepsilon, h)^{-1} = \frac{\Gamma\left(\frac{-1}{5}, \frac{\varepsilon^{-1}-1751}{495}, \frac{\varepsilon-776}{495}\right)^{-11}}{\Gamma\left(\frac{-11}{5}, \frac{\varepsilon^{-1}-1751}{45}, \frac{\varepsilon-776}{45}\right)^{-1}} \approx -27.5333588\dots - i \cdot 32.7146180\dots $$

with 1000 digits of precision and find it to be close to a root of the polynomial $x^{12}+57x^{11}+1956x^{10}+4640x^9+35415x^8-109818x^7+150139x^6-109818x^5+35415x^4+4640x^3+1956x^2+57x+1$ which defines $\mathbb{K}^+(\mathfrak{f})$. This example is developed in section 5.1.3 below.

In the cubic case, this conjecture has been tested numerically on hundreds of examples and we want to generalise this to ATR fields of degree $d \ge 4$ in some specific cases supported by numerical evidence. In section 4, we will describe a specific set of conditions under which we formulate our conjecture. To state it properly, we must discuss the choice of orientations, that is to say the choice of signs $\mu_\rho, \nu_\rho$. The cycle we use to evaluate our $G_r$ functions mimics the cycle used in [5], [27] in the context of explicit Shitani cones (or signed Shintani cones) in totally real fields, as well as in [22] in the context of the Eisenstein cocycle for totally real fields. In the context of number fields with exactly one complex place, explicit signed Shintani cones have been built in [10]. Therefore, we choose orientations to mimic the orientations given in [5]. Remember that we have fixed a set $\varepsilon_1, \dots, \varepsilon_r$ of fundamental units for $\mathcal{O}_{\mathfrak{f}}^{+\times}$. We have also fixed the ordering of the real embeddings so we may consider the sign of the regulator of these units.
We will now present a conjecture on special values of these arithmetic $G_r$ functions for specific choices of $(h_\rho)_{\rho \in \mathcal{S}_r}$ which we will describe in the next section.

## 3.2 Formulation of the conjecture

One aim of this article is to generalise the following conjecture of Bergeron, Charollois and García [1] on the special values of the elliptic Gamma function evaluated at points in complex cubic fields.

**Conjecture (Bergeron, Charollois and García):**

Suppose that $\mathbb{K}$ is a complex cubic field and $f \neq \mathcal{O}_{\mathbb{K}}$ is such that $\mathbb{K}^+(f)$ is totally complex. Take $\mathfrak{b}, \mathfrak{a}$ as before and further ask that $(\mathcal{N}(\mathfrak{a}), 6) = 1$. Then for any admissible vector $h \in L = \mathfrak{f}\mathfrak{b}^{-1}$:

* the number $\Gamma_{\mathfrak{f},\mathfrak{b},\mathfrak{a}}^{\pm}(\varepsilon; h)$ is independent of the choice of $h$ up to orientations and is the image in $\mathbb{C}$ of a unit $u_{L,\mathfrak{a}} \in \mathbb{K}^+(f)$ under a complex embedding $\sigma'_{\mathbb{C}}$ extending $\sigma_{\mathbb{C}}$.

* any complex embedding of $\mathbb{K}^+(f)$ above the real place of $\mathbb{K}$ sends $u_{L,\mathfrak{a}}$ to the unit circle.

* for $\mathfrak{c} \in I(f)$, the reciprocity law is explicitly given by $u_{L,\mathfrak{c}}^{\sigma_{\mathfrak{c}}} = u_{L^{c-1},\mathfrak{a}}$ where $\sigma_{\mathfrak{c}}$ is the corresponding element in $\text{Gal}(\mathbb{K}^+(f)/\mathbb{K})$.

Consider the following example for the field $\mathbb{K} = \mathbb{Q}(z)$ where $z = e^{2i\pi/3}10^{1/3}$ is the root of the polynomial $x^3 - 10$ in the upper half-plane, $f^3 = (5)$, $\mathfrak{b} = (1)$ and $\mathfrak{a}$ is the unique degree one prime above 11 in $\mathbb{K}$. The fundamental unit is $\varepsilon = (2z^2 - z - 7)/3$. Then, for $h = -(35z^2 + 20z + 35)/3$ we compute

$$ \Gamma_{\mathfrak{f},\mathfrak{b},\mathfrak{a}}^{+}(\varepsilon, h) = \frac{\Gamma\left(\frac{1}{5}, \frac{-\varepsilon^{-1}+1751}{495}, \frac{-\varepsilon+776}{495}\right)^{11}}{\Gamma\left(\frac{11}{5}, \frac{-\varepsilon^{-1}+1751}{45}, \frac{-\varepsilon+776}{45}\right)} \approx -27.5333588\dots - i \cdot 32.7146180\dots $$

with 1000 digits of precision and find it to be close to a root of the polynomial $x^{12} + 57x^{11} + 1956x^{10} + 4640x^9 + 35415x^8 - 109818x^7 + 150139x^6 - 109818x^5 + 35415x^4 + 4640x^3 + 1956x^2 + 57x + 1$ which defines $\mathbb{K}^+(f)$. This example is developed in section 5.1.3 below.

In the cubic case, this conjecture has been tested numerically on hundreds of examples and we want to generalise this to ATR fields of degree $d \ge 4$ in some specific cases supported by numerical evidence. In section 4, we will describe a specific set of conditions under which we formulate our conjecture.
**Definition 10:** The sign of the unit system $\epsilon_1, \dots, \epsilon_r$ is defined by:

$$sg(\epsilon_1, \dots, \epsilon_r) = \operatorname{sign}(\det(\log(\sigma_j(\epsilon_k))))_{1 \le j, k \le r})$$

Then, we define the following orientations

$$\mu_{\rho} = \nu_{\rho} = sg(\epsilon_1, \dots, \epsilon_r)\operatorname{sgn}(\rho)$$

where $\operatorname{sgn}(\rho)$ is the signature of the permutation $\rho$.

**Conjecture 11:** Suppose that $K$ is an ATR field of degree $d = r+2$. Suppose $f \neq O_K$ is an integral ideal such that $K^+(f)$ is totally complex. Suppose further that all units $\epsilon \in O_K^\times$ that are congruent to $1 \mod f$ have norm $+1$. Take $b, a$ two integral ideals as in Proposition 22. Suppose that there exists a system of fundamental units $\epsilon_1, \dots, \epsilon_r$ satisfying the conditions (*) and (**) associated to $f, b, a$ described in section 4.3. For any family of vectors $\underline{h}$ described in section 4.3, Proposition 22, and the orientations $\underline{\mu}, \underline{\nu}$ given in Definition 10, the following Kronecker type limit formula holds:

$$\mathcal{N}(a)\zeta'_{f}([b], 0) - \zeta'_{f}([ab], 0) = \frac{1}{\#\mathcal{Z}_{f}^{1}} \log \left| \prod_{(n, \epsilon) \in \mathcal{Z}_{f}^{1}} I_{r,f,b,a}(\epsilon_1, \dots, \epsilon_r; n\epsilon\underline{h}, \underline{\mu}, \underline{\nu}) \right|^2 \quad (16)$$

where the finite set $\mathcal{Z}_{f}^{1}$ is defined in section 4.3.4, Definition 25. In addition:

1. The product in the right-hand side is independent of the choice of vectors $\underline{h}$ in Proposition 22 up to a root of unity of order $m(r, N)$, where the number $m(r, N)$ is defined by $m(r, 2) = 4$, $m(r, N) = N$ for $3 \le N \le r+3$ and $m(r, N) = 1$ for $N > r+3$.

2. If $r \ge 1$, the product in the right-hand side, raised to the power $m(r, N)$ is the image in $\mathbb{C}$ of an algebraic unit $u_{L,a}$ in the class field $K^+(f)$ under a complex embedding $\sigma'_c$ extending $\sigma_c$.

3. Any embedding of $K^+(f)$ above a real embedding of $K$ sends $u_{L,a}$ to the unit circle.

**Remarks :**

1. The case $r = 0$ in the conjecture is already known as a consequence of the theory of complex multiplication and we refer to [20] for a presentation on the subject. Note that in this case the value

$$\theta_{f, b, a}^{\pm}(h)^{m(0,N)}$$
**Conjecture 10:** Suppose that $\mathbb{K}$ is an ATR field of degree $d = r+2$. Suppose $\mathfrak{f} \neq \mathcal{O}_{\mathbb{K}}$ is such that $\mathbb{K}^+(\mathfrak{f})$ is totally complex. Take $\mathfrak{b}, \mathfrak{a}$ as above. Suppose that there exists a system of fundamental units $\varepsilon_1, \dots, \varepsilon_r$ satisfying the conditions (*) and (**) associated to $\mathfrak{f}, \mathfrak{b}, \mathfrak{a}$ described in section 4.3. For all set of vectors $\underline{h}$ described in section 4.3, Proposition 22, there exists a choice of orientations $\underline{\mu}, \underline{\nu}$ such that

$$N \zeta'_{\mathfrak{f}}([\mathfrak{b}], 0) - \zeta'_{\mathfrak{f}}([\mathfrak{a}\mathfrak{b}], 0) = \log |I_{r,\mathfrak{f},\mathfrak{b},\mathfrak{a}}(\varepsilon_1, \dots, \varepsilon_r; \underline{h}, \underline{\mu}, \underline{\nu})|^2$$

In addition, if $r \ge 1$, there exists a positive integer $m(r, N)$ such that the value $I_{r,\mathfrak{f},\mathfrak{b},\mathfrak{a}}(\varepsilon_1, \dots, \varepsilon_r; \underline{h}, \underline{\mu}, \underline{\nu})^{m(r,N)}$ is the image in $\mathbb{C}$ of an algebraic unit $u_{L,\mathfrak{a},\underline{h}}$ in the class field $\mathbb{K}^+(\mathfrak{f})$ under a complex embedding $\sigma'_{\mathbb{C}}$ extending $\sigma_{\mathbb{C}}$. Moreover any embedding of $\mathbb{K}^+(\mathfrak{f})$ above a real embedding of $\mathbb{K}$ sends $u_{L,\mathfrak{a},\underline{h}}$ to the unit circle. Furthermore, $m(r, N)$ may be taken such that it is only divisible by prime numbers dividing $N$ and smaller than a bound $M(r)$.

**Remarks :**

1. The case $r = 0$ in the conjecture is already known as a consequence of the theory of complex multiplication and we refer to [17] for a presentation on the subject. Note that in this case the value

$$\theta_{\mathfrak{f},\mathfrak{b},\mathfrak{a}}^{\pm}(h)^{m(0,N)}$$

is not necessarily a unit but rather a $q$-unit where once again $q\mathbb{Z} = \mathfrak{f} \cap \mathbb{Z}$.

2. This conjecture is formulated with restrictions, but it has been successfully tested in many other cases outside of the conditions (*), (**). It is our aim to complete the formulation of this conjecture outside of these restrictions and to show that the unit $u_{L,\mathfrak{a},\underline{h}}$ is indeed independent of the choice of $\underline{h}$ as long as $\underline{h}$ is “admissible” in a way not yet fully understood. Once completed, it should be part of the conjecture that the explicit reciprocity law is given by $u_{L,\mathfrak{a}}^{\sigma'_{\mathbb{C}}} = u_{L,\mathfrak{c}^{-1},\mathfrak{a}}$.

3. The values of $m(r, N)$ in the cubic and quartic cases have been tested on dozens of examples. In the context of Proposition 22, the norm $N$ of the smoothing ideal $\mathfrak{a}$ is a prime number. The table below sums up our observations on the expected values for $m(r, N)$ in the cubic and quartic cases as well as the known values in the quadratic case.

<table><thead><tr><th>N</th><th>quartic<br>m(0, N)</th><th>cubic<br>m(1, N)</th><th>quartic<br>m(2, N)</th></tr></thead><tbody><tr><td>2</td><td>4</td><td>4</td><td>4</td></tr><tr><td>3</td><td>3</td><td>3</td><td>3</td></tr><tr><td>5</td><td>1</td><td>1</td><td>5</td></tr><tr><td>7 ≤</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>
is not necessarily a unit but rather a $q$-unit where once again $q\mathbb{Z} = f\cap\mathbb{Z}$. In the case $r = 1$, formula (16) was proven in [1] and the rest of the conjecture was also formulated in [1].

2. In the conjecture, the condition that all units congruent to $1 \pmod f$ should have norm $+1$ is not restrictive. Indeed, if a unit $\varepsilon$ existed with $\varepsilon \equiv 1 \pmod f$ and $\mathcal{N}(\varepsilon) = -1$ then all $L$-functions attached to the extension $\mathbb{K}^+(f)/\mathbb{K}$ would have a zero at $0$ of order at least $2$. Therefore, the left-hand side in formula (16) would vanish.

3. This conjecture is formulated with restrictions, but it has been successfully tested in many other cases outside of the conditions $(*)$, $(**)$.
It is our aim to complete the formulation of this conjecture outside of these restrictions and to extend the range of “compatible” vectors $\underline{h}$ for which the conjecture holds. Once completed, it should be part of the conjecture that the explicit reciprocity law is given by $u_{L,\mathfrak{a}}^{\sigma_c} = u_{L^{c-1},\mathfrak{a}}$ for an integral ideal $\mathfrak{c}$ in $I(f)$.

4. Our computations show a clear change of behavior between the smoothings with $N \le r+3$ and the smoothings with $N > r+3$. In the case of “small smoothings”, the product in the right-hand side of (16) should be an algebraic unit inside $\mathbb{K}^+(f)(\exp(2i\pi/m(r, N)))$ whereas in the case of “big smoothings”, it should be an algebraic unit inside $\mathbb{K}^+(f)$. We expect the exact bound $r+3$ because of the von Staudt-Clausen theorem on the denominator of Bernoulli numbers: a prime divisor $p$ of the denominator of one of the $B_1, B_2, \dots, B_{r+2}$ must satisfy $p \le r+3$. The value of $m(r, N)$ has been tested on many examples for $0 \le r \le 3$. The table below presents the expected values for $m(r, N)$ in these cases.

<table><thead><tr><th>N</th><th>quartic<br>m(0, N)</th><th>cubic<br>m(1, N)</th><th>quartic<br>m(2, N)</th><th>quintic<br>m(3, N)</th></tr></thead><tbody><tr><td>2</td><td>4</td><td>4</td><td>4</td><td>4</td></tr><tr><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td></tr><tr><td>5</td><td>1</td><td>1</td><td>5</td><td>5</td></tr><tr><td>7 &le;</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

In practice, the product in the conjecture often contains redundant terms, and it is thus already a power inside $\mathbb{K}^+(f)$. We therefore define the integer $\kappa$ as the maximal integer $k \ge 1$ dividing $\#\mathcal{Z}_f^1$, such that there exist subsets $(U_{\rho,j})_{\rho \in \mathfrak{S}_r, 1 \le j \le k}$ of $\mathcal{Z}_f^1$ satisfying

(i) for all $\rho \in \mathfrak{S}_r$, for all $1 \le j \le k$, $\#U_{\rho,j} = \#\mathcal{Z}_f^1/k$
This shows that there is a specific behavior for “small” smoothings and that most of the time $m(r, N) = 1$.

We give here an example supporting the conjecture with a “small” smoothing: take the quartic field $K = \mathbb{Q}(z)$ where $z$ is the root of the polynomial $x^4 - 6x^3 - x^2 - 3x + 1$ in the upper half-plane, and take $f$ the unique degree one prime above 2 in $K$. We choose $a$ the degree one prime above 5 in $K$. Then we may compute the two quotients

$$v_1 = \frac{G_2\left(\frac{1}{2}, \frac{5z^3-29z^2-15z-81}{70}, \frac{6z^3-39z^2+10z+5}{70}, \frac{-2z^3+13z^2-z+24}{70}\right)^{-5}}{G_2\left(\frac{5}{2}, \frac{5z^3-29z^2-15z-81}{14}, \frac{6z^3-39z^2+10z+5}{14}, \frac{-2z^3+13z^2-z-24}{14}\right)^{-1}},$$ $$v_2 = \frac{G_2\left(\frac{-1}{2}, \frac{-2z^3+13z^2-z+24}{70}, \frac{-5z^3+29z^2+15z+81}{70}, \frac{2z^3-13z^2-6z-101}{70}\right)^5}{G_2\left(\frac{-5}{2}, \frac{-2z^3+13z^2-z+24}{14}, \frac{-5z^3+29z^2+15z+81}{14}, \frac{2z^3-13z^2-6z-101}{14}\right)}$$

The values obtained are respectively $v_1 \approx -2.0167576... - i \cdot 5.8008598...$ and $v_2 \approx -0.4159958... + i \cdot 0.0018434...$. Their product $v_1v_2 \approx 0.8496565... - i \cdot 2.4094157...$ is not an algebraic integer in $K^+(f)$. Yet the fifth power $(v_1v_2)^5 \approx 108.0070738... - i \cdot 13.4979021...$ of this product coincides up to at least 1000 digits with a root of the polynomial $x^8 - 215x^7 + 11629x^6 + 11941x^5 + 3913x^4 + 11941x^3 + 11629x^2 - 215x + 1$ which defines $K^+(f)$. The constant term of this polynomial is 1, so the roots of this polynomial are units inside $K^+(f)$.

The way our conjecture was formulated is also inspired by the rank one abelian Stark conjecture in the ATR case:

**Rank one abelian Stark conjecture (ATR case, see [6]):**

Write $e_f$ for the number of roots of unity in $K^+(f)$. There is a unit $u_{Stark}$ in $K^+(f)$ such that

* for all class $[b]$ in $Cl^+(f)$ and corresponding $\sigma_b \in \text{Gal}(K^+(f)/K)$,

$$\zeta_f'([b], 0) = -\frac{1}{e_f} \log |u_{\text{Stark}}^{\sigma_b}|$$

* every complex embedding of $K^+(f)$ above a real embedding of $K$ sends $u_{\text{Stark}}$ to the unit circle.

* $K^+(f)(u_{\text{Stark}}^{1/e_f})$ is an abelian extension of $K$.

There are ways to make this statement a bit more precise so that the unit $u_{\text{Stark}}$ is unique and hopefully the unit $u_{L,a,\underline{h}}$ will be related to the Stark unit by the formula:

$$u_{\text{Stark}}^{N-\sigma_a} = u_{L,a,\underline{h}}$$
(ii) for all $\rho \in \mathfrak{S}_r$, $j \neq l$, $U_{\rho,j} \cap U_{\rho,l} = \emptyset$.

(iii) for all $1 \le j \le k$, the product

$$ \left( \prod_{\rho \in \mathfrak{S}_r} \prod_{(n,\varepsilon) \in U_{\rho,j}} G_{r,\mathfrak{f},\mathfrak{b},\mathfrak{a}}^{\mu_{\rho}}(\varepsilon_1, \dots, \varepsilon_r; n\varepsilon\underline{h})^{\nu_{\rho}} \right)^{m(r,N)} $$

is independent of $j$.

Conditions (i) and (ii) express the fact that the sets $U_{\rho,j}$ form a special partition of the set $(\mathcal{Z}_f^1)^{r!}$. Condition (iii) implies in particular that

$$ \prod_{(n,\varepsilon) \in \mathcal{Z}_f^1} I_{r,\mathfrak{f},\mathfrak{b},\mathfrak{a}}(\varepsilon_1, \dots, \varepsilon_r; n\varepsilon\underline{h}, \underline{\mu}, \underline{\nu}) = \left( \prod_{\rho \in \mathfrak{S}_r} \prod_{(n,\varepsilon) \in U_{\rho,1}} G_{r,\mathfrak{f},\mathfrak{b},\mathfrak{a}}^{\mu_{\rho}}(\varepsilon_1, \dots, \varepsilon_r; n\varepsilon\underline{h})^{\nu_{\rho}} \right)^{\kappa} $$

is already a $\kappa$-th power of what is expected to be an algebraic unit in $\mathbb{C}$. In the cubic case, it is expected that $\kappa = \#\mathcal{Z}_f^1$ and this is expected to be true in most simple examples for $d \ge 4$. However, the worst case $\kappa = 1$ with $\#\mathcal{Z}_f^1 > 1$ already appears for $d = 4$ (see section 5.4).

We give here a quartic example supporting the conjecture with a “small” smoothing: take the quartic field $K = \mathbb{Q}(z)$ where $z$ is the root of the polynomial $x^4 - 6z^3 - x^2 - 3x + 1$ in the upper half-plane and fix the ordering on the real embeddings of $K$ such that $\sigma_1(z) < \sigma_2(z)$. Take $\mathfrak{f}$ the unique degree one prime above $q = 2$ in $K$. A possible choice for the fundamental units is given by $\varepsilon_1 = (-2z^3 + 13z^2 - z + 3)/7$, $\varepsilon_2 = (-5z^3 + 29z^2 + 15z + 18)/7$. We choose $\mathfrak{a}$ the degree one prime above $N = 5$ in $K$ and $\mathfrak{b} = \mathcal{O}_K$. Here, $q = 2$ therefore $\mathcal{Z}_f^1 = \{(1,1)\}$ and $\kappa = \#\mathcal{Z}_f^1 = 1$. A possible choice of vectors $\underline{h}$ as in Proposition 22 is $h_1 = h_2 = (-18z^3 + 96z^2 + 82z + 62)/7$. We may compute the orientations $\underline{\mu} = \underline{\nu} = [-1,1]$ as in Definition 10 and the two quotients

$$ v_1 = \frac{G_2 \left( \frac{1}{2}, \frac{5z^3 - 29z^2 - 15z - 81}{70}, \frac{6z^3 - 39z^2 + 10z + 5}{70}, \frac{-2z^3 + 13z^2 - z + 24}{70} \right)^{-5}}{G_2 \left( \frac{5}{2}, \frac{5z^3 - 29z^2 - 15z - 81}{14}, \frac{6z^3 - 39z^2 + 10z + 5}{14}, \frac{-2z^3 + 13z^2 - z - 24}{14} \right)^{-1}}, \\ v_2 = \frac{G_2 \left( \frac{-1}{2}, \frac{-2z^3 + 13z^2 - z + 24}{70}, \frac{-5z^3 + 29z^2 + 15z + 81}{70}, \frac{2z^3 - 13z^2 - 6z - 101}{70} \right)^5}{G_2 \left( \frac{-5}{2}, \frac{-2z^3 + 13z^2 - z + 24}{14}, \frac{-5z^3 + 29z^2 + 15z + 81}{14}, \frac{2z^3 - 13z^2 - 6z - 101}{14} \right)^5} $$

The values obtained are respectively $v_1 \approx -2.0167576... - i \cdot 5.8008598...$ and $v_2 \approx -0.4159958... + i \cdot 0.0018434...$. Their product $v_1 v_2 \approx 0.8496565...$ -
Moreover, our conjecture relates to Hilbert's twelfth problem, allowing for a construction of certain class fields using specific multivariate analytic functions.

# 4 Computing the arithmetic $G_r$ functions

In order to understand how the choice of vectors $(h_\rho)_{\rho \in \mathcal{S}_r}$ should be made, we must carry out an explicit description of the geometric aspects of the construction of the arithmetic $G_{r,f,b,a}$ functions. Thus, we will identify key quantities in the construction, namely the parameters $t$ and $l$ defined in section 4.1 and then show how we can control these quantities. We then try to minimize the quantities that have the greatest impact on the computation time and describe an efficient algorithm to compute the arithmetic products $I_{r,f,b,a}(\varepsilon_1, \dots, \varepsilon_r; \underline{h}, \underline{\mu}, \underline{\nu})$ given in definition 9.

## 4.1 Geometric setup

Let $\mathbb{K}$ be an ATR field of degree $d = r+2$ and let $f, b, a$ be given as in section 3.1. Fix a basis $B$ of the $\mathbb{Q}$-vector space $\mathbb{K}$. Any basis $B'$ of the $\mathbb{Q}$-vector space $\mathbb{K}$ is called a positive basis if $\det_B(B') > 0$. We say that we have fixed a global orientation of $\mathbb{K}$, which is not to be confused with the orientations $\underline{\mu}$ and $\underline{\nu}$ associated to the product in definition 9. Put $L = \mathfrak{f}\mathfrak{b}^{-1}$. Fix a system of fundamental units $(\varepsilon_1, \dots, \varepsilon_r)$ of $\mathcal{O}_f^{+,\times}$. Write $(u_1, \dots, u_r) = [\varepsilon_{\rho(1)}|\dots|\varepsilon_{\rho(r)}]$ for any permutation $\rho \in \mathcal{S}_r$. For convenience, we will always write $u_0 = 1$. Choose $h \in L$ an admissible vector (see definition 7). Put $mh' = h$ where $m$ is a positive integer and $h'$ is primitive in $L$. This means that for all non-zero integer $n \in \mathbb{Z}$, $h'/n \in L$ implies that $n = \pm 1$.

We may choose a positive basis $B_L = (e_0 = h', e_1, \dots, e_{r+1})$ of $L$ such that $u_i h' = \sum_{j=0}^i c_{ij0} e_j$ where the coefficients $c_{ij0}$ are integers and $c_{ii0} > 0$. This means in particular that $\det_B(B_L) > 0$. To evaluate geometric $G_r$ functions we must fix an orientation form on the lattice $L$ and we consider the orientation form $\det_{B_L}$. In the basis $B_L$ we identify the units $u_i$ with the matrix $(c_{ijk})_{j,k} \in \text{SL}_d(\mathbb{Z})$ such that $u_i e_k = \sum_{j=0}^{r+1} c_{ijk} e_j$. Fix the dual basis $C = (f_0, \dots, f_{r+1})$ of $\Lambda = \text{Hom}_{\mathbb{Z}}(L, \mathbb{Z})$ such that $f_i(e_j) = \delta_{ij}$ where $\delta_{ij}$ is the Kronecker symbol. In this basis the linear form $a$ defined in (3.1) is exactly $f_{r+1}$. For $1 \le i \le r$, the composition of $a$ with multiplication by $u_i$ is written $au_i = \sum_{i=1}^{r+1} c_{i(r+1)k} f_k$. Then there is a unique primitive vector $\gamma \in L$ such that $\det_C(a, au_1, \dots, au_r, \cdot) = s\gamma = \pm sh'$ with $s$ a positive integer.

Next, we identify the family $\underline{a} = (a = au_0, au_1, \dots, au_r)$ with the matrix obtained by concatenation of the coefficients of the linear forms $au_i$ in the
i · 2.4094157... is not an algebraic integer in $K^+(f)$. Yet the fifth power $(v_1v_2)^5 \approx 108.0070738... - i \cdot 13.4979021...$ of this product coincides up to at least 1000 digits with a root of the polynomial $x^8 - 215x^7 + 11629x^6 + 11941x^5 + 3913x^4 + 11941x^3 + 11629x^2 - 215x + 1$ which defines $K^+(f)$. The constant term of this polynomial is 1, so the roots of this polynomial are units inside $K^+(f)$. We may also check formula (16) up to 1000 digits as:

$$ \mathcal{N}(\mathfrak{a})\zeta'_{f}([\mathfrak{b}], 0) - \zeta'_{f}([\mathfrak{ab}], 0) \approx \log |v_1v_2|^2 \approx 1.8759781... $$

We will present an example in the same conditions except for the smoothing which will be a “big smoothing” in section 5.2.1.

To end this section, we point out that the way our conjecture was formulated is also inspired by the rank one abelian Stark conjecture in the ATR case:

**Rank one abelian Stark conjecture (ATR case, see [7]):**

Write $e_f$ for the number of roots of unity in $K^+(f)$. There is a unit $u_{Stark}$ in $K^+(f)$ such that

* for all class $[\mathfrak{b}]$ in $Cl^+(f)$ and corresponding $\sigma_{\mathfrak{b}} \in \text{Gal}(K^+(f)/K)$,

$$ \zeta'_{f}([\mathfrak{b}], 0) = -\frac{1}{e_f} \log |u_{\text{Stark}}^{\sigma_{\mathfrak{b}}}|^2 $$

* every complex embedding of $K^+(f)$ above a real embedding of $K$ sends $u_{\text{Stark}}$ to the unit circle.

* $K^+(f)(u_{\text{Stark}}^{1/e_f})$ is an abelian extension of $K$.

There are ways to make this statement a bit more precise so that the unit $u_{\text{Stark}}$ is unique and hopefully the unit $u_{L,\mathfrak{a}}$ from Conjecture 11 will be related to the Stark unit by the formula:

$$ u_{\text{Stark}}^{m(r,N)(N-\sigma_{\mathfrak{a}})\#\mathbb{Z}_{f}^{1}} = u_{L,\mathfrak{a}}^{-e_f} $$

In the cases where $\zeta'_{f}([\mathfrak{b}], 0) \neq 0$ for some class $[\mathfrak{b}]$, our conjecture relates to Hilbert's twelfth problem, allowing for a construction of certain class fields using specific multivariate analytic functions.

# 4 Computing the arithmetic $G_r$ functions

In order to understand how the choice of vectors $(h_\rho)_{\rho \in \mathfrak{S}_r}$ should be made, we must carry out an explicit description of the geometric aspects of the
basis $C$:

$$ \underline{a} = \begin{pmatrix} au_0 \\ au_1 \\ \vdots \\ au_j \\ \vdots \\ au_r \end{pmatrix} = \begin{pmatrix} 0 & 0 & \dots & 0 & \dots & 1 \\ 0 & c_{1(r+1)1} & \dots & c_{1(r+1)k} & \dots & c_{1(r+1)(r+1)} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 0 & c_{j(r+1)1} & \dots & c_{j(r+1)k} & \dots & c_{j(r+1)(r+1)} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 0 & c_{r(r+1)1} & \dots & c_{r(r+1)k} & \dots & c_{r(r+1)(r+1)} \end{pmatrix} = (\mathbf{0} \quad \mathcal{A}) $$

This matrix has a first column filled with zeroes and the submatrix $\mathcal{A}$ is a square matrix of size $r+1$. The elementary divisors of $\mathcal{A}$ associated to its Smith normal form are the positive integers $[A_r, \dots, A_1, A_0 = 1]$ such that $A_i|A_{i+1}$. Then we define

$$ \lambda(u_1, \dots, u_r; h) = \lambda = \prod_{i=1}^{r} c_{ii0} $$

$$ s(u_1, \dots, u_r; h) = s = \prod_{i=1}^{r} A_i $$

$$ t(u_1, \dots, u_r; h) = t = A_r $$

so that

$$ \lambda a(u_1, \dots, u_r; h) = \lambda a = \det_{B_L}(h', u_1 h', \dots, u_r h', \cdot) \quad (15) $$

and $\det_C(a, au_1, \dots, au_r, \cdot) = \pm sh'$. Recalling the construction in Proposition 4, we must now consider positive dual families $\underline{\alpha} = (\alpha_0, \dots, \alpha_r) \subset L$ to $\underline{a}$. Recall that this means that for all $0 \le j \le r$:

$$ au_j(\alpha_j) > 0, \quad au_k(\alpha_j) = 0, \forall k \neq j $$

with once again the convention that $u_0 = 1$. Consider the matrix $\mathcal{B} = (\prod_{i=1}^r A_i^{-1})\mathrm{com}(\mathcal{A})^T$ which corresponds to the rescaled comatrix in formula (12). Then the product $\mathcal{A} \cdot \mathcal{B} = tI_{r+1}$. Write $\mathcal{B} = (b_{ij})_{1 \le i,j \le r+1}$. Then a possible choice for $\underline{\alpha}$ is given by $\alpha_j = \sum_{i=1}^{r+1} b_{ij} e_i \in L$. This family satisfies for all $0 \le j \le r$:

$$ au_j(\alpha_j) = t, \quad au_k(\alpha_j) = 0, \forall k \neq j \quad (16) $$

We write this as the multiplication of two matrices $\underline{a} \cdot \underline{\alpha} = tI_{r+1}$. We say that this choice of $\underline{\alpha}$ is a uniform positive dual family to $\underline{a}$ because the value $au_j(\alpha_j) = t$ is independent of $j$. We argue that this choice is minimal amongst the uniform positive dual families to $\underline{a}$ in $L$, as explained in the following lemma.
construction of the arithmetic $G_{r,f,b,a}$ functions. Thus, we will identify key quantities in the construction, namely the parameters $t$ and $l$ defined in section 4.1 and then show how we can control these quantities. We then try to minimize the quantities that have the greatest impact on the computation time and describe an efficient algorithm to compute the arithmetic products $I_{r,f,b,a}(\varepsilon_1, \dots, \varepsilon_r; \underline{h}, \underline{\mu}, \underline{\nu})$ given in Definition 9.

## 4.1 Geometric setup

Let $\mathbb{K}$ be an ATR field of degree $d = r+2$ and let $\mathfrak{f}, \mathfrak{b}, \mathfrak{a}$ be given as in section 3.1. Remember that we have fixed an ordering $\sigma_1, \dots, \sigma_r, \sigma_{r+1} = \sigma_{\mathbb{C}}, \overline{\sigma_{r+1}}$ on the embeddings of $\mathbb{K}$. Put $L = \mathfrak{f}\mathfrak{b}^{-1}$. Fix a system of fundamental units $(\varepsilon_1, \dots, \varepsilon_r)$ of $\mathcal{O}_f^{+,\times}$. Write $(u_1, \dots, u_r) = [\varepsilon_{\rho(1)}|\dots|\varepsilon_{\rho(r)}]$ for any permutation $\rho \in \mathfrak{S}_r$. For convenience, we will always write $u_0 = 1$. Choose $h \in L$ an admissible vector (see Definition 7). Put $mh' = h$ where $m$ is a positive integer and $h'$ is primitive in $L$. This means that for all non-zero integer $n \in \mathbb{Z}$, $h'/n \in L$ implies that $n = \pm 1$.

Recall that a basis $B = (e_0, \dots, e_{r+1})$ of the $\mathbb{Q}$-vector space $\mathbb{K}$ is called positive if $i \cdot \det((\sigma_j(e_{k-1}))_{1\le j,k\le d}) > 0$. We may choose a positive basis $B_L = (e_0 = h', e_1, \dots, e_{r+1})$ of $L$ such that $u_j h' = \sum_{k=0}^j c_{jk0} e_k$ where the coefficients $c_{jk0}$ are integers and $c_{jj0} > 0$. To evaluate geometric $G_r$ functions we fix the orientation form on the lattice $L$ given by $\det_{B_L}$. In the basis $B_L$ we identify the units $u_j$ with the matrix $(c_{jkl})_{k,l} \in \text{SL}_d(\mathbb{Z})$ such that $u_j e_l = \sum_{k=0}^{r+1} c_{jkl} e_k$. Fix the dual basis $C = (f_0, \dots, f_{r+1})$ of $\Lambda = \text{Hom}_{\mathbb{Z}}(L, \mathbb{Z})$ such that $f_j(e_k) = \delta_{jk}$ where $\delta_{jk}$ is the Kronecker symbol. In this basis the linear form $a$ defined in section 3.1 is exactly $f_{r+1}$. For $1 \le j \le r$, the composition of $a$ with multiplication by $u_j$ is written $au_j = \sum_{l=1}^{r+1} c_{j(r+1)l} f_l$. Then there is a unique primitive vector $\gamma \in L$ such that $\det_C(a, au_1, \dots, au_r, \cdot) = s\gamma = \pm sh'$ with $s$ a positive integer.

Next, we identify the family $\underline{a} = (a = au_0, au_1, \dots, au_r)$ with the matrix obtained by concatenation of the coefficients of the linear forms $au_i$ in the basis $C$:

$$ \underline{a} = \begin{pmatrix} au_0 \\ au_1 \\ \vdots \\ au_i \\ \vdots \\ au_r \end{pmatrix} = \begin{pmatrix} 0 & 0 & \dots & 0 & \dots & 1 \\ 0 & c_{1(r+1)1} & \dots & c_{1(r+1)k} & \dots & c_{1(r+1)(r+1)} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 0 & c_{i(r+1)1} & \dots & c_{i(r+1)k} & \dots & c_{i(r+1)(r+1)} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 0 & c_{r(r+1)1} & \dots & c_{r(r+1)k} & \dots & c_{r(r+1)(r+1)} \end{pmatrix} = (\mathbf{0} \quad \mathcal{A}) $$

This matrix has a first column filled with zeroes and the submatrix $\mathcal{A}$ is a square matrix of size $r+1$. The elementary divisors of $\mathcal{A}$ associated to its
**Lemma 11:** Let $\mathcal{B}' \in M_{r+1}(\mathbb{Z})$ be a square matrix of size $r+1$ such that $\mathcal{A} \cdot \mathcal{B}' = nI_{r+1}$. Then $t$ divides $n$. Consequently, any uniform positive dual family $\underline{\alpha}'$ to $\underline{\alpha}$ satisfies $\underline{\alpha} \cdot \underline{\alpha}' = nI_{r+1}$ where $t$ divides $n$. Furthermore, the minimal uniform positive dual family $\underline{\alpha}$ to $\underline{\alpha}$ described above satisfies the following property: for all prime $p$ there exists an index $0 \le j \le r$ such that $\forall m \in \mathbb{Z}, \alpha_j - m\gamma \notin pL$.

**Proof:**

Consider the matrices $U, V \in \text{GL}_{r+1}(\mathbb{Z})$ such that $U\mathcal{A}V$ is the diagonal matrix in Smith normal form $[A_r = t, \dots, A_1, 1]$. Then

$$nI_{r+1} = (U\mathcal{A}V) \cdot (V^{-1}\mathcal{B}'U^{-1}) = \begin{pmatrix} t & 0 & \dots & 0 \\ 0 & A_{r-1} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \dots & \dots & 1 \end{pmatrix} V^{-1}\mathcal{B}'U^{-1}$$

This shows that $t|n$. Consider now a uniform positive dual family $\underline{\alpha}' = (\alpha'_0, \dots, \alpha'_r)$ to $\underline{\alpha}$ in $L$ such that $\underline{\alpha} \cdot \underline{\alpha}' = nI_{r+1}$. Write the coordinates of the $\alpha'_j$ in the basis $B_L$ as $\alpha'_j = \sum_{i=0}^{r+1} b'_{ij}e_i$. Then the matrix $\mathcal{B}' = (b'_{ij})_{1 \le i,j \le r+1}$ satisfies $\mathcal{A} \cdot \mathcal{B}' = nI_{r+1}$ and $t$ divides $n$.

Consider now the minimal choice $\underline{\alpha}$ given above associated to the matrix $\mathcal{B} = (\prod_{i=1}^r A_i^{-1})\text{com}(\mathcal{A})^T$ such that $\underline{\alpha} \cdot \underline{\alpha} = tI_{r+1}$. The elementary divisors of $\mathcal{B}$ are given by the positive integers $[A_r, A_r/A_1, \dots, A_r/A_{r-1}, 1]$. The last elementary divisor is the $\text{gcd}$ of the coordinates of the $\alpha_j$ in the basis $B_L$. Therefore, for each prime $p$ there exists an integer $j$ and a coefficient $b_{ij}$ not divisible by $p$. For any $m \in \mathbb{Z}$, we get $\alpha_j - m\gamma = \pm me_0 + \sum_{i=1}^{r+1} b_{ij}e_i \notin pL$. $\square$

From now on we will always fix $\underline{\alpha} = \underline{\alpha}(u_1, \dots, u_r; h)$ the minimal uniform positive dual family $\underline{\alpha}$ to $\underline{\alpha}$ given above. Define $M = M(u_1, \dots, u_r; h) = \mathbb{Z}h' \oplus (\oplus_{j=0}^r \mathbb{Z}\alpha_j)$ in $L$ associated to this family. This sublattice has index $\det(\mathcal{B}) = t^{r+1}/s$ in $L$. Recalling the construction in section (3.1) and formula (11) for the geometric $G_{r,\underline{\alpha}}$ function we may identify $F(\underline{\alpha})/\mathbb{Z}\gamma \simeq L/M$ so that the geometric $G_{r,\underline{\alpha}}$ function is a product of $t^{r+1}/s$ ordinary elliptic $G_r$ functions. In the worst case scenario $t = s$ and $G_{r,\underline{\alpha}}$ is a product of $t^r$ ordinary elliptic $G_r$ functions.

Let us now analyse $\underline{\alpha}$ more closely. It follows from the definition of the linear form $a$ in formula (15) and lemma 11 that for $i \ne j$, $\alpha_i$ must satisfy

$$\det_{B_L}(u_0h', u_1h', \dots, u_rh', u_j\alpha_i) = 0$$

with once again the convention that $u_0 = 1$, so that $\alpha_i/h' = \sum_{k=0}^r d_{ijk}u_j^{-1}u_k$ with rational coefficients $d_{ijk}$. This quotient depends rather loosely upon $h'$
Smith normal form are positive integers denoted $[A_r, \dots, A_1, A_0 = 1]$ such that $A_i|A_{i+1}$. Then we define

$$ \lambda(u_1, \dots, u_r; h) = \lambda = \prod_{j=1}^{r} c_{jj0} $$

$$ s(u_1, \dots, u_r; h) = s = \prod_{j=1}^{r} A_j $$

$$ t(u_1, \dots, u_r; h) = t = A_r $$

so that

$$ \lambda a(u_1, \dots, u_r; h) = \lambda a = \det_{B_L}(h', u_1 h', \dots, u_r h', \cdot) \quad (17) $$

and $\det_C(a, au_1, \dots, au_r, \cdot) = \pm sh'$. Recalling the construction in Proposition 4, we must now consider positive dual families $\underline{\alpha} = (\alpha_0, \dots, \alpha_r) \subset L$ to $\underline{a}$. Recall that this means that for all $0 \le j \le r$:

$$ au_j(\alpha_j) > 0, \quad au_k(\alpha_j) = 0, \forall k \ne j $$

with once again the convention that $u_0 = 1$. Consider the matrix $\mathcal{B} = (\prod_{i=1}^r A_i^{-1})\mathrm{com}(\mathcal{A})^T$ which corresponds to the rescaled comatrix in formula (12). Then the product $\mathcal{A} \cdot \mathcal{B} = tI_{r+1}$. Write $\mathcal{B} = (b_{ij})_{1 \le i,j \le r+1}$. Then a possible choice for $\underline{\alpha}$ is given by $\alpha_j = \sum_{i=1}^{r+1} b_{ij}e_i \in L$. This family satisfies for all $0 \le j \le r$:

$$ au_j(\alpha_j) = t, \quad au_k(\alpha_j) = 0, \forall k \ne j \quad (18) $$

We write this as the multiplication of two matrices $\underline{a} \cdot \underline{a} = tI_{r+1}$. We say that this choice of $\underline{a}$ is a uniform positive dual family to $\underline{a}$ because the value $au_j(\alpha_j) = t$ is independent of $j$. We argue that this choice is minimal amongst the uniform positive dual families to $\underline{a}$ in $L$, as explained in the following lemma.

**Lemma 12:** Let $\mathcal{B}' \in M_{r+1}(\mathbb{Z})$ be a square matrix of size $r+1$ such that $\mathcal{A} \cdot \mathcal{B}' = nI_{r+1}$. Then $t$ divides $n$. Consequently, any uniform positive dual family $\underline{\alpha}'$ to $\underline{a}$ satisfies $\underline{a} \cdot \underline{\alpha}' = nI_{r+1}$ where $t$ divides $n$. Furthermore, the minimal uniform positive dual family $\underline{\alpha}$ to $\underline{a}$ described above satisfies the following property: for all prime $p$ there exists an index $0 \le j \le r$ such that $\forall n' \in \mathbb{Z}, \alpha_j - n'\gamma \notin pL$.

**Proof:**

Consider the matrices $U, V \in GL_{r+1}(\mathbb{Z})$ such that $UAV$ is the diagonal
and this leads us to investigate the case $h = 1$ (which is not an admissible vector). In what follows, we will write with a $\sim$ the counterpart of the quantities defined in the construction above for $h = 1$ and $\mathcal{O}_{\mathbb{K}}$ instead of the lattice $L$. There is a positive basis $B_{\mathbb{K}} = [1, \tilde{e}_1, \dots, \tilde{e}_{r+1}]$ of $\mathcal{O}_{\mathbb{K}}$ such that $u_i = \sum_{j=0}^{i} \tilde{c}_{ij0}\tilde{e}_j$ with $\tilde{c}_{ii0} > 0$. Consider

$$ \tilde{\lambda}\tilde{a} = \left( \prod_{i=1}^{r} \tilde{c}_{ii0} \right) \tilde{a} = \det_{B_{\mathbb{K}}} (1, u_1, \dots, u_r, \cdot) \quad (17) $$

and put $\tilde{a} = (\tilde{a}, \tilde{a}u_1, \dots, \tilde{a}u_r)$. Associate to $\tilde{a}$ a square matrix $\tilde{\mathcal{A}}$ of size $r+1$ as above for $\underline{a}$. Write as above the elementary divisors of this matrix $\tilde{\mathcal{A}}$ associated to $\tilde{a}$ as $[\tilde{A}_r, \dots, \tilde{A}_1, \tilde{A}_0 = 1]$ such that $\tilde{A}_i|\tilde{A}_{i+1}$. Then put $\tilde{s} = \prod_{i=1}^{r} \tilde{A}_i$, $\tilde{t} = \tilde{A}_r$ and construct as above $\tilde{\alpha}$ associated to the square matrix $\tilde{\mathcal{B}} = (\prod_{i=1}^{r} \tilde{A}_i^{-1})\mathrm{com}(\tilde{\mathcal{A}})^T$ such that $\tilde{a} \cdot \tilde{\alpha} = \tilde{t}I_{r+1}$. Recall that this means that for all $0 \le j \le r$:

$$ \tilde{a}u_j(\tilde{\alpha}_j) = \tilde{t}, \quad \tilde{a}u_k(\tilde{\alpha}_j) = 0, \forall k \neq j \quad (18) $$

In the case $r = 1$, $\tilde{\lambda}$ is the content of the smallest ring $\mathcal{O}$ containing the unit $\varepsilon$ in the sense of [[2], Definition 14] and $\tilde{t}$ measures how far the $\mathbb{Z}$-module $\mathbb{Z} + \mathbb{Z}\varepsilon$ is from being a ring. Therefore, we will call $\tilde{\lambda}$ the content of the unit system $(u_1, \dots, u_r)$ and $\tilde{t}$ the overflow of this unit system. We insist that these quantities only depend on the unit system $(u_1, \dots, u_r)$. We relate this general construction to the construction for a particular admissible vector $h$ in $L$ as follows:

**Lemma 12:** The family $\varepsilon\tilde{\alpha}h' = (\varepsilon\tilde{\alpha}_0h', \dots, \varepsilon\tilde{\alpha}_rh')$ is a uniform positive dual family to $\underline{a}$ in $L$ for some sign $\varepsilon = \pm 1$. Therefore there exists a positive integer $l$ such that $\underline{a} \cdot (\varepsilon\tilde{\alpha}h') = ltI_{r+1}$. The value of $l$ is given by the following equality which holds in $\mathbb{Z}$

$$ \varepsilon\lambda lt = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \tilde{\lambda}\tilde{t} \quad (19) $$

**Proof:**

We use formulae (15) relating $\underline{a}$ to the form $\det_{B_{\mathbb{K}}}$ and (17) relating $\tilde{a}$ to the form $\det_{B_L}$ to evaluate $\lambda au_k\tilde{\alpha}_j h'$.

$$ \lambda au_k \tilde{\alpha}_j h' = \det_{B_L}(h', u_1 h', \dots, u_r h', u_k \tilde{\alpha}_j h') $$

Using standard linear algebra and the definition of the norm of the fractional
matrix in Smith normal form $[A_r = t, \dots, A_1, 1]$. Then

$$ nI_{r+1} = (U\mathcal{A}V) \cdot (V^{-1}\mathcal{B}'U^{-1}) = \begin{pmatrix} t & 0 & \dots & 0 \\ 0 & A_{r-1} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \dots & \dots & 1 \end{pmatrix} V^{-1}\mathcal{B}'U^{-1} $$

This shows that $t|n$. Consider now a uniform positive dual family $\underline{\alpha}' = (\alpha'_0, \dots, \alpha'_r)$ to $\underline{a}$ in $L$ such that $\underline{a} \cdot \underline{\alpha}' = nI_{r+1}$. Write the coordinates of the $\alpha'_j$ in the basis $B_L$ as $\alpha'_j = \sum_{i=0}^{r+1} b'_{ij} e_i$. Then the matrix $\mathcal{B}' = (b'_{ij})_{1\le i,j \le r+1}$ satisfies $\mathcal{A} \cdot \mathcal{B}' = nI_{r+1}$ and $t$ divides $n$.

Consider now the minimal choice $\underline{\alpha}$ given above associated to the matrix $\mathcal{B} = (\prod_{i=1}^r A_i^{-1})\mathrm{com}(\mathcal{A})^T$ such that $\underline{a} \cdot \underline{\alpha} = tI_{r+1}$. The elementary divisors of $\mathcal{B}$ are given by the positive integers $[A_r, A_r/A_1, \dots, A_r/A_{r-1}, 1]$. The last elementary divisor is the $\mathrm{gcd}$ of the coordinates of the $\alpha_j$ in the basis $B_L$. Therefore, for each prime $p$ there exists an integer $j$ and a coefficient $b_{ij}$ not divisible by $p$. For any $n' \in \mathbb{Z}$, we get $\alpha_j - n'\gamma = \pm n'e_0 + \sum_{i=1}^{r+1} b_{ij}e_i \notin pL$. $\square$

From now on we will always fix $\underline{\alpha} = \underline{\alpha}(u_1, \dots, u_r; h)$ the minimal uniform positive dual family $\underline{\alpha}$ to $\underline{a}$ given above. Define $M = M(u_1, \dots, u_r; h) = \mathbb{Z}h' \oplus (\bigoplus_{j=0}^r \mathbb{Z}\alpha_j)$ in $L$ associated to this family. This sublattice has index $\mathrm{det}(\mathcal{B}) = t^{r+1}/s$ in $L$. Recalling the construction in section 3.1 and formula (11) for the geometric $G_{r,\underline{a}}$ function we may identify $F(\underline{\alpha})/\mathbb{Z}\gamma \simeq L/M$ so that the geometric $G_{r,\underline{a}}$ function is a product of $t^{r+1}/s$ ordinary elliptic $G_r$ functions. In the worst case scenario $t = s$ and $G_{r,\underline{a}}$ is a product of $t^r$ ordinary elliptic $G_r$ functions.

Let us now analyse $\underline{\alpha}$ more closely. It follows from the definition of the linear form $a$ in formula (17) and lemma 12 that for $j \neq k$, $\alpha_j$ must satisfy

$$ \det_{B_L}(u_0h', u_1h', \dots, u_rh', u_k\alpha_j) = 0 $$

with once again the convention that $u_0 = 1$, so that $\alpha_j/h' = \sum_{l=0}^r d_{jkl}u_k^{-1}u_l$ with rational coefficients $d_{jkl}$. This quotient depends rather loosely upon $h'$ and this leads us to investigate the case $h = 1$ (which is not an admissible vector). In what follows, we will write with a $\sim$ the counterpart of the quantities defined in the construction above for $h = 1$ and $\mathcal{O}_K$ instead of the lattice $L$. There is a positive basis $B_K = [1, \tilde{e}_1, \dots, \tilde{e}_{r+1}]$ of $\mathcal{O}_K$ such that $u_j = \sum_{k=0}^j \tilde{c}_{jk0}\tilde{e}_k$ with $\tilde{c}_{jj0} > 0$. Consider

$$ \tilde{\lambda}\tilde{a} = \left( \prod_{j=1}^{r} \tilde{c}_{jj0} \right) \tilde{a} = \det_{B_K}(1, u_1, \dots, u_r, \cdot) \quad (19) $$
ideal L we may rewrite this equality using det$_{B_K}$ instead of det$_{B_L}$ as

$$ \lambda au_k \tilde{\alpha}_j h' = \det_{B_L}(B_K) \det_{B_K}(h', u_1 h', \dots, u_r h', u_k \tilde{\alpha}_j h') $$

$$ \lambda au_k \tilde{\alpha}_j h' = \frac{1}{\mathcal{N}(L)} \det_{B_K}(h', u_1 h', \dots, u_r h', u_k \tilde{\alpha}_j h') $$

Then, using the definition of the norm $\mathcal{N}(h')$ of $h'$ we get

$$ \lambda au_k \tilde{\alpha}_j h' = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \det_{B_K}(1, u_1, \dots, u_k \tilde{\alpha}_j) $$

$$ \lambda au_k \tilde{\alpha}_j h' = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \tilde{\lambda} \tilde{a} u_k \tilde{\alpha}_j $$

We now use formula (18) to conclude that for $k \neq j$, $\lambda au_k \tilde{\alpha}_j h' = 0$ and for $k = j$:

$$ \lambda au_j \tilde{\alpha}_j h' = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \tilde{\lambda} \tilde{t} \in \mathbb{Z} $$

$$ au_j \tilde{\alpha}_j h' = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \frac{\tilde{\lambda}}{\lambda} \tilde{t} $$

The lattice L is a fractional ideal so that $\tilde{\alpha}h' \subset L$. The linear forms $au_k$ take integral values on L so the number $n = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \frac{\tilde{\lambda}}{\lambda} \tilde{t}$ is a non-zero integer. Write $\epsilon = \pm 1$ for the sign of $n$. Then $\epsilon \tilde{\alpha}h'$ is a uniform positive dual family to $\underline{a}$ in L such that $\underline{a} \cdot (\epsilon \tilde{\alpha}h') = |n|I_{r+1}$. Lemma 11 gives $n = \epsilon \mathit{lt}$ for a positive integer $l$ and

$$ \epsilon \lambda \mathit{lt} = \lambda n = \lambda au_j \tilde{\alpha}_j h' = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \tilde{\lambda} \tilde{t} $$

<figure><img src="image_2.png" /></figure>

We now give a precise statement on the relation between $\underline{a}$ and $\tilde{\alpha}$.

**Proposition 13:** Let $h$ be an admissible vector in L and consider a unit system $u_1, \dots, u_r$ for $O_{\mathfrak{f}}^{+,*}$. Let $\underline{a}$ and $\tilde{\alpha}$ be as defined in (16) and (18). Write $\epsilon$ for the sign in formula (19). Then for all $0 \le j \le r$, there exists an integer $m_j \in \mathbb{Z}$ such that $\epsilon l \alpha_j = \tilde{\alpha}_j h' + m_j h'$.

**Proof:**

Using lemma 2 for $\underline{a}$ and the two positive dual families $\underline{a}$, $\epsilon \tilde{\alpha}h'$ to $\underline{a}$ in L we get $\epsilon l \alpha_j = \tilde{\alpha}_j h' + m_j h'/n_j$ with $m_j/n_j$ a rational number in irreducible form. Write $p_j m_j + q_j n_j = 1$. Then

$$ p_j(\epsilon l \alpha_j - \tilde{\alpha}_j h') + q_j h' = h'/n_j \in L $$
and put $\tilde{a} = (\tilde{a}, \tilde{a}u_1, \dots, \tilde{a}u_r)$. Associate to $\tilde{a}$ a square matrix $\tilde{\mathcal{A}}$ of size $r+1$ as above for $\underline{a}$. Write as above the elementary divisors of this matrix $\tilde{\mathcal{A}}$ associated to $\tilde{a}$ as $[\tilde{A}_r, \dots, \tilde{A}_1, \tilde{A}_0 = 1]$ such that $\tilde{A}_i|\tilde{A}_{i+1}$. Then put $\tilde{s} = \prod_{i=1}^r \tilde{A}_i$, $\tilde{t} = \tilde{A}_r$ and construct as above $\tilde{\alpha}$ associated to the square matrix $\tilde{\mathcal{B}} = (\prod_{i=1}^r \tilde{A}_i^{-1})\mathrm{com}(\tilde{\mathcal{A}})^T$ such that $\tilde{a} \cdot \tilde{\alpha} = \tilde{t}I_{r+1}$. Recall that this means that for all $0 \le j \le r$:

$$ \tilde{a}u_j(\tilde{\alpha}_j) = \tilde{t}, \quad \tilde{a}u_k(\tilde{\alpha}_j) = 0, \forall k \neq j \qquad (20) $$

In the case $r = 1$, $\tilde{\lambda}$ is the content of the smallest ring $\mathcal{O}$ containing the unit $\varepsilon$ in the sense of [[2], Definition 14] and $\tilde{t}$ measures how far the $\mathbb{Z}$-module $\mathbb{Z} + \mathbb{Z}\varepsilon$ is from being a ring. Therefore, we will call $\tilde{\lambda}$ the content of the unit system $(u_1, \dots, u_r)$ and $\tilde{t}$ the overflow of this unit system. We insist that these quantities only depend on the unit system $(u_1, \dots, u_r)$. We relate this general construction to the one carried out for a particular admissible vector $h$ in $L$ as follows:

**Lemma 13:** The family $\epsilon\tilde{\alpha}h' = (\epsilon\tilde{\alpha}_0h', \dots, \epsilon\tilde{\alpha}_rh')$ is a uniform positive dual family to $\underline{a}$ in $L$ for some sign $\epsilon = \pm 1$. Therefore there exists a positive integer $l$ such that $\underline{a} \cdot (\epsilon\tilde{\alpha}h') = ltI_{r+1}$. The value of $l$ is given by the following equality which holds in $\mathbb{Z}$

$$ \epsilon\lambda lt = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \tilde{\lambda} \tilde{t} \qquad (21) $$

**Proof:**

We use formulae (17) relating $\underline{a}$ to the form $\det_{B_{\mathbb{K}}}$ and (19) relating $\tilde{a}$ to the form $\det_{B_L}$ to evaluate $\lambda au_k \tilde{\alpha}_j h'$.

$$ \lambda au_k \tilde{\alpha}_j h' = \det_{B_L}(h', u_1 h', \dots, u_r h', u_k \tilde{\alpha}_j h') $$

Using standard linear algebra and the definition of the norm of the fractional ideal $L$ we may rewrite this equality using $\det_{B_{\mathbb{K}}}$ instead of $\det_{B_L}$ as

$$ \lambda au_k \tilde{\alpha}_j h' = \det_{B_L}(B_{\mathbb{K}}) \det_{B_{\mathbb{K}}}(h', u_1 h', \dots, u_r h', u_k \tilde{\alpha}_j h') $$

$$ \lambda au_k \tilde{\alpha}_j h' = \frac{1}{\mathcal{N}(L)} \det_{B_{\mathbb{K}}}(h', u_1 h', \dots, u_r h', u_k \tilde{\alpha}_j h') $$

Then, using the definition of the norm $\mathcal{N}(h')$ of $h'$ we get

$$ \lambda au_k \tilde{\alpha}_j h' = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \det_{B_{\mathbb{K}}}(1, u_1, \dots, u_k \tilde{\alpha}_j) $$

$$ \lambda au_k \tilde{\alpha}_j h' = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \tilde{\lambda} \tilde{a} u_k \tilde{\alpha}_j $$
Because $h'$ is a primitive vector in $L$, we conclude that $n_j = 1$.

□

This proposition shows that the choice of $h$ only affects the value of $l$ and the values of the $m_j$ and not the main content of $\alpha/h'$. Indeed, the special value $G_{r,f,b,a}^{\pm}(u_1, \dots, u_r; h)$ we aim to compute is a product of $t^{r+1}/s$ ordinary elliptic $G_r$ functions with parameters $\alpha_0/h', \dots, \alpha_r/h'$, which we may write as $\epsilon(\tilde{\alpha}_0 + m_0)/l, \dots, \epsilon(\tilde{\alpha}_r + m_r)/l$. For instance in the cubic case, $\Gamma_{f,b,a}^{\pm}$ is a product of $t$ ordinary elliptic Gamma functions with parameters

$$ \tau = \pm \frac{\varepsilon + n_0}{l \tilde{\lambda}} = \frac{\tau_0}{l}, \sigma = \pm \frac{\varepsilon^{-1} + n_1}{l \tilde{\lambda}} = \frac{\sigma_0}{l} $$

for some integers $n_0, n_1$. Therefore, the integer $l$ can be thought of as a level and will be called the level of the value $G_{r,f,b,a}^{\pm}(u_1, \dots, u_r; h)$.

## 4.2 Computing the $G_r$ functions

### 4.2.1 Computing the ordinary elliptic $G_r$ functions

To compute the $G_r$ functions we make use of [[13], Proposition 3.6] which we write as

$$ G_r(z, \underline{\tau}) = \exp \left( - \sum_{j \ge 1} \frac{1}{j} \frac{|q|^j x^{-j} + (-1)^r x^j}{\prod_{k=0}^r (1 - q_k^j)} \right) \quad (20) $$

This formula is only valid for $\underline{\tau} \in \mathbb{H}^{r+1}$ and $0 < \mathfrak{F}(z) < \sum_{k=0}^r \mathfrak{F}(\tau_k)$. We call this domain the center strip. We now only need to make sure that we can use the properties of the $G_r$ functions to reach this domain. The first step in doing so, starting from $\underline{\tau} \in (\mathbb{C} - \mathbb{R})^{r+1}$ is to use the property

$$ G_r(z, \tau_0, \dots, \tau_{k-1}, -\tau_k, \tau_{k+1}, \dots, \tau_r) = G_r(z + \tau_k, \underline{\tau})^{-1} $$

to bring back all arguments in the upper half-plane. Then, to bring back $\mathfrak{F}(z)$ in the center strip, use recursively the property

$$ G_r(z + \tau_k, \underline{\tau}) = G_{r-1}(z, \underline{\tau} - [k]) G_r(z, \underline{\tau}) $$

Doing so will require the computation of lower degree functions, that is until we reach the case $r = 0$. Lastly, to compute $G_0 = \theta$ one may use Jacobi's triple product formula and compute directly $\theta$ using this fast-converging sum:

$$ \theta(z, \tau) = \frac{q^{1/24}}{\eta(\tau)} \sum_{n \in \mathbb{Z}} x^n (-1)^n q^{n(n-1)/2} $$
We now use formula (20) to conclude that for $k \neq j$, $\lambda au_k \tilde{\alpha}_j h' = 0$ and for $k = j$:

$$ \lambda au_j \tilde{\alpha}_j h' = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \tilde{\lambda} \tilde{t} \in \mathbb{Z} $$

$$ au_j \tilde{\alpha}_j h' = \frac{\mathcal{N}(h') \tilde{\lambda}}{\mathcal{N}(L) \tilde{\lambda}} \tilde{t} $$

The lattice $L$ is a fractional ideal so that $\underline{\alpha}h' \subset L$. The linear forms $au_k$ take integral values on $L$ so the number $n = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \tilde{\lambda} \tilde{t}$ is a non-zero integer. Write $\epsilon = \pm 1$ for the sign of $n$. Then $\epsilon \underline{\alpha}h'$ is a uniform positive dual family to $\underline{a}$ in $L$ such that $\underline{a} \cdot (\epsilon \underline{\alpha}h') = |n|I_{r+1}$. Lemma 12 gives $n = \epsilon lt$ for a positive integer $l$ and

$$ \epsilon \lambda lt = \lambda n = \lambda au_j \tilde{\alpha}_j h' = \frac{\mathcal{N}(h') \tilde{\lambda} \tilde{t}}{\mathcal{N}(L)} $$

□

We now give a precise statement on the relation between $\underline{a}$ and $\tilde{\underline{a}}$.

**Proposition 14:** Let $h$ be an admissible vector in $L$ and consider a unit system $u_1, \dots, u_r$ for $O_{\mathfrak{f}}^{+,\times}$. Let $\underline{a}$ and $\tilde{\underline{a}}$ be as defined in (18) and (20). Write $\epsilon$ for the sign in formula (21). Then for all $0 \le j \le r$, there exists an integer $m_j \in \mathbb{Z}$ such that $\epsilon l \alpha_j = \tilde{\alpha}_j h' + m_j h'$.

**Proof :**

Using lemma 2 for $\underline{a}$ and the two positive dual families $\underline{a}$, $\epsilon \tilde{\underline{a}}h'$ to $\underline{a}$ in $L$ we get $\epsilon l \alpha_j = \tilde{\alpha}_j h' + m_j h'/n_j$ with $m_j/n_j$ a rational number in irreducible form. Write $p_j m_j + q_j n_j = 1$. Then

$$ p_j(\epsilon l \alpha_j - \tilde{\alpha}_j h') + q_j h' = h'/n_j \in L $$

Because $h'$ is a primitive vector in $L$, we conclude that $n_j = 1$. □

This proposition shows that the choice of $h$ only affects the value of $l$ and the values of the $m_j$ and not the main content of $\underline{a}/h'$. Indeed, the special value $G_{r,f,b,a}^{\pm}(u_1, \dots, u_r; h)$ we aim to compute is a product of $t^{r+1}/s$ ordinary elliptic $G_r$ functions with parameters $\alpha_0/h', \dots, \alpha_r/h'$, which we may write as $\epsilon(\tilde{\alpha}_0 + m_0)/l, \dots, \epsilon(\tilde{\alpha}_r + m_r)/l$. For instance in the cubic case, $\Gamma_{\mathfrak{f},b,a}^{\pm}$ is a product of $t$ ordinary elliptic Gamma functions with parameters

$$ \tau = \pm \frac{\varepsilon + n_0}{l \tilde{\lambda}} = \frac{\tau_0}{l}, \sigma = \pm \frac{\varepsilon^{-1} + n_1}{l \tilde{\lambda}} = \frac{\sigma_0}{l} $$

for some integers $n_0, n_1$. Therefore, the integer $l$ can be thought of as a level and will be called the level of the value $G_{r,f,b,a}^{\pm}(u_1, \dots, u_r; h)$.
where $\eta(\tau)$ is the Dedeking $\eta$ function. Let us say a word about the complexity of such computations. Fix $\tau_0, \dots, \tau_r$ in the upper half-plane and suppose we want to compute the level $l$ function $G_r(z, \tau/l)$ with precision $\delta > 0$. Then the number of terms we need to compute in the sum (20) is $O(l|\log(\delta)|)$ where the constant explicitly depends on $\tau$ and $z$.

### 4.2.2 Computing the arithmetic $G_{r,f,b,a}^{\pm}(u_1, \dots, u_r; h)$ functions

We analyse the definition 8 of $G_{r,f,b,a}^{\pm}(u_1, \dots, u_r; h)$ using the geometric setup in section 4.1. For ease of presentation we treat the case of $G_{r,f,b,a}^{+}(u_1, \dots, u_r; h)$ and we insist that the computation for the other choice of sign is almost identical. Recall definition 8 for the arithmetic function:

$$G_{r,f,b,a}^{+}(u_1, \dots, u_r; h, \sigma_{\mathbb{C}}) = \frac{G_{r,(a,au_1,\dots,au_r)}\left(\frac{h}{q}, \sigma_{\mathbb{C}}, \mathrm{Hom}_{\mathbb{Z}}(\mathfrak{f}\mathfrak{b}^{-1}, \mathbb{Z})\right)^N}{G_{r,(a,au_1,\dots,au_r)}\left(\frac{h}{q}, \sigma_{\mathbb{C}}, \mathrm{Hom}_{\mathbb{Z}}(\mathfrak{f}(\mathfrak{b}\mathfrak{a})^{-1}, \mathbb{Z})\right)}$$

We drop once again the $\sigma_{\mathbb{C}}$ and we write using formula (11)

$$G_{r,f,b,a}^{+}(u_1, \dots, u_r; h) = \prod_{\delta \in F(\alpha)/\mathbb{Z}\gamma} \frac{G_r\left(\frac{h+\delta}{q\gamma}, \frac{\alpha_0}{\gamma}, \dots, \frac{\alpha_r}{\gamma}\right)^N}{G_r\left(\frac{Nh+N\delta}{q\gamma}, \frac{N\alpha_0}{\gamma}, \dots, \frac{N\alpha_r}{\gamma}\right)}$$

where $\alpha$ and $\gamma$ are defined in section 4.1. We use once again the fact that $F(\alpha)/\mathbb{Z}\gamma \simeq L/M$ where $M = M(u_1, \dots, u_r; h)$ is defined in section 4.1 to write

$$G_{r,f,b,a}^{+}(u_1, \dots, u_r; h) = \prod_{\delta \in L/M} \frac{G_r\left(\frac{h+\delta}{q\gamma}, \frac{\alpha_0}{\gamma}, \dots, \frac{\alpha_r}{\gamma}\right)^N}{G_r\left(\frac{Nh+N\delta}{q\gamma}, \frac{N\alpha_0}{\gamma}, \dots, \frac{N\alpha_r}{\gamma}\right)}$$

$$G_{r,f,b,a}^{+}(u_1, \dots, u_r; h) = \prod_{\delta \in L/M} \frac{G_r\left(\frac{h+\delta}{q\gamma}, \epsilon\frac{\tilde{\alpha}_0+m_0}{l}, \dots, \epsilon\frac{\tilde{\alpha}_r+m_r}{l}\right)^N}{G_r\left(\frac{Nh+N\delta}{q\gamma}, \epsilon\frac{N\tilde{\alpha}_0+Nm_0}{l}, \dots, \epsilon\frac{N\tilde{\alpha}_r+Nm_r}{l}\right)} \quad (21)$$

where the last expression is obtained using Proposition 13. This special value is a product of $t^{r+1}/s = \#|L/M|$ ordinary elliptic $G_r$ functions. In the worst case scenario we get $t = s$ and $t^{r+1}/s = t^r$. This means that the computation time of this value is $O(l|\log \delta|t^r)$ with a constant that depends on $u_1, \dots, u_r$. The parameters $l$ and $t$ which depend on $h$ for fixed $u_1, \dots, u_r$ are therefore crucial in our computations. For instance, if we want to compute this special value with a precision of 1000 digits (i.e. $|\log \delta| = 1000$) where $r = 2, l = 15$ and $t = 100$, then the computation requires the computation of $150.10^6$ terms appearing in the sum (20). With a personal computer it is generally
## 4.2 Computing the $G_r$ functions

### 4.2.1 Computing the ordinary elliptic $G_r$ functions

To compute the $G_r$ functions we make use of [[16], Proposition 3.6] which we write as

$$G_r(z, \underline{\tau}) = \exp \left( - \sum_{j \ge 1} \frac{1}{j} \frac{q_0^j \dots q_r^j x^{-j} + (-1)^r x^j}{\prod_{k=0}^r (1 - q_k^j)} \right) \quad (22)$$

This formula is only valid for $\underline{\tau} \in \mathbb{H}^{r+1}$ and $0 < \mathfrak{F}(z) < \sum_{k=0}^r \mathfrak{F}(\tau_k)$. We call this domain the center strip. We now only need to make sure that we can use the properties of the $G_r$ functions to reach this domain. The first step in doing so, starting from $\underline{\tau} \in (\mathbb{C} - \mathbb{R})^{r+1}$ is to use the property

$$G_r(z, \tau_0, \dots, \tau_{k-1}, -\tau_k, \tau_{k+1}, \dots, \tau_r) = G_r(z + \tau_k, \underline{\tau})^{-1}$$

to bring back all arguments in the upper half-plane. Then, to bring back $\mathfrak{F}(z)$ in the center strip, use recursively the property

$$G_r(z + \tau_k, \underline{\tau}) = G_{r-1}(z, \underline{\tau}^{-}[k])G_r(z, \underline{\tau})$$

Doing so will require the computation of lower degree functions, that is until we reach the case $r = 0$. Lastly, to compute $G_0 = \theta$ one may use Jacobi's triple product formula and compute directly $\theta$ using this fast-converging sum:

$$\theta(z, \tau) = \frac{q^{1/24}}{\eta(\tau)} \sum_{n \in \mathbb{Z}} x^n (-1)^n q^{n(n-1)/2}$$

where $\eta(\tau)$ is the Dedekind $\eta$ function. Let us say a word about the complexity of such computations. Fix $\tau_0, \dots, \tau_r$ in the upper half-plane and suppose we want to compute the level $l$ function $G_r(z, \underline{\tau}/l)$ with precision $\delta > 0$. Then the number of terms we need to compute in the sum (22) is $O(l|\log(\delta)|)$ where the constant explicitly depends on $\underline{\tau}$ and $z$.

The above algorithm will be useful to compute the value $G_{r,f,b,a}^{\mu}(u_1, \dots, u_r; h)$ when for all $0 \le j \le r$, $\sigma_{\mathbb{C}}(\tilde{\alpha}_j) \in \mathbb{C} - \mathbb{R}$. This is always the case except possibly when $\mathbb{K}$ is a field of even degree $d = 2d'$ and contains a real subfield $\mathbb{F}$ of degree $\le d'$. In this case, because the family $1, \tilde{\alpha}_0, \dots, \tilde{\alpha}_r$ is free over $\mathbb{Q}$, at most $d' - 1$ elements amongst the $\tilde{\alpha}_j$ may belong to $\mathbb{F}$. We will avoid this case for now under the (*) conditions below and discuss one example of such a case in section 5.2.4.
an overwhelming computation. In the next section, we describe how we can choose the vector $h$ to obtain efficient computations. Namely, because the parameter $t$ has the biggest impact on computation time, we will show that under some assumptions we may choose a vector $h$ such that $t = 1$ at the cost of a moderate elevation of the level $l$.

### 4.3 The choice of vectors $(h_\rho)_{\rho \in \mathfrak{S}_r}$

In view of the above estimation on the computation time for the special value $G_{r,f,b,a}^{\pm}([\varepsilon_{\rho(1)}|\dots|\varepsilon_{\rho(r)}];h_\rho)$ for a permutation $\rho \in \mathfrak{S}_r$ we must be careful in choosing the vector $h_\rho$ to perform efficient computations. Also, because we want to build a product of these values for $\rho \in \mathfrak{S}_r$, we must find a way to unify our choices for each permutation $\rho$. To this extent, we will fix the unit system $\varepsilon_1, \dots, \varepsilon_r$ and the permutation $\rho$. Write once again $(u_1, \dots, u_r) = [\varepsilon_{\rho(1)}, \dots, \varepsilon_{\rho(r)}]$. Our aim is to reduce at all costs the value of $t = t(u_1, \dots, u_r; h)$ by choosing $h$ carefully. Formula (19)

$$ \pm \lambda t = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \tilde{\lambda} \tilde{t} $$

will reveal useful to understand how to minimize the value of $t$. The unit system $(u_1, \dots, u_r)$ is fixed and we therefore have to deal with the fixed constants $\tilde{\lambda}, \tilde{t}$ defined in section 4.1. In what follows, a goal will be to understand how the prime factors of $\tilde{\lambda}$ and $\tilde{t}$ transfer to the left-hand side of this equation. From our computations it seems that the value of $\tilde{\lambda}$ is a hard limit for the value of $t$, which means that for all admissible vector $h$, $\tilde{\lambda}$ divides $t$. We also believe that this value of $\tilde{\lambda}$ is exactly the minimal achievable value for $t$. We are far from being able to prove this statement, but it will be proven in the case $\tilde{\lambda} = 1$ under the conditions $(\ast)$ and $(\ast\ast)$ below. The rest of this section contains the study of the $p$-adic valuations of the quantities appearing in formula (19) first in the cases where $p$ divides $\tilde{t}$ and then in the case where $p$ does not divide $\tilde{t}$. Under some conditions we show that in the first case the $p$-part of $h$ (as in definition 14 below) is best chosen as specific ideals which we call *target ideals* (see definition 18 and lemma 19 below). In the second case, it will be shown how a vector $h$ may have a non-trivial $p$-part while not affecting the value of $t$ using *helper ideals* (see lemma 21 below). In the rest of this section, we will discuss a uniform choice for the vectors $h_\rho, \rho \in \mathfrak{S}_r$ under the following conditions, referred to as the $(\ast)$ conditions: suppose that $N(f) = q = \ell^k$ is a power of a prime number $\ell$ and that the choice of fundamental units $\varepsilon_1, \dots, \varepsilon_r$ for $O_f^{+,*}$ is such that for all $\rho \in \mathfrak{S}_r$:

$$ \bullet \quad \tilde{\lambda}_{\rho} = 1 $$
### 4.2.2 Computing the arithmetic $G_{r,f,b,a}^{\mu}(u_1, \dots, u_r; h)$ functions

We analyse Definition 8 for $G_{r,f,b,a}^{\mu}(u_1, \dots, u_r; h)$ using the geometric setup in section 4.1 for the choice of sign $\mu = \pm 1$. Recall Definition 8 for the arithmetic function:

$$G_{r,f,b,a}^{\mu}(u_1, \dots, u_r; h, \sigma_{\mathbb{C}}) = \frac{G_{r,(\mu a, \mu a u_1, \dots, \mu a u_r)}\left(\frac{h}{q}, \sigma_{\mathbb{C}}, \mathrm{Hom}_{\mathbb{Z}}(\mathfrak{f}\mathfrak{b}^{-1}, \mathbb{Z})\right)^N}{G_{r,(\mu a, \mu a u_1, \dots, \mu a u_r)}\left(\frac{h}{q}, \sigma_{\mathbb{C}}, \mathrm{Hom}_{\mathbb{Z}}(\mathfrak{f}(\mathfrak{b}a)^{-1}, \mathbb{Z})\right)}$$

We drop once again the $\sigma_{\mathbb{C}}$ and we write using formula (11):

$$G_{r,f,b,a}^{\mu}(u_1, \dots, u_r; h) = \prod_{\delta \in F(\underline{\alpha})/\mathbb{Z}\gamma} \frac{G_r\left(\frac{h+q\mu\delta}{q\mu^{r+1}\gamma}, \frac{\mu\alpha_0}{\mu^{r+1}\gamma}, \dots, \frac{\mu\alpha_r}{\mu^{r+1}\gamma}\right)^N}{G_r\left(\frac{N(h+q\mu\delta)}{q\mu^{r+1}\gamma}, \frac{N\mu\alpha_0}{\mu^{r+1}\gamma}, \dots, \frac{N\mu\alpha_r}{\mu^{r+1}\gamma}\right)}$$

where $\underline{\alpha}$ and $\gamma$ are defined in section 4.1. We use formula (8) as well as the fact that $F(\underline{\alpha})/\mathbb{Z}\gamma \simeq L/M$ where $M = M(u_1, \dots, u_r; h)$ is defined in section 4.1 to write

$$G_{r,f,b,a}^{\mu}(u_1, \dots, u_r; h)^{\mu r} = \prod_{\delta \in L/M} \frac{G_r\left(\frac{\mu h + q \delta}{q \gamma}, \frac{\alpha_0}{\gamma}, \dots, \frac{\alpha_r}{\gamma}\right)^N}{G_r\left(\frac{N(\mu h + q \delta)}{q \gamma}, \frac{N \alpha_0}{\gamma}, \dots, \frac{N \alpha_r}{\gamma}\right)}$$

$$G_{r,f,b,a}^{\mu}(u_1, \dots, u_r; h)^{\mu r} = \prod_{\delta \in L/M} \frac{G_r\left(\frac{\mu h + q \delta}{q \gamma}, \epsilon \frac{\tilde{\alpha}_0 + m_0}{l}, \dots, \epsilon \frac{\tilde{\alpha}_r + m_r}{l}\right)^N}{G_r\left(\frac{N(\mu h + q \delta)}{q \gamma}, \epsilon \frac{N(\tilde{\alpha}_0 + m_0)}{l}, \dots, \epsilon \frac{N(\tilde{\alpha}_r + m_r)}{l}\right)} \quad (23)$$

where the last expression is obtained using Proposition 14. This special value is a product of $t^{r+1}/s = \#|L/M|$ ordinary elliptic $G_r$ functions. In the worst case scenario we get $t = s$ and $t^{r+1}/s = t^r$. This means that the computation time of this value is $O(l|\log \delta|t^r)$ with a constant that depends on $u_1, \dots, u_r$. The parameters $l$ and $t$ which depend on $h$ for fixed $u_1, \dots, u_r$ are therefore crucial in our computations. For instance, if we want to compute this special value with a precision of 1000 digits (i.e. $|\log \delta| = 1000$) where $r = 2, l = 15$ and $t = 100$, then the computation requires the computation of $150.10^6$ terms appearing in the sum (22). With a personal computer it is generally an overwhelming computation. In the next section, we describe how we can choose the vector $h$ to obtain efficient computations. Namely, because the parameter $t$ has the biggest impact on computation time, we will show that under some assumptions we may choose a vector $h$ such that $t = 1$ at the cost of a moderate elevation of the level $l$.
• $\tilde{t}_{\rho}$ and $qNN(\mathfrak{b})$ are relatively coprime. (*)

• $\tilde{s}_{\rho} = \tilde{t}_{\rho} \neq 0$

• for all $0 \le j \le r$, $\tilde{\alpha}_{\rho,j}$ is not a totally real element of $\mathbb{K}$

where the subscript $\rho$ indicates that the quantities are related to the unit system $[\varepsilon_{\rho(1)}|\dots|\varepsilon_{\rho(r)}]$. The last condition is relevant only if $\mathbb{K}$ contains a totally real subfield and it is there to ensure that we compute $G_r$ functions with parameters outside of $\mathbb{R}$. Note that the value of $\tilde{t}_{\rho}$ does not depend on $\mathfrak{b}$ or $\mathfrak{a}$ which means that in the second line, we only require $\tilde{t}_{\rho}$ to be coprime to $q$ and we decide to choose $\mathfrak{b}$ and $\mathfrak{a}$ conveniently afterwards, so that this part is not restrictive. We will review the other assumptions when we discuss (**) below.

### 4.3.1 Target ideals

While choosing the vector $h = h_{\rho}$, we must be very careful of the value of the overflow $\tilde{t} = \tilde{t}_{\rho}$. Indeed, for most admissible vectors $h$, the constant $\tilde{t}$ depending only on $u_1, \dots, u_r$ will divide $t = t(u_1, \dots, u_r; h)$ and the computations will require $t^r$ evaluations of ordinary $G_r$ functions, which can become quickly overwhelming. We now introduce the notion of target ideal to understand how to choose $h$ such that $t = 1$. Explicitly, we explain that there is a tradeoff with an increase of the level $l$ as we diminish the value of $t$. In the rest of this section we will make extensive use of the following definitions:

**Definition 14:** Let $x \in \mathbb{K}$ and $I$ be an ideal such that $N(I)$ is a power of the prime number $p$. We say that the $p$-part of $x \in \mathbb{K}$ is the ideal $I$ if for all prime ideal $\mathfrak{P}$ above $p$, $v_{\mathfrak{P}}(x) = v_{\mathfrak{P}}(I)$.

**Definition 15:** Let $I$ be a non zero integral ideal in $O_{\mathbb{K}}$ such that $I \cap \mathbb{Z} = n\mathbb{Z}$. Then, for all $\alpha \in O_{\mathbb{K}}$ congruent to an integer modulo $I$ we write $\text{rmd}(\alpha, I)$ for the unique integer $0 \le \text{rmd}(\alpha, I) < |n|$ such that $\alpha - \text{rmd}(\alpha, I) \in I$.

The number $\text{rmd}(\alpha, I)$ is naturally called the remainder of $\alpha \mod I$. Note that if $I$ is a product of degree one primes above different prime numbers, then all elements $\alpha \in O_{\mathbb{K}}$ are congruent to integers mod $I$.

**Definition 16:** Let $I$ be a non zero integral ideal in $O_{\mathbb{K}}$. Then, there is a unique primitive ideal $I^c$ and a unique integer $n > 0$ such that $II^c = nO_{\mathbb{K}}$. We call $I^c$ the complementary ideal of $I$.
## 4.3 The choice of vectors $(h_\rho)_{\rho \in \mathfrak{S}_r}$

In view of the above estimation on the computation time for the special value $G_{r,f,b,a}^{\pm}([\varepsilon_{\rho(1)}|\dots|\varepsilon_{\rho(r)}]; h_\rho)$ for a permutation $\rho \in \mathfrak{S}_r$ we must be careful in choosing the vector $h_\rho$ to perform efficient computations. Also, because we want to build a product of these values for $\rho \in \mathfrak{S}_r$, we must find a way to unify our choices for each permutation $\rho$. To this end, we will fix the unit system $\varepsilon_1, \dots, \varepsilon_r$ and the permutation $\rho$. Write once again $(u_1, \dots, u_r) = [\varepsilon_{\rho(1)}|\dots|\varepsilon_{\rho(r)}] = (\varepsilon_{\rho(1)}, \dots, \prod_{j=1}^r \varepsilon_{\rho(j)})$. Our aim is to reduce at all costs the value of $t = t(u_1, \dots, u_r; h)$ by choosing $h$ carefully. Formula (21)

$$ \pm \lambda t = \frac{\mathcal{N}(h')}{\mathcal{N}(L)} \tilde{\lambda} \tilde{t} $$

will be fundamental to understand how to minimize the value of $t$. The unit system $(u_1, \dots, u_r)$ is fixed and we therefore have to deal with the fixed constants $\tilde{\lambda}, \tilde{t}$ defined in section 4.1. In what follows, a goal will be to understand how the prime factors of $\tilde{\lambda}$ and $\tilde{t}$ transfer to the left-hand side of this equation. From our computations it seems that the value of $\tilde{\lambda}$ is a hard limit for the value of $t$, which means that for all admissible vector $h$, $\tilde{\lambda}$ divides $t$. We also believe that this value of $\tilde{\lambda}$ is exactly the minimal achievable value for $t$. We are far from being able to prove this statement, but it will be proven in the case $\tilde{\lambda} = 1$ under the conditions (*) and (**) below. The rest of this section contains the study of the $p$-adic valuations of the quantities appearing in formula (21) first in the cases where $p$ divides $\tilde{t}$ and then in the case where $p$ does not divide $\tilde{t}$. Under some conditions we show that in the first case the $p$-part of $h$ (as in definition 15 below) is best chosen as specific ideals which we call *target ideals* (see definition 18 and lemma 19 below). In the second case, it will be shown how a vector $h$ may have a non-trivial $p$-part while not affecting the value of $t$ using *helper ideals* (see lemma 21 below). In the rest of this section, we will discuss a uniform choice for the vectors $h_\rho, \rho \in \mathfrak{S}_r$ under the following conditions, referred to as the (*) conditions: suppose that the ideal $f = \mathfrak{l}^k$ is a power of a degree one prime ideal $\mathfrak{l}$ above a prime number $\ell$ and that the choice of fundamental units $\varepsilon_1, \dots, \varepsilon_r$ for $\mathcal{O}_f^{+,*}$ is such that for all $\rho \in \mathfrak{S}_r$:

* $\tilde{\lambda}_\rho = 1$

* $\tilde{t}_\rho$ and $qNN(\mathfrak{b})$ are relatively coprime (*)

* for all $0 \le j \le r$, $\sigma_{\mathbb{C}}(\tilde{\alpha}_{\rho,j}) \notin \mathbb{R}$

where the subscript $\rho$ indicates that the quantities are related to the unit system $[\varepsilon_{\rho(1)}|\dots|\varepsilon_{\rho(r)}]$. Note that the value of $\tilde{t}_\rho$ does not depend on $\mathfrak{b}$ or $\mathfrak{a}$
Before we define the notion of target ideal, we prove a lemma which we will need to control the p-adic valuation of the integer $\lambda$ in the construction of section 4.1.

**Lemma 17:** Take $h = mh'$ an admissible element in $L$. Suppose that there is a linear combination

$$ \left( u_j + \sum_{i=0}^{j-1} n_i u_i \right) h' \in p^k L $$

where $j \ge 1$ and the coefficients $n_i$ are integers. Then $p^k$ divides the integer $c_{jj0}$ and therefore $p^k$ divides the integer $\lambda$.

**Proof:**

Recall that the basis $B_L = (e_0, \dots, e_{r+1})$ defined at the beginning of section 4.1 is such that $e_0 = h'$ and for all $1 \le i \le r$, $u_i h' = \sum_{k=0}^i c_{ik0} e_k$ where the $c_{ik0}$ are integers and $c_{ii0} > 0$. This means that

$$ \left( u_j + \sum_{i=0}^{j-1} n_i u_i \right) h' = c_{jj0} e_j + \left( \sum_{k=0}^{j-1} c_{jk0} e_k \right) + \sum_{i=0}^{j-1} \left( \sum_{k=0}^{i} n_i c_{ik0} e_k \right) \in p^k L $$

so the coordinate of this element on the vector $e_j$ is exactly $c_{jj0}$ and $p^k$ divides $c_{jj0}$. Recalling that $\lambda = \prod_{j=1}^r c_{jj0}$, we get $p^k$ divides $\lambda$. $\square$

<figure><img src="image_3.png" /></figure>

We now define the notion of target ideal to deal with the p-adic valuation of the overflow $\tilde{t}$.

**Definition 18:** Let $I$ be a non zero primitive integral ideal of $O_K$ such that $II^c = p^{k_0}O_K$. We say that $I$ is a target ideal associated to the unit system $u_0 = 1, u_1, \dots, u_r$ if the two following conditions are satisfied:

* there is a linear combination $u_r + \sum_{j=0}^{r-1} n_j u_j \in I^c$ with $n_j \in \mathbb{Z}$

* for all $0 \le j \le r$, $\tilde{\alpha}_j$ is congruent to an integer $n'_j$ modulo $I^c$

Note that if the first condition is satisfied or if the second condition is satisfied for some $j$ when $I$ is a degree one prime then necessarily $p|\tilde{t}$. Indeed, in the first case, $\tilde{\alpha}_r - \text{rmd}(\tilde{\alpha}_r, I) \in I$ so that $(u_r + \sum_{j=0}^{r-1} n_j u_j)(\tilde{\alpha}_r - \text{rmd}(\tilde{\alpha}_r, I)) \in pO_K$ and

$$ \tilde{t} = \tilde{a} u_r \tilde{\alpha}_r = \tilde{a} \left( u_r + \sum_{j=0}^{r-1} n_j u_j \right) (\tilde{\alpha}_r - \text{rmd}(\tilde{\alpha}_r, I)) \in p\mathbb{Z} $$
which means that in the second line, we only require $\tilde{t}_{\rho}$ to be coprime to $q$ and we decide to choose **b** and **a** conveniently afterwards, so that this part is not restrictive. The last assumption is only relevant if $\mathbb{K}$ is of even degree $2d' = d$ and contains a real subfield $\mathbb{F}$ of degree $\le d'$, in which case at most $d' - 1$ of the $\tilde{\alpha}_{\rho,j}$ may belong to $\mathbb{F}$. This particular situation is discussed in section 5.2.4. We will review the other assumptions when we discuss (**) below.

### 4.3.1 Target ideals

While choosing the vector $h = h_{\rho}$, we must be very careful of the value of the overflow $\tilde{t} = \tilde{t}_{\rho}$. Indeed, for most admissible vectors $h$, the constant $\tilde{t}$ depending only on $u_1, \dots, u_r$ will divide $t = t(u_1, \dots, u_r; h)$ and the computations will require $t^r$ evaluations of ordinary $G_r$ functions, which can become quickly overwhelming. We now introduce the notion of target ideal to understand how to choose $h$ such that $t = 1$. Explicitly, we explain that there is a tradeoff with an increase of the level $l$ as we diminish the value of $t$. In the rest of this section we will make extensive use of the following definitions:

**Definition 15:** Let $x \in \mathbb{K}$ and $I$ be an ideal such that $\mathcal{N}(I)$ is a power of the prime number $p$. We say that the $p$-part of $x \in \mathbb{K}$ is the ideal $I$ if for all prime ideal $\mathfrak{P}$ above $p$, $v_{\mathfrak{P}}(x) = v_{\mathfrak{P}}(I)$.

**Definition 16:** Let $I$ be a non zero integral ideal in $\mathcal{O}_{\mathbb{K}}$ such that $I \cap \mathbb{Z} = n\mathbb{Z}$. Then we define the complementary ideal $I^c$ of $I$ to be the integral ideal $n\mathcal{O}_{\mathbb{K}}/I$ so that $II^c = n\mathcal{O}_{\mathbb{K}}$. We call a non zero integral ideal primitive if $I$ is not contained in any ideal of the form $m\mathcal{O}_{\mathbb{K}}$ with $m \in \mathbb{Z}, m \ge 2$ or equivalently if $(I^c)^c = I$.

Before we define the notion of target ideal, we prove a lemma which we will need to control the $p$-adic valuation of the integer $\lambda$ in the construction of section 4.1.

**Lemma 17:** Take $h = mh'$ an admissible element in $L$. Suppose that there is a linear combination

$$ \left( u_j + \sum_{i=0}^{j-1} n_i u_i \right) h' \in p^k L $$

where $j \ge 1$ and the coefficients $n_i$ are integers. Then $p^k$ divides the integer $c_{jj0}$ and therefore $p^k$ divides the integer $\lambda$.
In the second case, $u_j - \text{rmd}(u_j, I) \in I$, so that $(u_j - \text{rmd}(u_j, I))(\tilde{\alpha}_j - n'_j) \in p\mathcal{O}_K$ and $\tilde{a}(u_j - \text{rmd}(u_j, I))(\tilde{\alpha}_j - n'_j) = \tilde{a}u_j\tilde{\alpha}_j = \tilde{t} \in p\mathbb{Z}$. The importance of target ideals is expressed by the following lemma.

**Lemma 19:** Suppose (*). Let $p$ be a prime number such that $v_p(\tilde{t}) > 0$ and suppose that $I$ is a target ideal with $II^c = p^{k_0}\mathcal{O}_K$ where $k_0 \ge v_p(\tilde{t})$ for the unit system $u_0, \dots, u_r$. Suppose further that $\mathcal{N}(I) = p^{2k_0-v_p(\tilde{t})}$. Take $h$ an admissible vector with $p$-part exactly $I$. Then $v_p(\lambda) = v_p(l) = k_0$ and $v_p(t) = 0$.

**Proof:**

The ideal $I$ is a target ideal therefore there are integers $n_0, \dots, n_{r-1}$ such that $(u_r + \sum_{j=0}^{r-1} n_j u_j)h' \in I^c I = p^{k_0}\mathcal{O}_K$. Using lemma 17 we obtain that $p^{k_0}$ divides $c_{rr0}$ and so $p^{k_0}$ divides $\lambda$. On the other hand, from lemma 11 there is an index $j$ such that $p$ does not divide $\alpha_j + mh'$ for all $m \in \mathbb{Z}$ and from Proposition 13 there is an integer $m_j$ such that $\ell l\alpha_j = \tilde{\alpha}_j h' + m_j h'$ for some sign $\epsilon = \pm 1$. The ideal $I^c$ is a target ideal, therefore there is an integer $n'_j$ such that $\tilde{\alpha}_j - n'_j \in I^c$. This implies that $\ell l\alpha_j - m_j h' - n'_j h' = (\tilde{\alpha}_j - n'_j)h' \in p^{k_0}\mathcal{O}_K$. Suppose for now that $p$ does not divide $l$. Then $\exists m \in \mathbb{Z}$ such that $\ell lm - m_j - n'_j \in p\mathbb{Z}$. Therefore $\ell l(\alpha_j - mh') = (\tilde{\alpha}_j - n'_j)h' + (m_j + n'_j - \ell lm)h' \in p\mathcal{O}_K$ but this is a contradiction. Therefore $p$ must divide the integer $l$. We then prove by induction that $p^2, \dots, p^{k_0}$ divide $l$. Indeed, if $p^k|l$ with $k < k_0$ then $\ell l\alpha_j - (m_j + n'_j)h' \in p^{k_0}\mathcal{O}_K$ which means that $m_j + n'_j \in p^k\mathbb{Z}$. Suppose that $l \notin p^{k+1}\mathbb{Z}$. Then there is an integer $m \in \mathbb{Z}$ such that $\ell lm - m_j - n'_j \in p^{k+1}\mathbb{Z}$ and then $\ell l(\alpha_j - mh') \in p^{k+1}\mathcal{O}_K$ which yields again a contradiction. Now, using formula (19) one finds

$$v_p(\lambda) + v_p(l) + v_p(t) = v_p(\mathcal{N}(h')) - v_p(\mathcal{N}(L)) + v_p(\tilde{\lambda}) + v_p(\tilde{t})$$

$$v_p(\lambda) + v_p(l) + v_p(t) = (2k_0 - v_p(\tilde{t})) - 0 + 0 + v_p(\tilde{t}), \quad v_p(\lambda) \ge k_0, v_p(l) \ge k_0$$

And this leads to $v_p(\lambda) = k_0$, $v_p(l) = k_0$ and $v_p(t) = 0$. Note once again the most important case where $I$ is a degree one prime and $v_p(\tilde{t}) = 1$. Then we have $v_p(\tilde{\lambda}) = 1$, $v_p(l) = 1$ and $v_p(t) = 0$.

□

This allows us to strengthen the conditions under which we might understand special values of arithmetic $G_r$ functions. We add to the conditions (*) the following conditions (**): for all $\rho \in \mathfrak{S}_r$, for all $p|\tilde{t}_\rho$, there exists a target ideal $I$ such that $II^c = p^{k_0}\mathcal{O}_K$ and $\mathcal{N}(I) = p^{2k_0-v_p(\tilde{t}_\rho)}$ for the unit system $[\varepsilon_{\rho(1)}|\dots|\varepsilon_{\rho(r)}]$. We discuss briefly what we expect from the conditions (*) and (**). In regards to our computations for now, the most restrictive conditions would be the content condition $\tilde{\lambda}_\rho = 1$, as well as the conditions
**Proof :**

Recall that the basis $B_L = (e_0, \dots, e_{r+1})$ defined at the beginning of section 4.1 is such that $e_0 = h'$ and for all $1 \le i \le r$, $u_i h' = \sum_{k=0}^i c_{ik0} e_k$ where the $c_{ik0}$ are integers and $c_{ii0} > 0$. This means that

$$ \left( u_j + \sum_{i=0}^{j-1} n_i u_i \right) h' = c_{jj0} e_j + \left( \sum_{k=0}^{j-1} c_{jk0} e_k \right) + \sum_{i=0}^{j-1} \left( \sum_{k=0}^{i} n_i c_{ik0} e_k \right) \in p^k L $$

so the coordinate of this element on the vector $e_j$ is exactly $c_{jj0}$ and $p^k$ divides $c_{jj0}$. Recalling that $\lambda = \prod_{j=1}^r c_{jj0}$, we get $p^k$ divides $\lambda$. $\square$

We now define the notion of target ideal to deal with the p-adic valuation of the overflow $\tilde{t}$.

**Definition 18:** Assume $p$ is a prime number such that $v_p(\tilde{t}) > 0$. Let $I$ be a non zero primitive integral ideal of $O_{\mathbb{K}}$ such that $II^c = p^{v_p(\tilde{t})}O_{\mathbb{K}}$. We say that $I$ is a target ideal associated to the unit system $u_0 = 1, u_1, \dots, u_r$ and to the prime $p$ if $\mathcal{N}(I) = p^{v_p(\tilde{s})}$ and the two following conditions are satisfied:

* there are distinct integers $1 \le m_1 < \dots < m_{k_0} \le r$ and there are linear combinations $u_{m_k} + \sum_{j=0}^{m_k-1} n_{k,j} u_j \in J_k$ with $n_{k,j} \in \mathbb{Z}$ and $J_k$ a divisor of $I^c$ such that $IJ_k \subset p^{w_k}O_{\mathbb{K}}$ for some valuations $w_k \in \mathbb{Z}_{>0}$ satisfying $\sum_{k=1}^{k_0} w_k = v_p(\tilde{s})$.

* for all $0 \le j \le r$, $\tilde{\alpha}_j$ is congruent to an integer $n'_j$ modulo $I^c$

The importance of target ideals is expressed by the following lemma.

**Lemma 19:** Suppose $(*)$. Let $p$ be a prime number such that $v_p(\tilde{t}) > 0$ and suppose that $I$ is a target ideal for the unit system $u_0, \dots, u_r$. Take $h = mh'$ an admissible vector such that $h'$ has $p$-part exactly $I$. Then:

$$ v_p(\lambda) = v_p(\tilde{s}), \quad v_p(l) = v_p(\tilde{t}), \quad v_p(t) = 0 $$

**Proof :**

The ideal $I$ is a target ideal therefore there are distinct integers $1 \le m_1 < \dots < m_{k_0} \le r$ and linear combinations $(u_{m_k} + \sum_{j=0}^{m_k-1} n_{k,j} u_j)h' \in IJ_kL \subset p^{w_k}L$ with $n_{k,j} \in \mathbb{Z}$. Using lemma 17 we obtain that $p^{w_k}$ divides $c_{m_k,m_k,0}$ and therefore $p^{v_p(\tilde{s})} = \prod_{k=1}^{k_0} p^{w_k}$ divides $\lambda$.

On the other hand, from lemma 12 there is an index $j$ such that $p$ does not divide $\alpha_j - nh'$ for all $n \in \mathbb{Z}$ and from Proposition 14 there is an integer
$\mathcal{N}(f) = q = l^k$ and $\tilde{t}_\rho$ coprime to $q$ as together they rule out many possible
ideals $f$. The condition $\tilde{s}_\rho = \tilde{t}_\rho$ is expected to be true quite often. Finally,
we expect the condition $(**)$ to be true generally under the conditions $(*)$.
Note that there may be more than one target ideal for each prime number $p$
dividing the overflow $\tilde{t}_\rho$.

**4.3.2 Helper ideals**

Target ideals helped us understand how to choose the *p*-part of *h* when *p*|*t*.
It is not clear however how to choose an admissible vector with this specific
*p*-part. Especially when the class number of $\mathbb{K}$ is not 1, it is most likely
that the *p*-part of an admissible vector *h* for some *p* not dividing $\tilde{t}$ will be
non-trivial, in which case it must be carefully chosen. To understand the
best case scenario, we introduce the notion of *helper ideals*.

**Definition 20:** A primitive ideal *I* is a helper ideal if its complementary ideal *I*<sup>*c*</sup> is a product of degree one primes with *N*(*I*<sup>*c*</sup>) squarefree.

These ideals are interesting because they do not impact the value of *t* as
explained by the following lemma.

**Lemma 21:** Let $p$ be a prime number not dividing $\tilde{t}qN\mathcal{N}(\mathfrak{b})$. Suppose that $h = mh'$ is an admissible vector generating an ideal whose $p$-part is a helper ideal dividing $p\mathcal{O}_{\mathbb{K}}$. This means that there is a prime ideal $\mathfrak{P}$ of norm $p$ such that $h'\mathfrak{P} \in pL$ and $h \notin pL$. Then under the assumptions $(*)$, $v_p(l) = 1$ and $v_p(t) = 0$.

**Proof :**

We use once again formula (19). Under the assumptions $(*)$, we have $\tilde{\lambda} = 1$.
Here, by assumption we also have $v_p(\tilde{t}) = 0$, $v_p(\mathcal{N}(h')/\mathcal{N}(L)) = d-1$. Yet,
because $u_j-\text{rmd}(u_j, \mathfrak{P}) \in \mathfrak{P}$ for all $1 \le j \le r$ one gets $(u_j-\text{rmd}(u_j, \mathfrak{P}))h' \in pL$. Using lemma 17 successively for all indices $1 \le j \le r$ with integers $n_0 = -\text{rmd}(u_j, \mathfrak{P})$ and $n_i = 0$ for $1 \le i \le j-1$ we get that $p$ divides $c_{jj0}$. Therefore, $v_p(\lambda) \ge r = d-2$ and using formula (19) we get $v_p(lt) \le 1$. We now want to prove that $v_p(l) = 1$ so that $v_p(t) = 0$. This is essentially the same proof as for target ideals. There exists $0 \le j \le r$ such that $p$ does not divide $\alpha_j + mh'$ for all $m \in \mathbb{Z}$ (in fact this is true for all $j$ here). There are an integer $k_j$ and a sign $\epsilon = \pm 1$ such that $\epsilon l \alpha_j = (\tilde{\alpha}_j - \text{rmd}(\tilde{\alpha}_j, \mathfrak{P}))h' + k_j h'$. Suppose that $p$ does not divide $l$. Then there is an integer $m$ such that $\epsilon lm + k_j \in p\mathbb{Z}$ and $\epsilon l(\alpha_j + mh') = (\tilde{\alpha}_j - \text{rmd}(\tilde{\alpha}_j, \mathfrak{P}))h' + (\epsilon lm + k_j)h' \in pL$. This is once again a contradiction and we must conclude that $p$ divides $l$ and $p$ does not divide $t$.

$\square$

<figure><img src="image_4.png" /></figure>
$m_j$ such that $\epsilon l\alpha_j = \tilde{\alpha}_j h' + m_j h'$ for some sign $\epsilon = \pm 1$. The ideal $I$ is a target ideal, therefore there is an integer $n'_j$ such that $\tilde{\alpha}_j - n'_j \in I^c$. This implies that $\epsilon l\alpha_j - m_j h' - n'_j h' = (\tilde{\alpha}_j - n'_j)h' \in p^{v_p(\tilde{t})}L$. Suppose for now that $p$ does not divide $l$. Then $\exists n \in \mathbb{Z}$ such that $\epsilon l n - m_j - n'_j \in p\mathbb{Z}$. Therefore $\epsilon l(\alpha_j - nh') = (\tilde{\alpha}_j - n'_j)h' + (m_j + n'_j - \epsilon l n)h' \in pL$ but this is a contradiction. Therefore $p$ must divide the integer $l$. We then prove by induction that $p^2, \dots, p^{v_p(\tilde{t})}$ divide $l$. Indeed, if $p^k|l$ with $k < v_p(\tilde{t})$ then $\epsilon l\alpha_j - (m_j + n'_j)h' \in p^{v_p(\tilde{t})}L$ which means that $m_j + n'_j \in p^k\mathbb{Z}$. Suppose that $l \notin p^{k+1}\mathbb{Z}$. Then there is an integer $n \in \mathbb{Z}$ such that $\epsilon l n - m_j - n'_j \in p^{k+1}\mathbb{Z}$ and then $\epsilon l(\alpha_j - nh') \in p^{k+1}L$ which yields again a contradiction. Now, using formula (21) one finds

$$v_p(\lambda) + v_p(l) + v_p(t) = v_p(\mathcal{N}(h')) - v_p(\mathcal{N}(L)) + v_p(\tilde{\lambda}) + v_p(\tilde{t})$$

$$v_p(\lambda) + v_p(l) + v_p(t) = v_p(\tilde{s}) - 0 + 0 + v_p(\tilde{t}), \quad v_p(\lambda) \ge v_p(\tilde{s}), v_p(l) \ge v_p(\tilde{t})$$

And this leads to $v_p(\lambda) = v_p(\tilde{s}), v_p(l) = v_p(\tilde{t})$ and $v_p(t) = 0$. $\square$

<figure><img src="image_2.png" /></figure>

This allows us to strengthen the conditions under which we might understand special values of arithmetic $G_r$ functions. We add to the conditions (*) the following conditions (**):

* for all $\rho \in \mathfrak{S}_r$, for all $p|\tilde{t}_\rho$, there exists a target ideal $I_{\rho,p}$ for the unit system $[\varepsilon_{\rho(1)}|\dots|\varepsilon_{\rho(r)}]$ and for the prime $p$ (**)

* for all $\rho \in \mathfrak{S}_r$, there exists a positive integer $k_\rho$ coprime to $q$ such that the class of $k_\rho \prod_{p|\tilde{t}} I_{\rho,p}$ in the wide class group $Cl(f)$ is independent of $\rho$

We discuss briefly what we expect from the conditions (*) and (**). In regards to our computations for now, the most restrictive conditions would be the content condition $\tilde{\lambda}_\rho = 1$, as well as the conditions $f = l^k$ and $\tilde{t}_\rho$ coprime to $q$ as together they rule out many possible ideals $f$. We expect the target ideal condition (***) to be true generally under the conditions (*) and we expect the target ideals to be uniquely determined. Finally, the condition on the classes represented by $\prod_{p|\tilde{t}} I_{\rho,p}$ is expected to be often true.

### 4.3.2 Helper ideals

Target ideals helped us understand how to choose the $p$-part of $h$ when $p|\tilde{t}$. It is not clear however how to choose an admissible vector with this specific $p$-part. Especially when the class number of $\mathbb{K}$ is not 1, it is most likely that the $p$-part of an admissible vector $h$ for some $p$ not dividing $\tilde{t}$ will be
### 4.3.3 Using target and helper ideals to choose $(h_\rho)_{\rho \in \mathfrak{S}_r}$

We now use both target ideals and helper ideals to describe how we choose the vectors $h_\rho$ in conjecture 10.

**Proposition 22:** Suppose that $\varepsilon_1, \dots, \varepsilon_r$ is a set of fundamental units for $\mathcal{O}_f^{+, \times}$ satisfying (*) and (**). For all $\rho \in \mathfrak{S}_r$, write $p_{\rho,j}$ the prime numbers dividing $\tilde{t}_\rho$ for $1 \le j \le j_\rho$ and $I_{\rho,j}$ for any corresponding target ideal in (**). Put $D = q \prod_\rho \prod_{j=1}^{j_\rho} p_{\rho,j}$. Choose for the smoothing ideal $\mathfrak{a}$ a degree one prime ideal of norm $N$ coprime to $D$. Choose a representative integral ideal $\mathfrak{b}$ for a class in $Cl^+(\mathfrak{f})$ coprime to $DN$. Then there is an ideal $\mathfrak{J}_\rho$ which is either $\mathcal{O}_K$ or a helper ideal coprime to $DN\mathcal{N}(\mathfrak{b})$ such that the ideal

$$q \frac{N}{\mathfrak{a}\mathfrak{b}} \mathfrak{J}_{\rho} \prod_{j=1}^{j_{\rho}} I_{\rho,j}$$

is principal, with an admissible generator $h_\rho$. For all orientations $\mu, \underline{\nu}$, the value $I_{r,\mathfrak{f},\mathfrak{b},\mathfrak{a}}(\varepsilon_1, \dots, \varepsilon_r; \underline{h}, \underline{\mu}, \underline{\nu})$ is a product of $r!$ smoothed ordinary elliptic $G_r$ functions.

**Proof:**

We want to pick an admissible generator of the ideal $q \frac{N}{\mathfrak{a}\mathfrak{b}} \prod_{j=1}^{j_\rho} I_{\rho,j}$ but this ideal may not be principal nor possess an admissible generator. We use helper ideals to deal with those as follows. By Cebotarev's density theorem (see [[5], Theorem 8.17]), there are infinitely many degree one prime ideals in each class of $Cl^+(\mathfrak{f})$ and therefore infinitely many helper ideals in each class of $Cl^+(\mathfrak{f})$. Therefore, there is an ideal $\mathfrak{J}_\rho$ which is either $\mathcal{O}_K$ or a helper ideal prime to $DN\mathcal{N}(\mathfrak{b})$ such that the ideal

$$\frac{N}{\mathfrak{a}\mathfrak{b}} \mathfrak{J}_{\rho} \left( \prod_{j=1}^{j_{\rho}} I_{\rho,j} \right)$$

lies in the trivial class of $Cl^+(\mathfrak{f})$. In particular, this ideal has a generator $g_\rho$ congruent to $1 \mod \mathfrak{f}$ and we put $h_\rho = qg_\rho$. Then $h_\rho$ is admissible because $h_\rho/q \equiv 1 \mod L$ and $h_\rho/N \in \mathfrak{a}^{-1}L - L$ is a generator of the cyclic group $\mathfrak{a}^{-1}L/L$ of prime order $N$.

Write $\mathcal{N}(I_{\rho,j}) = p_{\rho,j}^{2k_{\rho,j}-w_{\rho,j}}$ where $w_{\rho,j} = v_{p_{\rho,j}}(\tilde{t}_\rho)$. Write also $n_\rho^{d-1} = \mathcal{N}(\mathfrak{J}_\rho)$. Then, using lemma 19 we get $v_{p_{\rho,j}}(l_\rho) = k_{\rho,j}$ and $v_{p_{\rho,j}}(t_\rho) = 0$ for all $\rho, j$. Using lemma 21 we get $n_\rho|l_\rho$ and $(n_\rho, t_\rho) = 1$. Now we have to understand what happens for the $\ell$-adic valuations in formula (19) where $\mathcal{N}(\mathfrak{f}) = q = \ell^k$. In this setting, we get $v_\ell(\lambda_\rho) = rk = (d-2)k$, $v_\ell(l_\rho) = k$, and