# Action of the automorphism group on the Jacobian of Klein’s quartic curve
II: Invariant theta functions

Dimitri Markushevich  Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé,
59000 Lille, France<EMAIL_ADDRESS>and  Anne Moreau
Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, Rue
Michel Magat, Bât. 307, 91405 Orsay, France  anne.moreau@universite-paris-
saclay.fr

  *     * scAbstract. Bernstein–Schwarzman conjectured that the quotient of a complex affine space by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture was proved by Schwarzman and Tokunaga–Yoshida in dimension $2$ for almost all such groups, and for all crystallographic reflection groups of Coxeter type by Looijenga, Bernstein–Schwarzman and Kac–Peterson in any dimension. We prove that the conjecture is true for the crystallographic reflection group in dimension $3$ for which the associated collineation group is Klein’s simple group of order $168$. In this case, the quotient is the $3$-dimensional weighted projective space with weights $1,2,4,7$. The main ingredient in the proof is the computation of the algebra of invariant theta functions. Unlike in the Coxeter case, the invariant algebra is not free polynomial, and this was the major stumbling block.

scKeywords. Finite quotient of abelian variety, complex crystallographic
reflection group, Klein’s simple group, weighted projective space, invariant
theta function

sc2020 Mathematics Subject Classification. 14B05, 11F22, 20D06, 14H45, 20H15

  *
cDecember 3, 2023Received by the Editors on June 28, 2023.
Accepted on February 7, 2024.

Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, 59000 Lille, France

sce-mail<EMAIL_ADDRESS>
Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, Rue
Michel Magat, Bât. 307, 91405 Orsay, France

sce-mail<EMAIL_ADDRESS>
D.M. was partially supported by the PRCI SMAGP (ANR-20-CE40-0026-01) and the
Labex CEMPI (ANR-11-LABX-0007-01). A.M. is partially supported by the European
Research Council (ERC) under the European Union’s Horizon 2020 research
innovation programme (ERC-2020-SyG-854361-HyperK).

© by the author(s) This work is licensed under
http://creativecommons.org/licenses/by-sa/4.0/

###### Contents

  1. 1 Introduction
  2. 2 The group, the lattice and the associated theta functions
  3. 3 Theta transformation formula
  4. 4 Unitary action of $\boldsymbol{G}$ on theta functions and its character
  5. 5 Hypersurface in a weighted projective space of dimension 4
  6. 6 Degree 8 hypersurfaces in $\boldsymbol{{\mathbb{P}}(1,1,2,4,7)}$ with correct singularities
  7. 7 Relation to quotients by the Weyl group for the root systems $\boldsymbol{B_{3}}$ and $\boldsymbol{C_{3}}$

## 1\. Introduction

A general conjecture of Bernstein and Schwarzman [BS06] claims that the
quotient of ${\mathbb{C}}^{n}$ by the action of an irreducible complex
crystallographic group generated by reflections (complex crystallographic
reflection group for short) is a weighted projective space. The conjecture is
proved for almost all complex crystallographic reflection groups for $n=2$, as
well as for irreducible complex crystallographic reflection groups of Coxeter
type of any rank $n\geqslant 2$. A complex crystallographic reflection group
is said to be of Coxeter type if it is obtained by complexification of a real
crystallographic group or, in other words, if the group of its linear parts is
conjugate to a finite subgroup of the orthogonal group $\mathbf{O}(n)$. See
[MM23] for more historical comments and further references.

Since the 1980s, the conjecture remained widely open for any irreducible
complex crystallographic reflection group of rank $n\geqslant 3$ which is
genuinely complex, that is, not of Coxeter type. In the present paper we prove
the conjecture for the rank $3$ complex crystallographic reflection group of
type $[K_{24}]$ in the classification of Popov [Pop82]. We show that the
quotient $X={\mathbb{C}}^{3}/\Gamma$ is isomorphic to a weighted projective
space.(1)(1)(1)The proof of the Bernstein–Schwarzman conjecture for $[K_{24}]$
appeared in the second version of the present article, posted on arXiv in
November 2022, and at that moment it was the only non-Coxeter complex
crystallographic reflection group of rank greater than $2$ for which the
conjecture was established. In March 2023, Eric Rains posted the preprint
[Rai23], in which he proves the conjecture in full generality. More
explicitly, the main result of the article is the following.

######  Main Theorem (Theorem 6.3).

The quotient variety ${\pazocal J}/G$, where ${\pazocal J}={\pazocal J}(C)$ is
the $3$-dimensional Jacobian of the plane Klein quartic curve $C$ and $G$ is
the full automorphism group of order $336$, is isomorphic to the weighted
projective space ${\mathbb{P}}(1,2,4,7)$.

The group $[K_{24}]$ is in several regards the most intriguing one among the
rank $3$ complex crystallographic reflection groups. Firstly, it is the only
one whose projectivized group of linear parts is simple, namely, is equal to
Klein’s group $H$ of order $168$. The groups of linear parts in all the other
cases are solvable. Secondly, the quotient ${\mathbb{C}}^{3}/\Gamma$ is
isomorphic to the quotient of the Jacobian ${\pazocal J}(C)$ of Klein’s
quartic curve

$C:=\\{x^{3}y+y^{3}z+z^{3}x=0\\}\subset{\mathbb{P}}^{2}$

by the full group

$G=\\{\pm 1\\}\times H$

of its automorphisms as a principally polarized abelian variety. As follows
from Hurwitz’ bound, $C$ has a maximal number of automorphisms for a curve of
genus $3$. Algebraic varieties acted on by Klein’s group, not only curves, are
a recurrent subject of interest in algebraic geometry. Thirdly, $[K_{24}]$
encloses a very rich number-theoretic content, for Klein’s curve $C$ is
nothing but the modular curve $X(7)$, or else it can be viewed as a Shimura
curve, while its Jacobian is isomorphic, as an abstract abelian variety, to
the cube of the elliptic modular curve $X_{0}(49)$. The representation of the
translation lattice of that we use was given in [Maz86].

An important ingredient of our proof for $[K_{24}]$ is the computation of the
Hilbert function of the algebra of -invariant theta functions. The proofs of
the conjecture in the Coxeter case obtained in the 1980s passed through
showing that the invariant algebra is free. As the quotient variety is the
spectrum of the invariant algebra, the freeness of the latter implies that the
quotient is a weighted projective space. But this idea does not work for
genuinely complex crystallographic reflection groups since the invariant
algebra is not free anymore. In the case of $[K_{24}]$, we succeed to
understand the structure of relations between generators of this algebra. It
turns out that the ideal of relations is principal and defines a hypersurface
of degree $8$ in the $4$-dimensional weighted projective space
${\mathbb{P}}(1,1,2,4,7)$.

On the other hand, in [MM23], we determined the singularities of $X$: they are
images of the orbits whose stabilizers are not generated by reflections. We
observed that the singularities of $X$ are analytically equivalent to those of
${\mathbb{P}}(1,2,4,7)$, which prompted us to conjecture that $X$ is
isomorphic to this weighted projective space. The latter also embeds in
${\mathbb{P}}(1,1,2,4,7)$ as an octic hypersurface! It can be given by the
equation $y_{0}y_{4}=y_{3}^{2}.$ The last step of our proof is to show that
all degree $8$ hypersurfaces in ${\mathbb{P}}(1,1,2,4,7)$ whose singularities
are those of ${\mathbb{P}}(1,2,4,7)$ are equivalent to $y_{0}y_{4}=y_{3}^{2}$
by automorphisms of ${\mathbb{P}}(1,1,2,4,7)$.

As a byproduct, we see that ${\mathbb{P}}(1,2,4,7)$ possesses a non-trivial
deformation, obtained by varying the coefficients of the octic in the
$4$-dimensional weighted projective space. This deformation turns out to be
versal and is a partial smoothing, so that the general member of the
deformation family is a 2-Gorenstein Fano 3-fold with Picard group
${\mathbb{Z}}$ and only rigid isolated singularities. Deformations of
1-Gorenstein weighted projective spaces in dimension $3$ have been studied in
[DS23].

Another noteworthy feature of our quotient $X$ is the existence of a natural
double cover $\overline{Y}\to X$, a Calabi–Yau orbifold that can be used as a
target space for the superstring compactification; in particular, an
interesting question is what its mirror family is. This double cover is
obtained as the quotient $\overline{Y}={\mathbb{C}}^{3}/{}_{0}$, where 0 is
the subgroup of index $2$ in , the unimodular part of . It can also be
realized as an anticanonical hypersurface in ${\mathbb{P}}(1,2,4,7,14)$ of
dimension $4$. However, the known procedure of constructing mirrors of generic
hypersurfaces in a weighted projective space does not apply to this case, for
$\overline{Y}$ is by no means generic; it is a very special member of the
anticanonical system on this weighted projective space having non-isolated
singularities. We hope to return to the study of this Calabi–Yau orbifold in
the future.

Let us now detail our strategy to prove the main theorem. In general, a
complex crystallographic group of rank $n$ is a group of affine
transformations of ${\mathbb{C}}^{n}$ which fits in an exact triple

$0\longrightarrow L\longrightarrow\Gamma\longrightarrow\hskip
1.29167pt\mathrm{d}\Gamma\longrightarrow 1,$

where $L\simeq{\mathbb{Z}}^{2n}$ is a lattice of maximal rank $2n$ in
${\mathbb{C}}^{n}$, acting by translations, and $\hskip
1.29167pt\mathrm{d}\Gamma$ is a finite subgroup of the unitary group
$\mathbf{U}(n)$. A complex crystallographic group is a complex
crystallographic reflection group if it is generated by complex affine
reflections, where an affine transformation of ${\mathbb{C}}^{n}$ is called a
reflection if it is of finite order and its fixed locus is an affine
hyperplane. In our case $\hskip 1.29167pt\mathrm{d}\Gamma=G$; there is a
unique $G$-invariant rank $6$ lattice $L$ in ${\mathbb{C}}^{3}$, which turns
out to be the period lattice of Klein’s quartic $C$, and the above extension
is necessarily split, so that $\Gamma=\Lambda\rtimes G$.

The quotient ${\mathbb{C}}^{n}/\Gamma$ can be thought of as the quotient of
the abelian variety ${\mathbb{C}}^{n}/L=A$ by the induced action of the finite
group of linear parts: ${\mathbb{C}}^{n}/\Gamma\simeq A/\hskip
1.29167pt\mathrm{d}\Gamma$. The idea applied in the case of irreducible
complex crystallographic reflection groups of Coxeter type in [BS06, Loo76] is
to represent $A$ as the Proj of the graded algebra $S({\mathcal{L}})$ of
sections of the powers of an ample line bundle ${\mathcal{L}}$ on $A$,
linearizable by the action of $\hskip 1.29167pt\mathrm{d}\Gamma$, and then
${\mathbb{C}}^{n}/\Gamma$ is the Proj of the $\hskip
1.29167pt\mathrm{d}\Gamma$-invariant part:

$S({\mathcal{L}})=\bigoplusop\displaylimits_{k=0}^{\infty}H^{0}\left(A,{\mathcal{L}}^{k}\right),\quad
A=\operatorname{Proj}\nolimits
S({\mathcal{L}}),\quad{\mathbb{C}}^{n}/\Gamma\simeq A/\hskip
1.29167pt\mathrm{d}\Gamma=\operatorname{Proj}\nolimits\left(S({\mathcal{L}})^{\hskip
0.90417pt\mathrm{d}\Gamma}\right).$

The sections of the powers of ${\mathcal{L}}$ are given by theta functions for
the lattice $L$, and it is proven that the invariant part is a polynomial
algebra in $n+1$ free generators.

Mimicking this approach, we introduce the theta functions
$\theta_{m,k}\colon{\mathbb{C}}^{3}\to{\mathbb{C}}$ of degree $k\geqslant 0$
for the lattice in such a way that $\\{\theta_{m,k}\\}_{m\in P_{k}}$ is a
basis of the space $H^{0}({\pazocal J},{\mathcal{L}}^{k})$ of global sections
of the $k^{\mathrm{th}}$ power of an appropriate line bundle ${\mathcal{L}}$
on ${\pazocal J}={\mathbb{C}}^{3}/\Lambda$ defining a principal polarization,
where $m$ runs over a set $P_{k}$ of representatives of
$\frac{1}{k}{\mathbb{Z}}^{3}/{\mathbb{Z}}^{3}$. Unlike in the Coxeter case,
the thus defined line bundle ${\mathcal{L}}$ is not ${\hskip
1.29167pt\mathrm{d}\Gamma}$-invariant, only even powers of ${\mathcal{L}}$ can
be $\hskip 1.29167pt\mathrm{d}\Gamma$-linearized, and thus we have to work
with the second Veronese subalgebra $S({\mathcal{L}}^{2})$ of
$S({\mathcal{L}})$. We have

$X=A/\hskip 1.29167pt\mathrm{d}\Gamma\simeq\operatorname{Proj}\nolimits
S\left({\mathcal{L}}^{2}\right)^{\hskip 0.90417pt\mathrm{d}\Gamma},\quad
S\left({\mathcal{L}}^{2}\right)^{\hskip
0.90417pt\mathrm{d}\Gamma}=\bigoplusop\displaylimits_{p=0}^{\infty}H^{0}\left(A,{\mathcal{L}}^{2p}\right)^{\hskip
0.90417pt\mathrm{d}\Gamma}.$

Thus one cannot expect $S({\mathcal{L}}^{2})^{\hskip
0.90417pt\mathrm{d}\Gamma}$ to be polynomial; what we prove is the isomorphism
between $S({\mathcal{L}}^{2})^{\hskip 0.90417pt\mathrm{d}\Gamma}$ and the
second Veronese algebra of ${\mathbb{P}}(1,2,4,7)$, the latter being non-
polynomial.

Now we describe the content of the paper by sections. Section 2 gathers
definitions and preliminary results on the complex crystallographic reflection
group , the lattice and associated theta functions.

In Section 3, we compute the transformation formula for the action of the
elements of the modular group on our theta functions (Theorem 3.4).

In Section 4, we determine the Hilbert function of the algebra
$S({\mathcal{L}}^{2})^{G}$, proving that it coincides with the Hilbert
function of the second Veronese algebra of ${\mathbb{P}}(1,2,4,7)$; see
Theorem 4.3. This is the second step of the proof of the isomorphism
$X\simeq{\mathbb{P}}(1,2,4,7)$; the first one was the study of the
singularities of $X$, done in [MM23, Theorem 4.3].

Next, we prove that $X$ admits an embedding in ${\mathbb{P}}(1,1,2,4,7)$ as a
degree $8$ hypersurface. This is done in Section 5. We first show that there
exist four homogeneous elements $\varphi_{0},\ldots,\varphi_{3}$ of
$S({\mathcal{L}}^{2})^{G}$ of degrees $1,1,2,4$ which are algebraically
independent. The four invariant theta functions $\varphi_{i}$ are chosen in an
ad hoc way in Lemma 5.2, and their algebraic independence is proved by
evaluating their Jacobian. Using the known Hilbert function, we deduce from
this that there is a fifth element $\varphi_{4}$ of $S({\mathcal{L}}^{2})^{G}$
of degree $7$ such that the five functions $\varphi_{i}$ generate
$S({\mathcal{L}}^{2})^{G}$, and this makes $X$ into a degree $8$ hypersurface
in ${\mathbb{P}}(1,1,2,4,7)$ (see Theorem 5.1).

Section 6 contains the last step of the proof of the main result: in
Proposition 6.2, we show that the degree $8$ hypersurfaces in
${\mathbb{P}}(1,1,2,4,7)$ whose singularities are those of
${\mathbb{P}}(1,2,4,7)$ form just one orbit under the action of the group of
coordinate changes in ${\mathbb{P}}(1,1,2,4,7)$. Thus $X$ is equivalent,
modulo a coordinate change, to ${\mathbb{P}}(1,2,4,7)$ embedded in
${\mathbb{P}}(1,1,2,4,7)$ as a degree $8$ hypersurface. This implies the main
result of the paper (Theorem 6.3). In Remark 6.4, we discuss the non-trivial
deformations of ${\mathbb{P}}(1,2,4,7)$ smoothing out the non-isolated
singularity, provided by the family of weighted octics in
${\mathbb{P}}(1,1,2,4,7)$.

In Section 7, we look at the quotient ${\mathbb{C}}^{3}/\Lambda\rtimes W$,
where $W$ is a maximal real reflection subgroup of $G$, of order $48$. The
group $\Lambda\rtimes W$ is complex crystallographic but is not generated by
reflections. It is plausible that this quotient is a weak weighted projective
space (see Proposition 7.3), and we formulate a conjecture, generalizing this
example, which says that if and 1 are commensurable complex crystallographic
groups with the same linear parts, such that is complex crystallographic
reflection, then ${\mathbb{C}}^{n}/{}_{1}$ is a weak weighted projective space
(see Conjecture 7.4). We also provide in Section 7 some heuristic explanation
of our ad hoc choice for generators of $S({\mathcal{L}}^{2})^{G}$ (see Remark
7.1).

## Acknowledgements

The authors thank the referee for the useful comments. They also wish to thank
Thomas Dedieu and Xavier Roulleau for interesting discussions.

## 2\. The group, the lattice and the associated theta functions

Let the complex vector space $V={\mathbb{C}}^{3}$, with coordinates
$(z_{1},z_{2},z_{3})$, be endowed with its standard Hermitian product
$\langle-|-\rangle$ and its standard symmetric bilinear form $(-|-)$, and
consider the complex root system in it, consisting of the 42 vectors obtained
from $(2,0,0)$, $(0,\alpha,\alpha)$ and $(1,1,\overline{\alpha})$, where
$\alpha=\frac{1+i\sqrt{7}}{2}$, by sign changes and permutations of
coordinates. For each $\phi\in{\Phi}$, the reflection of order $2$

$r_{\phi}\colon V\longrightarrow V,\quad z\longmapsto
z-2\frac{\langle\phi|z\rangle}{\langle\phi|\phi\rangle}\phi$

sends onto . As $r_{\phi}=r_{-\phi}$, we obtain 21 reflections in this way. We
choose

$\phi_{1}=(0,\alpha,-\alpha),\quad\phi_{2}=(0,0,2),\quad\phi_{3}=(1,1,\overline{\alpha})$

as the basic roots and write $r_{i}=r_{\phi_{i}}$, $i=1,2,3$:

$r_{1}={\scriptsize\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}},\quad
r_{2}={\scriptsize\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&{-1}\end{pmatrix}},\quad
r_{3}=\tfrac{1}{2}{\scriptsize\begin{pmatrix}1&{-1}&{-\alpha}\\\
{-1}&1&{-\alpha}\\\ -\overline{\alpha}&-\overline{\alpha}&0\end{pmatrix}}.$

We define the lattice as the root lattice $Q(\Phi)$ of , that is,
$\Lambda=\sumop\displaylimits_{\phi\in\Phi}{\mathbb{Z}}\phi$. We have

$\Lambda={\mathcal{O}}\phi_{1}+{\mathcal{O}}\phi_{2}+{\mathcal{O}}\phi_{3},\ \
{\mathcal{O}}={\mathbb{Z}}[\alpha]={\mathbb{Z}}+\alpha{\mathbb{Z}}.$

By [Maz86, Section III.3, pp. 235–236], is the period lattice of Klein’s
quartic; it can also be represented in the form

$\Lambda=\left\\{(z_{1},z_{2},z_{3})\in{{\mathcal{O}}}^{3}\colon z_{1}\equiv
z_{2}\equiv z_{3}\ \mathrm{mod}\ \alpha,\ z_{1}+z_{2}+z_{3}\equiv 0\
\mathrm{mod}\ \overline{\alpha}\right\\}.$

The group $G$ is the full group of complex-linear automorphisms of . It
contains 21 reflections, all of order $2$, and is generated by the three basic
reflections: $G=\langle r_{1},r_{2},r_{3}\rangle$. According to [ST54,
Equation (10.1)], the following relations are defining for $G$:

(2.1)
$r_{1}^{2}=r_{2}^{2}=r_{3}^{2}=(r_{1}r_{2})^{4}=(r_{2}r_{3})^{4}=(r_{3}r_{1})^{3}=(r_{1}r_{2}r_{1}r_{3})^{3}=1.$

Klein’s simple group of order $168$ is the unimodular part of $G$:

$H=\\{h\in G\colon\det(h)=1\\}.$

It is generated by the antireflections $\rho_{\phi}:=-r_{\phi}$; of course,
the antireflections $\rho_{i}=-r_{i}$ associated to the three basic roots
$\phi_{i}$ ($i=1,2,3$) suffice to generate $H$. According to [Pop82], there is
a unique extension of by $G$, the split one, or the semi-direct product
$\Gamma=\Lambda\rtimes G$, and it is a complex crystallographic reflection
group.

Note that contains the sublattice $\alpha Q$, homothetic to the root lattice
$Q=Q(C_{3})$ of the real root system $C_{3}$, and thus $G$ contains the Weil
group $W=W(C_{3})$ of order $48$. The latter consists of all the monomial
matrices of size $3$ whose only non-zero elements are $\pm 1$. We have
$W=G\cap\mathbf{O}(3)$, and $G$ is the union of seven cosets $g_{7}^{i}W$
($i=0,\ldots,6$) for some element $g_{7}$ of order $7$. We can choose

$g_{7}=\rho_{1}\rho_{2}\rho_{3}=-r_{1}r_{2}r_{3}=\tfrac{1}{2}\left(\begin{matrix}{-1}&1&\alpha\\\
-\overline{\alpha}&-\overline{\alpha}&0\\\ 1&{-1}&\alpha\end{matrix}\right).$

This is a handy way to enumerate all the elements of $G$, say, for computer
checks of some properties.

Let $M$ denote the weight lattice of $C_{3}$,
$M=Q^{*}={\mathbb{Z}}^{3}+\frac{1}{2}(1,1,1)$. Then
$\Lambda=2\overline{\alpha}M+\alpha Q$ is generated by the columns of the
6-by-3 period matrix $\Omega=(\omega_{1}\,|\,\omega_{2})$, where
$\omega_{1},\omega_{2}$ are square blocks of size 3, the columns of
$\omega_{2}$ being $\alpha$ times the elements of a basis of $C_{3}$, and we
set

(2.2) $\omega_{2}=\alpha C,\quad\omega_{1}=-2\overline{\alpha}({C}^{{\sf
T}})^{-1},\quad C=\begin{pmatrix}1&0&0\\\ -1&1&0\\\ 0&-1&2\\\ \end{pmatrix}.$

The normalized period matrix of ${\pazocal J}$ is obtained by multiplying on
the left by $\omega_{2}^{-1}$:

(2.3) $\omega_{2}^{-1}\Omega=(Z\,|\,I),\quad Z=\omega_{2}^{-1}\omega_{1}=\tau
B,$

where $\tau=-\overline{\alpha}^{\,2}$ and

(2.4) $B=\left({C}^{{\sf T}}C\right)^{-1}=\frac{1}{4}\begin{pmatrix}4&4&2\\\
4&8&4\\\ 2&4&3\\\ \end{pmatrix}$

is real symmetric and positive definite, so that $Z=\tau B\in\mathfrak H_{3}$,
where $\mathfrak H_{r}$ denotes the Siegel half-space of complex symmetric
matrices of size $r$ with positive-definite imaginary part.

Let $c_{1},c_{2}\in{\mathbb{R}}^{r}$ be two vectors, $k\in{\mathbb{Z}}$,
$k\geqslant 0$, $Z\in\mathfrak H_{r}$. The classical theta function with
period $Z$ and characteristic $c_{1}Z+c_{2}$ and of degree $k$ is the complex-
valued function

${\mathbb{C}}^{r}\ni
v\longmapsto\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{c_{1}}{c_{2}}$}(v,Z)=\sumop\displaylimits_{u\in{\mathbb{Z}}^{r}}e^{2\pi
ik[{(v+c_{2})}^{{\sf T}}(u+c_{1})+\frac{1}{2}{(u+c_{1})}^{{\sf
T}}Z(u+c_{1})]}.$

When $k=1$, the subscript $k$ is usually omitted. For $k=1$ and
$c_{1}=c_{2}=0$, the function
$\theta\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{0}{0}$}(\bullet,Z)$
represents a section of a uniquely determined line bundle ${\mathcal{L}}$ on
the principally polarized abelian variety
$A={\mathbb{C}}^{r}/(Z{\mathbb{Z}}^{r}+{\mathbb{Z}}^{r})$, and then for any
$k,c_{1},c_{2}$, the function
$\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{c_{1}}{c_{2}}$}(\bullet,Z)$
represents a section of the line bundle
$T_{c_{1}Z+c_{2}}^{*}({\mathcal{L}}^{k})$, the pullback of the tensor power
${\mathcal{L}}^{k}$ of ${\mathcal{L}}$ by the translation by the point
$c_{1}Z+c_{2}{\,\mathrm{mod}\,}(Z{\mathbb{Z}}^{r}+{\mathbb{Z}}^{r})$ of $A$.
If we choose a set $P_{k}$ of representatives of
$\frac{1}{k}{\mathbb{Z}}^{r}/{\mathbb{Z}}^{r}$, then the $k^{3}$ theta
functions from
$\\{\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{c_{1}}{c_{2}}$}(\bullet,Z)\\}_{m\in
P_{k}}$ represent a basis of
$H^{0}(A,T_{c_{1}Z+c_{2}}^{*}({\mathcal{L}}^{k}))$. See e.g. [BL04].

######  Definition 2.1.

Let be as above, $Z\in\mathfrak H_{3}$ as in (2.3), $r=3$, $k\in{\mathbb{Z}}$
with $k\geqslant 1$ and $m\in\frac{1}{k}{\mathbb{Z}}^{3}$. We define the theta
function for of degree $k$ with characteristic $m$ by the formula

$\theta_{m,k}(z)=\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}(\omega_{2}^{-1}z,Z)\quad\text{for
any}\ z\in V.$

There is a unique line bundle ${\mathcal{L}}$ on ${\pazocal J}=V/\Lambda\simeq
V/(Z{\mathbb{Z}}^{3}+{\mathbb{Z}}^{3})$ defining a principal polarization such
that the functions $\\{\theta_{m,k}\\}_{m\in P_{k}}$ form a basis of
$H^{0}({\pazocal J},{\mathcal{L}}^{k})$, where $P_{k}$ is a set of
representatives of $\frac{1}{k}{\mathbb{Z}}^{3}/{\mathbb{Z}}^{3}$, for all
$k\geqslant 1$.

We denote by $\boldsymbol{Sp}(2r,{\mathbb{Z}})$ the symplectic group of
automorphisms of ${\mathbb{Z}}^{2r}$ preserving the skew-symmetric bilinear
form given by the matrix $E$:

$\boldsymbol{Sp}(2r,{\mathbb{Z}})=\left\\{A\in
M_{2r}({\mathbb{Z}})\colon{A}^{{\sf T}}EA=E\right\\},\quad
E=E_{r}=\begin{pmatrix}0&-I_{r}\\\ I_{r}&0\end{pmatrix},$

where $I$ (or $I_{r}$) denotes the identity matrix (of size $r$). We represent
$E$ and the matrices $\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\boldsymbol{Sp}(2r,{\mathbb{Z}})$ by their blocks of size
$r$. A crucial ingredient of our computation of the action of $G$ on the theta
functions $\theta_{m,k}$ is a transformation formula under modular
transformations. The action of the modular transformation $\gamma$ on a
classical theta function is defined by

(2.5)
$\left(\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{c_{1}}{c_{2}}$}\right)^{\gamma}(v,Z)=\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{c_{1}}{c_{2}}$}\left(\left({(cZ+d)}^{{\sf
T}}\right)^{-1}v,(aZ+b)(cZ+d)^{-1}\right).$

The following is a particular case of Igusa’s theorem for a polarization of
type $(k,k,\ldots,k)$.

######  Theorem 2.2 (cf. [Igu72, Theorem II.5.6]).

For every $(v,Z)\in{\mathbb{C}}^{r}\times\mathfrak H_{r}$,
$c_{1},c_{2}\in{\mathbb{R}}^{g}$, $m\in P_{k}$ and
$\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\boldsymbol{Sp}(2r,{\mathbb{Z}})$, the matrix $cZ+d$ is
invertible, and we have

$\left(\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m+c^{\prime}_{1}}{c^{\prime}_{2}}$}\right)^{\gamma}(v,Z)=e^{\pi
ik{v}^{{\sf
T}}{(cZ+d)}^{-1}cv}\det(cZ+d)^{\frac{1}{2}}\cdot\sumop\displaylimits_{m^{\prime}\in
P_{k}}u_{m,m^{\prime}}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m^{\prime}+c_{1}}{c_{2}}$}(v,Z),$

where

$\begin{bmatrix}c_{1}^{\prime}\\\
c_{2}^{\prime}\end{bmatrix}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{bmatrix}c_{1}\\\
c_{2}\end{bmatrix}+\frac{1}{2}\begin{bmatrix}(c{d}^{{\sf T}})_{0}\\\
(a{b}^{{\sf T}})_{0}\end{bmatrix},$

$(u_{m,m^{\prime}})\in\mathbf{U}(k^{r})$ and $(h)_{0}$ denotes the column
vector of the diagonal elements of $h$ for any square matrix $h$.

In the next section, we will represent the automorphisms from $G$ by modular
transformations and compute explicitly the matrices $(u_{m,m^{\prime}})$ for
even $k$. As we will see, the presence of the half-integer inhomogeneous term
in the transformation formula for the characteristics $c_{1},c_{2}$ implies
the non-invariance of ${\mathcal{L}}$ under the action of $G$; however, the
even powers of ${\mathcal{L}}$ are $G$-invariant.

## 3\. Theta transformation formula

We keep the notation of the previous section. The elements of $G$ are complex
3-by-3 matrices leaving invariant the lattice . As is generated by the columns
of the 6-by-3 matrix $\Omega=(\omega_{1}\,|\,\omega_{2})$, we can associate to
each $g\in G$ a matrix $\gamma=\gamma_{g}=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\boldsymbol{Sp}(6,{\mathbb{Z}})$ in such a way
that

$(\omega_{1}\,|\,\omega_{2})\begin{pmatrix}{a}^{{\sf T}}&{c}^{{\sf T}}\\\
{b}^{{\sf T}}&{d}^{{\sf T}}\end{pmatrix}=g(\omega_{1}\,|\,\omega_{2}).$

Obviously, the map $g\mapsto\gamma_{g}$ is a group homomorphism.

######  Lemma 3.1.

Let $g\in G$, and let $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)$ be as above. Then the following properties hold:

  1. (i)

$(aZ+b)(cZ+d)^{-1}=Z$;

  2. (ii)

$\det(cZ+d)=\det g=\pm 1$;

  3. (iii)

$\det d=\pm 1$.

###### Proof.

(i) We have

${Z}^{{\sf
T}}=Z=\omega_{2}^{-1}\omega_{1}=(g\omega_{2})^{-1}(g\omega_{1})=(\omega_{1}{c}^{{\sf
T}}+\omega_{2}{d}^{{\sf T}})^{-1}\cdot(\omega_{1}{a}^{{\sf
T}}+\omega_{2}{b}^{{\sf T}})=\\\ (Z{c}^{{\sf T}}+{d}^{{\sf
T}})^{-1}(Z{a}^{{\sf T}}+{b}^{{\sf T}})={\left((aZ+b)(cZ+d)^{-1}\right)}^{{\sf
T}}.$

(ii) By the definition of $\gamma$, $Z{c}^{{\sf T}}+{d}^{{\sf
T}}={(cZ+d)}^{{\sf T}}$ is the matrix of $g$ in the basis of
${\mathbb{C}}^{3}$ given by the columns of $\omega_{2}$, so $\det
g=\det{(cZ+d)}^{{\sf T}}$.

(iii) This is verified by a direct computation of the matrices $\gamma_{g}$
for all elements $g\in G$. ∎

######  Corollary 3.2.

For $z\in V$, we have

$g\cdot\theta_{0,1}(z)=\left(\theta\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{0}{0}$}\right)^{\gamma_{g}}(v,Z)=\chi_{g}\theta\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{\nu^{\prime}}{\nu^{\prime\prime}}$}(v,Z),$

where

$v=\omega_{2}^{-1}z,\quad\chi_{g}=e^{\pi i{v}^{{\sf T}}(cZ+d)^{-1}cv}(\det
g)^{1/2},\quad\begin{bmatrix}{\nu^{\prime}}\\\
{\nu^{\prime\prime}}\end{bmatrix}=-\frac{1}{2}\begin{pmatrix}{d}^{{\sf
T}}&-{b}^{{\sf T}}\\\ -{c}^{{\sf T}}&{a}^{{\sf
T}}\end{pmatrix}\begin{bmatrix}{({c}{d}^{{\sf T}})_{0}}\\\ {({a}{b}^{{\sf
T}})_{0}}\end{bmatrix}.$

###### Proof.

This immediately follows from Lemma 3.1, Igusa’s theorem with $k=1$ and the
inversion formula

$\begin{pmatrix}{a}&{b}\\\ {c}&{d}\end{pmatrix}^{-1}=\begin{pmatrix}{d}^{{\sf
T}}&-{b}^{{\sf T}}\\\ -{c}^{{\sf T}}&{a}^{{\sf T}}\end{pmatrix}$

for matrices $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\boldsymbol{Sp}(2r,{\mathbb{Z}})$. ∎

We see that the theta function $\theta_{0,1}$, representing a section of
${\mathcal{L}}$, acquires a half-integer characteristic upon the action by an
element $g\in G$ whenever the diagonal elements of the integer matrices
$c{d}^{{\sf T}}$ and $a{b}^{{\sf T}}$ are not all even. In this case,
$g^{*}{\mathcal{L}}$ is not ${\mathcal{L}}$ but the translation of
${\mathcal{L}}$ by a point of order $2$.

######  Corollary 3.3.

  1. (i)

For $g\in G$, we have the equivalence
$g^{*}{\mathcal{L}}\simeq{\mathcal{L}}\Leftrightarrow g\in W$, where
$W=W(C_{3})=G\cap\mathbf{O}(3)$ is the subgroup of real matrices in $G$.

  2. (ii)

$g^{*}{\mathcal{L}}^{k}\simeq{\mathcal{L}}^{k}$ for all $g\in G\Leftrightarrow
k$ is even.

###### Proof.

By a direct computation, we verify that for $g\in G$, the diagonal elements of
the integer matrices $c{d}^{{\sf T}}$ and $a{b}^{{\sf T}}$ are all even if and
only if $g\in W$. This implies both assertions. ∎

Thus the problem of calculating the action of $G$ on $H^{0}({\pazocal
J},{\mathcal{L}}^{k})$ has sense only for even $k$. We are now going to
calculate, in a particular case and only for even $k$, the matrix
$(u_{m,m^{\prime}})$ from Igusa’s theorem up to proportionality; we denote the
matrix we find by $(\widetilde{u}_{m,m^{\prime}})$, as it is a multiple of
Igusa’s matrix $(u_{m,m^{\prime}})$ which is not necessarily unitary.

For $\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\boldsymbol{Sp}(2r,{\mathbb{Z}})$ with $\det
d=\pm 1$, we define

$\widetilde{a}=a-bd^{-1}c=({d}^{{\sf
T}})^{-1},\quad\widetilde{b}={\widetilde{b}}^{{\sf T}}=b{\widetilde{a}}^{{\sf
T}},\quad\widetilde{c}={\widetilde{c}}^{{\sf T}}=-c{d}^{{\sf T}}.$

######  Theorem 3.4.

Let $k$ be a positive even integer, $P_{k}$ a set of representatives of
$\frac{1}{k}{\mathbb{Z}}^{r}/{\mathbb{Z}}^{r}$,
$\gamma\in\boldsymbol{Sp}(2r,{\mathbb{Z}})$ such that $\det d=\pm 1$ and
$Z=\tau B$ for a real symmetric positive-definite matrix $B$ of size $r$,
where $\tau\in{\mathbb{C}}$, $\operatorname{Im}\nolimits\tau>0$. Then
${\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}}^{\gamma}=\chi\sumop\displaylimits_{m^{\prime}\in
P_{k}}\widetilde{u}_{m,m^{\prime}}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m^{\prime}}{0}$}$,
where $\chi=\chi_{\gamma}(v,Z)$ is a nowhere-vanishing analytic function on
${\mathbb{C}}^{r}\times\mathfrak H_{r}$, depending on $\gamma$, and

$\widetilde{u}_{m,m^{\prime}}=e^{\pi
ik\widetilde{b}[m]}\sumop\displaylimits_{\hat{m}\in P_{k}}e^{2\pi
ik{\left(m-dm^{\prime}+\frac{1}{2}\widetilde{c}\hat{m}\right)}^{{\sf
T}}\hat{m}}.$

###### Proof.

We decompose $\gamma$ in a product of elementary transformations as follows:

$\gamma=\raisebox{-6.88889pt}{$\begin{subarray}{c}\begin{pmatrix}1&\widetilde{b}\\\
0&1\end{pmatrix}\\\ \,\rotatebox{90.0}{$\,=$}\\\
\sigma_{1}\end{subarray}$}\raisebox{-6.88889pt}{$\begin{subarray}{c}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\\\ \,\rotatebox{90.0}{$\,=$}\\\
\sigma_{2}\end{subarray}$}\raisebox{-6.88889pt}{$\begin{subarray}{c}\begin{pmatrix}1&\widetilde{c}\\\
0&1\end{pmatrix}\\\ \,\rotatebox{90.0}{$\,=$}\\\
\sigma_{3}\end{subarray}$}\raisebox{-6.88889pt}{$\begin{subarray}{c}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\\\ \,\rotatebox{90.0}{$\,=$}\\\
\sigma_{4}\end{subarray}$}\raisebox{-6.88889pt}{$\begin{subarray}{c}\begin{pmatrix}\widetilde{a}&0\\\
0&d\end{pmatrix}\\\ \,\rotatebox{90.0}{$\,=$}\\\ \sigma_{5}\end{subarray}$}$

and apply the factors of this decomposition successively.

Step 1\.
$\left(\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}\right)^{\sigma_{1}}(v,Z)=\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}(v,Z+\widetilde{b})=$

$\sumop\displaylimits_{u\in{\mathbb{Z}}^{r}}e^{2\pi ik({v}^{{\sf
T}}(u+m)+\frac{1}{2}{(u+m)}^{{\sf T}}Z(u+m))}\cdot e^{\pi
ik\widetilde{b}[u]}\cdot e^{2\pi ik{m}^{{\sf T}}\widetilde{b}u}\cdot e^{\pi
ik\widetilde{b}[m]}=e^{\pi
ik\widetilde{b}[m]}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}(v,Z),$

where we use the notation $A[u]={u}^{{\sf T}}Au$ for any symmetric matrix $A$
of size $r$ and any vector $u\in{\mathbb{C}}^{r}$, and $e^{\pi
ik\widetilde{b}[u]}=e^{2\pi ik{m}^{{\sf T}}\widetilde{b}u}=1$ for any
$u\in{\mathbb{Z}}^{r}$, $k$ even.

Step 2\. We have

$\left(\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}\right)^{\sigma_{1}\sigma_{2}}(v,Z)=e^{\pi
ik\widetilde{b}[m]}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}(-Z^{-1}v,-Z^{-1})=e^{\pi
ik\widetilde{b}[m]}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{-m}{0}$}(Z^{-1}v,-Z^{-1}),$
$\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{-m}{0}$}(Z^{-1}v,-Z^{-1})=\sumop\displaylimits_{u\in{\mathbb{Z}}^{r}}e^{\frac{2\pi
ik}{\tau}({v}^{{\sf T}}B^{-1}(u-m)-\frac{1}{2}{(u-m)}^{{\sf
T}}B^{-1}(u-m))}=e^{\frac{\pi
ik}{\tau}A[v]}\sumop\displaylimits_{u\in{\mathbb{Z}}^{r}}e^{-\frac{\pi
ik}{\tau}A[u-m-v]},$

where $A=B^{-1}$ and we used the hypothesis that $Z=\tau B$. To transform the
latter expression, we apply the Jacobi inversion formula (see for example
[Gun62, Chapter VI]):

$\sumop\displaylimits_{u\in{\mathbb{Z}}^{r}}e^{\pi
itA[x+u]}=\frac{1}{\sqrt{(-it)^{r}\det
A}}\sumop\displaylimits_{u\in{\mathbb{Z}}^{r}}e^{-\frac{\pi
i}{t}A^{-1}[u]+2\pi i{x}^{{\sf T}}u}$

for $x\in{\mathbb{R}}^{r},\ t\in{\mathbb{C}},\ \operatorname{Im}\nolimits
t>0,\ A\in M_{r}({\mathbb{R}}),\ {A}^{{\sf T}}=A,\ A>0$. We set $t=-k/\tau$
and $x=-m-v$, and we obtain

$\left(\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}\right)^{\sigma_{2}}(v,Z)=\chi_{0}\sumop\displaylimits_{u\in{\mathbb{Z}}^{r}}e^{\pi
ikZ[\frac{u}{k}]-2\pi ik{(m+v)}^{{\sf
T}}\frac{u}{k}}=\chi_{0}\sumop\displaylimits_{u^{\prime}\in\frac{1}{k}{\mathbb{Z}}^{r}}e^{\pi
ikZ[u^{\prime}]-2\pi ik{(m+v)}^{{\sf T}}u^{\prime}},$

where $\chi_{0}(v,Z)=\frac{e^{\pi ikZ^{-1}[v]}}{\sqrt{(-ik)^{r}\det Z^{-1}}}$.
Now each $u^{\prime}\in\frac{1}{k}{\mathbb{Z}}^{r}$ has a unique
representation $u^{\prime}=u+m^{\prime}$ with $u\in{\mathbb{Z}}^{r}$ and
$m^{\prime}\in P_{k}$, and we obtain

$\left(\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}\right)^{\sigma_{1}\sigma_{2}}(v,Z)=\chi_{0}e^{\pi
ik\widetilde{b}[m]}\sumop\displaylimits_{m^{\prime}\in P_{k}}e^{2\pi
ik{m}^{{\sf
T}}m^{\prime}}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m^{\prime}}{0}$}(v,Z),$

where $\chi_{0}=\chi_{0}(v,Z)$ is the nowhere-vanishing analytic function in
$v,Z$ defined above.

Step 3\. We have

$\left(\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}\right)^{\sigma_{1}\sigma_{2}\sigma_{3}}(v,Z)=\chi_{1}e^{\pi
ik\widetilde{b}[m]}\sumop\displaylimits_{m^{\prime}\in P_{k}}e^{2\pi
ik({m}^{{\sf
T}}m^{\prime}+\frac{1}{2}\widetilde{c}[m^{\prime}])}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m^{\prime}}{0}$}(v,Z),$

where $\chi_{1}(v,Z)=\chi_{0}(v,Z+\widetilde{c})$.

Step 4\. The calculation is similar to that in Step 2:

$\displaystyle\left(\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}\right)^{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}(v,Z)$
$\displaystyle=\chi_{1}(Z^{-1}v,-Z^{-1})e^{\pi
ik\widetilde{b}[m]}\sumop\displaylimits_{m^{\prime}\in P_{k}}e^{2\pi
ik{(m+\frac{1}{2}\widetilde{c}m^{\prime})}^{{\sf
T}}m^{\prime}}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}(Z^{-1}v,-Z^{-1})$
$\displaystyle=\chi_{2}\sumop\displaylimits_{m^{\prime}\in
P_{k}}\sumop\displaylimits_{m^{\prime\prime}\in
P_{k}}\zeta_{m^{\prime},m^{\prime\prime}}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m^{\prime\prime}}{0}$}(v,Z),$

where $\zeta_{m^{\prime},m^{\prime\prime}}=e^{\pi ik\widetilde{b}[m]}e^{2\pi
ik{(m+\frac{1}{2}\widetilde{c}m^{\prime}-m^{\prime\prime})}^{{\sf
T}}m^{\prime}}$ and $\chi_{2}(v,Z)=\chi_{1}(Z^{-1}v,-Z^{-1})\chi_{0}(v,Z)$.

Step 5\.
$\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m}{0}$}^{\gamma}(v,Z)=\chi{\displaystyle\sumop\displaylimits_{m^{\prime},m^{\prime\prime}}}\zeta_{m^{\prime},m^{\prime\prime}}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m^{\prime\prime}}{0}$}^{\left(\begin{smallmatrix}\widetilde{a}&0\\\
0&d\end{smallmatrix}\right)}(v,Z)=\chi{\displaystyle\sumop\displaylimits_{m^{\prime},m^{\prime\prime}}}\zeta_{m^{\prime},m^{\prime\prime}}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{{\widetilde{a}}^{{\sf
T}}m^{\prime\prime}}{0}$}(v,Z)$

$=\chi{\displaystyle\sumop\displaylimits_{m^{\prime},m^{\prime\prime}}}\zeta_{m^{\prime},({\widetilde{a}}^{{\sf
T}})^{-1}m^{\prime\prime}}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m^{\prime\prime}}{0}$}(v,Z)=\chi{\displaystyle\sumop\displaylimits_{m^{\prime\prime}}}\widetilde{u}_{m,m^{\prime\prime}}\theta_{k}\raisebox{2.0pt}{$\genfrac{[}{]}{0.0pt}{2}{m^{\prime\prime}}{0}$}(v,Z),\hphantom{=\;\,}$

where $\widetilde{u}_{m,m^{\prime\prime}}$ is as in the statement of the
theorem and $\chi(v,Z)=\chi_{2}(({d}^{{\sf T}})^{-1}v,({d}^{{\sf
T}})^{-1}Zd^{-1})$. ∎

Together with Igusa’s theorem and Lemma 3.1, Theorem 3.4 obviously implies the
following corollary.

######  Corollary 3.5.

For every even $k\geqslant 0$, the application of Theorem 3.4 to the theta
functions $\theta_{m,k}$ for the lattice introduced in Definition 2.1 provides
a map $g\mapsto\widetilde{U}_{g}$, where $\widetilde{U}_{g}$ is the complex
matrix $(\widetilde{u}_{m,m^{\prime}})$ of size $k^{3}$ defined in the
statement of Theorem 3.4 with $\gamma_{g}$ in place of $\gamma$, and this map
provides a group homomorphism from $G$ to the projective unitary group
$\mathbf{P}\mathbf{U}(k^{3})=\mathbf{U}(k^{3})/\text{$($homotheties$)$}$.

## 4\. Unitary action of $\boldsymbol{G}$ on theta functions and its character

Neither Igusa’s theorem nor our approach applied in Theorem 3.4 allows us to
conclude that the matrices $\widetilde{U}_{g}$ can be normalized by
multiplying by some constants $\epsilon_{g}$ in such a way that the normalized
map $g\mapsto U_{g}=\epsilon_{g}\widetilde{U}_{g}$ is a group homomorphism
$G\to\mathbf{U}(k^{3})$. However, we managed to find convenient constants
$\epsilon_{g}$ by trial and error. We will write
$U_{g}^{(k)},\widetilde{U}_{g}^{(k)}$ when we want to specify the degree $k$
of theta functions on which $U_{g}$, $\widetilde{U}_{g}$ act.

######  Proposition 4.1.

Let $r_{1},r_{2},r_{3}$ be the basic reflections generating $G$ introduced in
Section 2. Set $U_{j}=\frac{1}{k^{3}}\widetilde{U}_{r_{j}}$ for $j=1,2$ and
$U_{3}=\frac{1}{ik^{3}}\widetilde{U}_{r_{3}}$. Then
$U_{1},U_{2},U_{3}\in\mathbf{U}(k^{3})$. We denote by $U_{G}^{(k)}$ the
subgroup of $\mathbf{U}(k^{3})$ generated by these three matrices.

  1. (i)

The $U_{j}$ satisfy the same relations (2.1) as the basic reflections $r_{j}$:

$U_{1}^{2}=U_{2}^{2}=U_{3}^{2}=(U_{1}U_{2})^{4}=(U_{2}U_{3})^{4}=\\\
(U_{3}U_{1})^{3}=(U_{1}U_{2}U_{1}U_{3})^{3}=1.$

  2. (ii)

$U_{G}^{(2)}\simeq H$ and $U_{G}^{(k)}\simeq G$ for all even $k\geqslant 4$.

###### Proof.

We verify this for $k=2,4,6$ by direct computation using the computer algebra
system Macaulay2 [Mac2]; the result for all even $k$ follows from the fact
that the algebra of even-degree theta functions on a p.p.a.v. is generated in
degrees at most $6$. ∎

We thus have a unitary representation $\boldsymbol{\rho}_{k}$ of $G$ on the
space $H^{0}({\pazocal J},{\mathcal{L}}^{k})$ of dimension $k^{3}$ for each
even $k\geqslant 0$, defined by substituting the $U_{j}$ for the $r_{j}$ in
the words in the $r_{j}$ defining all the elements of $G$. We denote by
$\boldsymbol{\chi}_{k}$ the character of this representation. In order to
determine it, we start by fixing the choice of representatives of the
conjugacy classes of $G$. Klein’s simple group $H$ has six conjugacy classes,
represented by the following elements:

$g_{1}=1,\quad g_{2}=\rho_{1},\quad
g_{3}=\rho_{1}\rho_{3}\rho_{1}\rho_{2},\quad g_{4}=\rho_{1}\rho_{2},\quad
g_{7}=\rho_{1}\rho_{2}\rho_{3},\quad g_{7}^{-1},$

where the $\rho_{i}=-r_{i}$ are the basic antireflections and the subscript
$p$ in $g_{p}$ stands for the order of $g_{p}$.

The conjugacy classes of $G$ are deduced from these in an obvious way: to
every conjugacy class $\operatorname{Cl}\nolimits_{H}(g)$ in $H$ correspond
two conjugacy classes in $G$ of the same length:
$\operatorname{Cl}\nolimits_{G}(g)=\operatorname{Cl}\nolimits_{H}(g)$ and
$\operatorname{Cl}\nolimits_{G}(-g)=-\operatorname{Cl}\nolimits_{H}(g)$. Also,
to each irreducible representation $f$ of $H$ correspond two irreducible
representations of $G$, $\widetilde{f}=f\circ\pi$ and
$\widetilde{f}\otimes\det$, where $\pi\colon G\to H\simeq G/\langle-1\rangle$
is the natural surjection.

The lengths of the conjugacy classes of $H$ are given by the following table,
providing the characters of $H$. The characters of $G$ are easily deduced from
it.

$g$ | $g_{1}$ | $g_{2}$ | $g_{3}$ | $g_{4}$ | $g_{7}$ | $g_{7}^{-1}$
---|---|---|---|---|---|---
$|\mathrm{Cl}_{H}(g)|$ | 1 | 21 | 56 | 42 | 24 | 24
$\chi_{\mathbf{1}}$ | 1 | 1 | 1 | 1 | 1 | 1
$\chi_{\mathbf{3}}$ | 3 | $-1$ | 0 | 1 | $-\overline{\alpha}$ | $-\alpha$
$\chi_{\overline{\boldsymbol{3}}}$ | 3 | $-1$ | 0 | 1 | $-\alpha$ | $-\overline{\alpha}$
$\chi_{\mathbf{6}}$ | 6 | 2 | 0 | 0 | $-1$ | $-1$
$\chi_{\mathbf{7}}$ | 7 | $-1$ | 1 | $-1$ | 0 | 0
$\chi_{\mathbf{8}}$ | 8 | 0 | $-1$ | 0 | 1 | 1

######  Theorem 4.2.

For any even $k>0$, the character $\boldsymbol{\chi}_{k}$ of
$\boldsymbol{\rho}_{k}$ takes the following values on the above
representatives of the conjugacy classes:

$g_{1}$ | $-g_{1}$ | $g_{2}$ | $-g_{2}$ | $g_{3}$ | $-g_{3}$ | $g_{4}$ | $-g_{4}$ | $g_{7}$ | $-g_{7}$ |  $g_{7}^{-1}$ |  $-g_{7}^{-1}$
---|---|---|---|---|---|---|---|---|---|---|---
$k^{3}$ | $8$ | $2k$ | $k^{2}$ | $k$ | $2$ | $k$ |  $3+(-1)^{\frac{k}{2}}$ | ${\genfrac{(}{)}{1.0pt}{1}{k}{7}\ {\rm if}\ 7\nmid k,\rule{0.0pt}{11.19443pt}\atop{-i\sqrt{7}\ {\rm if}\ 7\mid k\rule[-3.61664pt]{0.0pt}{0.0pt}}}$ | $1$ | ${\genfrac{(}{)}{1.0pt}{1}{k}{7}\ {\rm if}\ 7\nmid k,\rule{0.0pt}{11.19443pt}\atop{i\sqrt{7}\ {\rm if}\ 7\mid k\rule[-3.61664pt]{0.0pt}{0.0pt}}}$ | $1$

where

$\genfrac{(}{)}{1.0pt}{0}{k}{7}=\left\\{\begin{array}[]{lll}1&\mbox{\rm
if}&k\equiv 1,2\ \mbox{\rm or}\ 4{\,\mathrm{mod}\,}7,\\\ -1&\mbox{\rm
if}&k\equiv 3,5\ \mbox{\rm or}\ 6{\,\mathrm{mod}\,}7,\\\ 0&\mbox{\rm
if}&k\equiv 0{\,\mathrm{mod}\,}7\end{array}\right.$

is the Legendre symbol.

###### Proof.

The result is obvious for $g_{1}$ and needs some reasoning, following the same
pattern, in the other cases. We will illustrate this reasoning on the example
of $g=g_{7}$, where the details of the calculation are the most involved. We
first observe that the normalization constant for $g=g_{7}$ is
$\epsilon_{g}=\frac{1}{ik^{3}}$, so that

$\boldsymbol{\chi}_{k}(g)=\operatorname{tr}\nolimits
U_{g}=\frac{1}{ik^{3}}{}_{k},\quad\text{where}\
{}_{k}=\sumop\displaylimits_{m\in P_{k}}\widetilde{u}_{m,m}.$

By Theorem 3.4,

${}_{k}=\sumop\displaylimits_{m,m^{\prime}\in P_{k}}e^{\pi
ik\left(2{m^{\prime}}^{{\sf
T}}(\operatorname{id}\nolimits-d)m+\widetilde{b}[m]+\widetilde{c}[m^{\prime}]\right)}=\sumop\displaylimits\limits_{0\leqslant
x_{i}\leqslant k-1}e^{\frac{\pi i}{k}K[x]},$

where we pass to the summation over the integer column vector
$x={(x_{1},\ldots,x_{6})}^{{\sf T}}=k{(m^{\prime},m)}^{{\sf
T}}\in{\mathbb{Z}}^{6}$, $K[x]={x}^{{\sf T}}Kx$ as before and $K$ is the
following integer matrix of size 6:

$K=\begin{pmatrix}\widetilde{c}&I_{3}-d\\\ I_{3}-{d}^{{\sf
T}}&\widetilde{b}\end{pmatrix}.$

Explicitly, we have

$\gamma_{g}=\begin{pmatrix}\begin{smallmatrix}-1&0&1\rule{0.0pt}{6.02777pt}\\\
-1&1&0\\\
0&0&1\rule[-2.41112pt]{0.0pt}{0.0pt}\end{smallmatrix}&\vline&\begin{smallmatrix}0&-2&-1\rule{0.0pt}{6.02777pt}\\\
0&-4&-2\\\ -1&-3&-2\rule[-2.41112pt]{0.0pt}{0.0pt}\end{smallmatrix}\\\
\hline\cr\begin{smallmatrix}0&0&0\rule{0.0pt}{4.52083pt}\\\ -1&1&-1\\\
1&-1&2\rule[-4.52083pt]{0.0pt}{0.0pt}\end{smallmatrix}&\vline&\begin{smallmatrix}-1&-1&0\rule{0.0pt}{4.52083pt}\\\
1&0&0\\\
-1&-1&-1\rule[-4.52083pt]{0.0pt}{0.0pt}\end{smallmatrix}\end{pmatrix},\quad
K=\begin{pmatrix}\begin{smallmatrix}0&0&0\rule{0.0pt}{6.02777pt}\\\ 0&1&-1\\\
0&-1&2\rule[-2.41112pt]{0.0pt}{0.0pt}\end{smallmatrix}&\vline&\begin{smallmatrix}2&1&0\rule{0.0pt}{6.02777pt}\\\
-1&1&0\\\ 1&1&2\rule[-2.41112pt]{0.0pt}{0.0pt}\end{smallmatrix}\\\
\hline\cr\begin{smallmatrix}2&-1&1\rule{0.0pt}{4.52083pt}\\\ 1&1&1\\\
0&0&2\rule[-4.52083pt]{0.0pt}{0.0pt}\end{smallmatrix}&\vline&\begin{smallmatrix}\>1\
&2\ &1\rule{0.0pt}{4.52083pt}\\\ 2&4&2\\\
1&2&2\rule[-4.52083pt]{0.0pt}{0.0pt}\end{smallmatrix}\end{pmatrix}.$

By Gauss diagonalization over ${\mathbb{Z}}$, we reduce the quadratic form
$x\mapsto K[x]$ to the diagonal representation $y=(y_{1},\ldots,y_{6})\mapsto
y_{1}^{2}+y_{2}^{2}+y_{3}^{2}-y_{4}^{2}-y_{5}^{2}-7y_{6}^{2}$. Thus

k $\displaystyle=\sumop\displaylimits_{0\leqslant x_{i}\leqslant
k-1}e^{\frac{\pi i}{k}K[x]}=\sumop\displaylimits_{0\leqslant y_{i}\leqslant
k-1}e^{\frac{\pi
i}{k}(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}-y_{4}^{2}-y_{5}^{2}-7y_{6}^{2})}$
$\displaystyle=\left(\sumop\displaylimits_{y=0}^{k-1}e^{\frac{\pi
i}{k}y^{2}}\right)^{3}\left(\sumop\displaylimits_{y=0}^{k-1}e^{-\frac{\pi
i}{k}y^{2}}\right)^{2}\left(\sumop\displaylimits_{y=0}^{k-1}e^{-\frac{7\pi
i}{k}y^{2}}\right).$

The exponential sums in the last line belong to the class of Gauss sums. We
use the following formula for Gauss sums from [BEW98]:

$G(q,r)=\sumop\displaylimits_{n=0}^{r-1}e^{\frac{2\pi
iq}{r}n^{2}}=(1+i)\kappa_{q}^{-1}\sqrt{r}\genfrac{(}{)}{1.0pt}{0}{r}{q}$

if $2\nmid q,\ 4|r$ and gcd$(q,r)=1$, where
$\kappa_{q}=\left\\{\begin{array}[]{lll}1&\text{if}&q\equiv
1{\,\mathrm{mod}\,}4,\\\ i&\text{if}&q\equiv
3{\,\mathrm{mod}\,}4\end{array}\right.$ and $\genfrac{(}{)}{1.0pt}{0}{r}{q}$
is the Jacobi symbol (which coincides with the Legendre symbol when
$q\rule{0.0pt}{12.91663pt}$ is prime). Applying this formula, we obtain

$\displaystyle\sumop\displaylimits_{y=0}^{k-1}e^{\frac{\pi
i}{k}y^{2}}=\frac{1}{2}G(1,2k)=\frac{1+i}{\sqrt{2}}\sqrt{k},$
$\displaystyle\sumop\displaylimits_{y=0}^{k-1}e^{-\frac{7\pi
i}{k}y^{2}}=\begin{cases}\displaystyle{\frac{1}{2}\overline{G(7,2k)}=\frac{1+i}{\sqrt{2}}\genfrac{(}{)}{1.0pt}{0}{k}{7}\sqrt{k}}&\text{if}\
7\nmid k,\\\\[12.91663pt]
\displaystyle{\frac{7}{2}\,\overline{G(1,2k_{1})}=\frac{1-i}{\sqrt{2}}\sqrt{7k}}&\text{if}\
k=7k_{1}.\end{cases}$

We thus finish the calculation of k and obtain the value of
$\boldsymbol{\chi}_{k}(g)$ given in the table. ∎

By taking the scalar product of $\boldsymbol{\chi}_{k}$ with the trivial
character, we obtain the following.

######  Theorem 4.3.

The Hilbert function of the algebra of invariant theta functions

$S\left({\mathcal{L}}^{2}\right)^{G}=\bigoplusop\displaylimits_{p=0}^{\infty}H^{0}\left({\pazocal
J},{\mathcal{L}}^{2p}\right)^{G}$

is given by

$h_{S({\mathcal{L}}^{2})^{G}}(\tfrac{k}{2})=\tfrac{1}{336}\left[k^{3}+21\,k^{2}+140\,k+294+(-1)^{\frac{k}{2}}\times
42+48\genfrac{(}{)}{1.0pt}{1}{k}{7}\right].$

This coincides with the Hilbert function of the second Veronese algebra of
${\mathbb{P}}(1,2,4,7)$.

###### Proof.

The formula for the Hilbert function is an immediate consequence of Theorem
4.2. The Hilbert series for ${\mathbb{P}}={\mathbb{P}}(1,2,4,7)$ is given by

${\mathpzc h}_{\mathbb{P}}(t)=\frac{1}{(1-t)(1-t^{2})(1-t^{4})(1-t^{7})},$

and that of the second Veronese of ${\mathbb{P}}$ is

${\mathpzc h}^{(2)}_{\mathbb{P}}(t)=\frac{1}{2}\left({\mathpzc
h}_{\mathbb{P}}\left(t^{1/2}\right)+{\mathpzc
h}_{\mathbb{P}}\left(-t^{1/2}\right)\right)=\frac{1+t^{4}}{\left(1-t\right)^{2}\left(1-t^{2}\right)\left(1-t^{7}\right)}.$

Denote by $h_{p}$ the coefficients of the latter series, so that ${\mathpzc
h}^{(2)}_{\mathbb{P}}(t)=\sumop\displaylimits_{p\geqslant 0}h_{p}t^{p}$; by
[Sta86, Theorem 4.4.1], the Hilbert function $p\mapsto h_{p}$ is a rational
quasi-polynomial of degree $3$ whose coefficients are periodic with period 14.
The dominant coefficient being constant, it suffices to compare the initial
segments of the sequences $h_{S({\mathcal{L}}^{2})^{G}}(p)$ and $h_{p}$ of
length 42 to see that they coincide for all $p\geqslant 0$. In this way, we
conclude the proof. ∎

## 5\. Hypersurface in a weighted projective space of dimension 4

For any theta function $\theta$ of even degree $k>0$, we denote by $\pazocal
R_{G}^{(k)}$ the Reynolds operator

(5.1) $\pazocal R_{G}^{(k)}(\theta)=\frac{1}{336}\sumop\displaylimits_{g\in
G}U_{g}^{(k)}(\theta).$

We will call $\pazocal R_{G}^{(k)}(\theta)$ the $G$-average of $\theta$; for
given $k$, the $G$-averages of theta functions of degree $k$ represent
sections of ${\mathcal{L}}^{k}$ which generate $H^{0}({\pazocal
J},{\mathcal{L}}^{k})^{G}$ over ${\mathbb{C}}$.

We will identify sections of ${\mathcal{L}}^{k}$ with regular functions on the
total space $\mathbf{V}({\mathcal{L}}^{-1})$ of the line bundle
$\mathcal{L}^{-1}$ which are fiber-homogeneous of degree $k$. Explicitly,
$\mathbf{V}({\mathcal{L}}^{-1})$ is the quotient of the trivial line bundle
${\mathbb{C}}\times V$ over $V={\mathbb{C}}^{3}$ by the action of
$\Lambda\simeq\\{Zu^{\prime}+u^{\prime\prime}\ |\
u^{\prime},u^{\prime\prime}\in{\mathbb{Z}}^{3}\\}$:

$Zu^{\prime}+u^{\prime\prime}\colon(t,v)\longmapsto\left(e^{2\pi
i\left({v}^{{\sf T}}u^{\prime}+\frac{1}{2}Z[u^{\prime}]\right)}t,\
v+Zu^{\prime}+u^{\prime\prime}\right),$

and to each degree $k$ theta function $\theta$, we associate the -invariant
function

(5.2) $\widetilde{\theta}\colon{\mathbb{C}}\times
V\to{\mathbb{C}},\quad(t,v)\longmapsto t^{k}\theta(v),$

which descends to a fiber-homogeneous function of degree $k$ on
$\mathbf{V}({\mathcal{L}}^{-1})$ denoted by the same symbol
$\widetilde{\theta}$.

######  Theorem 5.1.

There exist five $G$-invariant theta functions $\varphi_{i}=\varphi_{i,k_{i}}$
of degrees $k_{i}$, where $(k_{0},\ldots,k_{4})=(2,2,4,8,14)$, such that the
associated sections of the powers ${\mathcal{L}}^{k_{i}}$ of ${\mathcal{L}}$
generate $S({\mathcal{L}}^{2})^{G}$. The ideal of relations between these
generators is generated by a single relation of weighted degree $16$, so that
the quotient $X={\pazocal J}/G$ is isomorphic to a hypersurface in the
weighted projective space ${\mathbb{P}}(1,1,2,4,7)$ of weighted degree $8$.

###### Proof.

It suffices to show that we can find $G$-invariant theta functions
$\varphi_{0},\varphi_{1},\varphi_{2},\varphi_{3}$ of respective degrees
$2,2,4,8$ such that the associated fiber-homogeneous functions
$\widetilde{\varphi}_{i}$ on $\mathbf{V}({\mathcal{L}}^{-1})$ are
algebraically independent. This is done below in Lemma 5.2. Then the functions
$\varphi_{i}$, $i=0,\ldots,3$, generate the free polynomial subalgebra
${\mathcal{S}}^{\prime}$ in ${\mathcal{S}}=S({\mathcal{L}}^{2})^{G}$, and
comparing the initial segments of their Hilbert functions,

$(h_{{\mathcal{S}}}(p))_{p\geqslant 0}=(1,2,4,6,10,14,20,27,36,46,58,\ldots),$
$(h_{{\mathcal{S}}^{\prime}}(p))_{p\geqslant
0}=(1,2,4,6,10,14,20,26,35,44,56,\ldots),$

we see that the four $\varphi_{i}$ generate ${\mathcal{S}}$ in degrees less
than $7$ and $h_{{\mathcal{S}}}(7)=h_{{\mathcal{S}}^{\prime}}(7)+1$, so that
one extra generator of degree $7$, say $\varphi_{4}$, is needed to generate
${\mathcal{S}}$ up to degree $7$. Furthermore, there are 35 monomials in the
$\varphi_{i}$, $i=0,\ldots,3$, of degree $8$, and two more monomials
$\varphi_{0}\varphi_{4}$, $\varphi_{1}\varphi_{4}$ involving $\varphi_{4}$,
and as $h_{{\mathcal{S}}}(7)=36$ is one less than the number of monomials of
degree $8$, there is precisely one linear relation $F_{8}=0$ between the 37
monomials of degree $8$. One easily verifies that
$h_{{\mathcal{S}}}(p)=h_{R}(p)-h_{R}(p-8)$ for all $p\in{\mathbb{Z}}$, where
$R$ denotes a free polynomial algebra with generators of degrees $1,1,2,4,7$,
so that the natural map $R/F_{8}R\to{\mathcal{S}}$ is an isomorphism. Thus $X$
is isomorphic to a hypersurface $F_{8}=0$ of degree $8$ in the weighted
projective space ${\mathbb{P}}(1,1,2,4,7)=\operatorname{Proj}\nolimits R$. ∎

For the algebraic independence of
$\widetilde{\varphi}_{0},\ldots,\widetilde{\varphi}_{3}$, it is necessary and
sufficient that the Jacobian $J=J(\widetilde{\varphi}_{i})$ is not identically
zero. We will present explicitly an ad hoc example of $\varphi_{i}(v)$ lying
in the images of the Reynolds operators $\pazocal R_{G}^{(k_{i})}$ and
satisfying this property; some motivation for the choice of the example is
given in Remark 7.1. We will verify that $J(t_{0},v_{0})\neq 0$ at a specific
point $(t_{0},v_{0})$ by computing the differentials $\hskip
1.29167pt\mathrm{d}\widetilde{\varphi}_{i}(t_{0},v_{0})$ approximately as
partial sums of their Fourier series, which converge very rapidly. We recall
that the period matrix of ${\pazocal J}$ is $Z=\tau B$, where $B$ is given by
(2.4) and $\tau=\frac{3+i\sqrt{7}}{2}$, and we set $q=e^{2\pi
i\tau}=-e^{-\pi\sqrt{7}}$; we also adopt the convention that $q^{r}=e^{2\pi
i\tau r}$ for any $r\in{\mathbb{Q}}$.

######  Lemma 5.2.

Let the vectors $\xi_{i}$ ($i=0,\ldots,3$) of ${\mathbb{Z}}^{3}$ be defined by

$\xi_{0}=0,\quad\xi_{1}=\frac{1}{2}(0,0,1),\quad\xi_{2}=\frac{1}{4}(1,1,0),\quad\xi_{3}=\frac{1}{8}(2,1,1),$

the first three being equal to the vectors $\mu_{i}$ in (7.1) in Section 7,
with the basis $(b_{i})$ of $M$ used to identify $M$ with ${\mathbb{Z}}^{3}$,
and $\xi_{3}=\frac{7}{8}\mu_{3}$. Define four $G$-invariant theta functions on
${\pazocal J}$ by

$\varphi_{0}=\pazocal
R_{G}^{(2)}(\theta_{\xi_{0},2}),\quad\varphi_{1}=\pazocal
R_{G}^{(2)}(\theta_{\xi_{1},2}),\quad\varphi_{2}=\pazocal
R_{G}^{(4)}(\theta_{\xi_{2},4}),\quad\varphi_{3}=\pazocal
R_{G}^{(8)}(\theta_{\xi_{3},8}),$

where the operators $\pazocal R_{G}^{(k)}$ are defined in (5.1). Associate to
the $\varphi_{i}$ the -invariant functions $\widetilde{\varphi}_{i}$ on
${\mathbb{C}}\times V$, as in (5.2), and denote by $J$ the Jacobian
$J(\widetilde{\varphi}_{i})$. Set
$(t_{0},v_{0})=\left(1,\left(\frac{1}{8},\frac{1}{16},\frac{1}{4}\right)\right)$.
Then $J(t_{0},v_{0})\neq 0$.

###### Proof.

Let us choose $P_{k}=\left\\{\tfrac{\nu}{k}\ |\
\nu\in\\{0,1,\ldots,k-1\\}^{3}\right\\}$ for a set of representatives of
$\frac{1}{k}{\mathbb{Z}}^{3}/{\mathbb{Z}}^{3}$. We have

$\theta_{\xi_{i},k_{i}}=\sumop\displaylimits_{u\in{\mathbb{Z}}^{3}+\xi_{i}}q^{\frac{k_{i}}{2}B\left[u\right]}e^{2k_{i}\pi
i{v}^{{\sf T}}u},\quad i=0,1,2,3,\ (k_{i})=(2,2,4,8).$

We compute the $G$-averages of these four theta functions, using formula
(5.1). By Proposition 4.1 and Theorem 3.4, the elements
$u^{(g,k_{i})}_{m^{\prime},m}$ ($m^{\prime},m\in P_{k_{i}}$) of the matrices
$U_{g}^{(k_{i})}$ belong to the cyclotomic field
${\mathbb{Q}}\left(e^{\frac{\pi i}{k_{i}}}\right)$. We introduce the matrices
of the Reynolds operators on the theta functions of degree $k_{i}$:

$\pazocal R_{G}^{(k)}=\left(r^{(k)}_{m^{\prime},m}\right)_{m^{\prime},m\in
P_{k}},\quad r^{(k)}_{m^{\prime},m}=\frac{1}{336}\sumop\displaylimits_{g\in
G}u^{(g,k)}_{m^{\prime},m},\quad k=2,4,8.$

The exact values of the elements of the matrices $U_{g}^{(k_{i})}$, belonging
to ${\mathbb{Q}}\left(e^{\frac{\pi i}{8}}\right)$, and the resulting Reynolds
matrices were computed with Macaulay2; see [Mac2]. We have

$\varphi_{i}=\sumop\displaylimits_{m\in
P_{k_{i}}}r^{(k_{i})}_{\xi_{i},m}\theta_{m,k_{i}}=\sumop\displaylimits_{u\in\frac{1}{k_{i}}{\mathbb{Z}}^{3}}r^{(k_{i})}_{\xi_{i},u}q^{\frac{k_{i}}{2}B\left[u\right]}e^{2k_{i}\pi
i{v}^{{\sf T}}u},$

where $r^{(k_{i})}_{\xi_{i},u}$ is defined to be $r^{(k_{i})}_{\xi_{i},m}$ for
the unique $m\in P_{k_{i}}$ such that $u\equiv
m{\,\mathrm{mod}\,}{\mathbb{Z}}^{3}$. Replacing further $u$ with
$\frac{1}{k_{i}}u$ with $u$ running over ${\mathbb{Z}}^{3}$ and
differentiating, we obtain

$\displaystyle\partial\widetilde{\varphi}_{i}/\partial
t=2k_{i}t^{k_{i}-1}\sumop\displaylimits_{u\in{\mathbb{Z}}^{3}}q^{\frac{1}{2k_{i}}B\left[u\right]}r^{(k_{i})}_{\xi_{i},\frac{1}{k}u}e^{2\pi
i{v}^{{\sf T}}u},$ $\displaystyle\partial\widetilde{\varphi}_{i}/\partial
v_{j}=2\pi
it^{k_{i}}\sumop\displaylimits_{u\in{\mathbb{Z}}^{3}}q^{\frac{1}{2k_{i}}B\left[u\right]}r^{(k_{i})}_{\xi_{i},\frac{1}{k}u}u_{j}e^{2\pi
i{v}^{{\sf T}}u},\quad j=1,2,3.$

As $B$ is positive definite, these infinite sums of powers of $q$ contain only
a finite number of summands $N_{i}(c)$ with exponent at most $c$ for any given
constant $c$, $N_{i}(c)\sim\frac{4\pi}{3\sqrt{\det
B}}(2k_{i}c)^{\frac{3}{2}}$, and, since $|q|<1$, the sums converge uniformly
on compact subsets of ${\mathbb{C}}\times V$. The convergence is in fact very
rapid. For example, in the case of the slowest convergence, when
$i=3,k_{i}=8$, if we stop summation at the terms of order
$q^{7/2}\approx-2.3211\,i\cdot 10^{-13}$, then $N_{3}(3.5)=3527$, and all the
vectors $u\in{\mathbb{Z}}^{3}$ occurring in the truncated sum are in the block
$|u_{1}|\leqslant 10,|u_{2}|\leqslant 10,|u_{3}|\leqslant 14$. Computing the
determinant of the thus obtained approximate Jacobian matrix at the point
$(t_{0},v_{0})$, we obtain $J(t_{0},v_{0})\approx
0.000064967853+0.000075028580\,i$, all the shown decimal digits being exact. ∎

## 6\. Degree 8 hypersurfaces in $\boldsymbol{{\mathbb{P}}(1,1,2,4,7)}$ with
correct singularities

The goal of this section is to show that a hypersurface in
${\mathbb{P}}(1,1,2,4,7)$ defined by a degree 8 homogeneous polynomial whose
singularities are those of ${\mathbb{P}}(1,2,4,7)$ is actually isomorphic to
${\mathbb{P}}(1,2,4,7)$. To achieve this, we classify the degree $8$
hypersurfaces in ${\mathbb{P}}(1,1,2,4,7)$ whose singularities are those of
${\mathbb{P}}(1,2,4,7)$.

Recall that ${\mathbb{P}}(1,2,4,7)$ embeds in ${\mathbb{P}}(1,1,2,4,7)$ as the
degree $8$ hypersurface given by the equation

$F_{8}^{0}=y_{3}^{2}-y_{0}y_{4}.$

Its singular locus is the union of two irreducible components,
${\mathbb{P}}^{1}=\ell$ and an isolated point $p$. The singularity at $p$ is
of analytic type $\frac{1}{7}(1,2,4)$. Here, for a cyclic group
${\boldsymbol{\mu}}_{d}$ of order $d$, we denote by
$\frac{1}{d}(\nu_{1},\nu_{2},\nu_{3})$ the (analytic equivalence class of the)
cyclic quotient singularity ${\mathbb{C}}^{3}/{\boldsymbol{\mu}}_{d}$, where
the generator $c_{d}$ of ${\boldsymbol{\mu}}_{d}$ acts by
$c_{d}\colon(z_{1},z_{2},z_{3})\mapsto(\epsilon^{\nu_{1}}z_{1},\epsilon^{\nu_{2}}z_{2},\epsilon^{\nu_{3}}z_{3})$,
$\epsilon=\exp\left(\frac{2\pi i}{d}\right)$. At all but one point of $\ell$,
the singularity of $X$ is of type $\frac{1}{2}(1,0,1)$, that is,
${\mathbb{C}}\times A_{1}$, the Cartesian product of ${\mathbb{C}}$ with a
surface singularity ${\mathbb{C}}^{2}/\langle-1\rangle$ of type $A_{1}$. The
type of the singularity at $q$, the unique point of $\ell$ where the type of
singularity changes, is $\frac{1}{4}(1,2,3)$.

Let us describe in affine charts the hypersurface $\\{F_{8}^{0}=0\\}$ in
${\mathbb{P}}(1,1,2,4,7)$. Write $y_{0},\ldots,y_{4}$ for the coordinates in
${\mathbb{P}}(1,1,2,4,7)$ with weights

$n_{0}=1,\quad n_{1}=1,\quad n_{2}=2,\quad n_{3}=4,\quad n_{4}=7.$

Set $y_{i}=1$, and quotient ${\mathbb{C}}^{4}$ with coordinates
$(y_{0},\ldots,\widehat{y_{i}},\ldots,y_{4})$ by the action of the cyclic
group ${\boldsymbol{\mu}}_{n_{i}}$ with weights
$(n_{0},\ldots,\widehat{n_{i}},\ldots,n_{4})$ to obtain the affine chart
$U_{i}$ of ${\mathbb{P}}(1,1,2,4,7)$. Note that $U_{0}$ and $U_{1}$ are smooth
charts isomorphic to ${\mathbb{C}}^{4}$ since $n_{0}=n_{1}=1$.

The restriction of $F_{8}^{0}$ to $U_{0}$ is $\\{y_{3}^{2}-y_{4}=0\\}$. The
hypersurface $F_{8}^{0}|_{U_{0}}=0$ is a smooth hypersurface parameterized by
$(y_{1},y_{2},y_{3})$, isomorphic to ${\mathbb{C}}^{3}$.

The restriction of $F_{8}^{0}$ to $U_{1}$ is $y_{3}^{2}-y_{0}y_{4}$. The
singular locus of $F_{8}^{0}|_{U_{1}}=0$ is the line $y_{0}=y_{3}=y_{4}$ with
type ${\mathbb{C}}\times A_{1}$.

The restriction of $F_{8}^{0}$ to $U_{2}$ is
$\\{y_{3}^{2}-y_{0}y_{4}=0\\}/\frac{1}{2}(1,1,0,1)$. Let us consider the
quotient ${\mathbb{C}}^{3}/\frac{1}{2}(1,0,1)$, where the coordinates of
${\mathbb{C}}^{3}$ are denoted by $(u,v,w)$. One can identify this quotient
with the quadric $y_{0}y_{4}-y_{3}^{2}=0$ in ${\mathbb{C}}^{4}$ through the
mapping $(u,v,w)\mapsto(y_{0}=u^{2},y_{1}=v,y_{3}=uw,y_{4}=w^{2})$. From this
we observe that $\\{y_{0}y_{4}-y_{3}^{2}=0\\}/\frac{1}{2}(1,1,0,1)$ is the
same as ${\mathbb{C}}^{3}/\frac{1}{4}(1,2,3)$. As a result, the restriction of
$F_{8}^{0}$ to $U_{2}$ is isomorphic to ${\mathbb{C}}^{3}/\frac{1}{4}(1,2,3)$.
Note that this is the affine chart $x_{2}=1$ in ${\mathbb{P}}(1,2,4,7)$ with
coordinates $(x_{0},x_{1},x_{2},x_{3})$.

The restriction of $F_{8}^{0}$ to $U_{3}$ is
$\\{y_{0}y_{4}=1\\}/\frac{1}{4}(1,1,2,3)$. This chart is contained in
$U_{0}\cup U_{4}$.

The restriction of $F_{8}^{0}$ to $U_{4}$ is
$\\{y_{3}^{2}-y_{0}=0\\}/\frac{1}{7}(1,1,2,4)$, which is isomorphic to
${\mathbb{C}}^{3}/\frac{1}{7}(1,2,4)$, that is, the affine chart $x_{3}=1$ in
${\mathbb{P}}(1,2,4,7)$.

Let us now consider a hypersurface $X$ in ${\mathbb{P}}(1,1,2,4,7)$ defined by
any degree $8$ homogeneous equation $F_{8}=0$ such that the singularities of
$X$ are those of ${\mathbb{P}}(1,2,4,7)$.

######  Lemma 6.1.

Write

$\displaystyle F_{8}=\;$ $\displaystyle
y_{4}\varphi_{1}(y_{0},y_{1})+c_{0}y_{3}^{2}+c_{1}y_{2}^{2}y_{3}+c_{2}y_{2}^{4}+y_{2}y_{3}\varphi_{2}(y_{0},y_{1})+y_{2}^{3}\psi_{2}(y_{0},y_{1})$
$\displaystyle+y_{3}\varphi_{4}(y_{0},y_{1})+y_{2}^{2}\psi_{4}(y_{0},y_{1})+y_{2}\varphi_{6}(y_{0},y_{1})+\varphi_{8}(y_{0},y_{1}),$

with $c_{j}\in{\mathbb{C}}$ and $\varphi_{i},\psi_{i}$ homogeneous polynomials
of degree $i$. Then both $c_{0}$ and $\varphi_{1}$ are non-zero.

###### Proof.

First assume that $c_{0}=0$. Then $Q_{3}=(0,0,0,1,0)$ is in $X$. It is the
origin of the chart $U_{3}={\mathbb{C}}^{4}/{\frac{1}{4}(1,1,2,3)}$ of
${\mathbb{P}}(1,1,2,4,7)$. Denote by
$\widetilde{X}_{3}\subseteq{\mathbb{C}}^{4}$ the hypersurface defined by
$\\{F_{8}|_{y_{3}=1}=0\\}$, and let $\widetilde{Q}_{3}\in\widetilde{X}_{3}$ be
such that the image of $\widetilde{Q}_{3}$ in the quotient
${\mathbb{C}}^{4}/{{\boldsymbol{\mu}}_{4}}$ is $Q_{3}$. The germ of the
canonical sheaf $\omega_{\widetilde{X}_{3},\widetilde{Q}_{3}}$ is generated by

$\sigma={\rm res}_{\widetilde{X}_{3}}\left(\dfrac{dy_{0}\wedge dy_{1}\wedge
dy_{2}\wedge dy_{4}}{F_{8}|_{y_{3}=1}}\right),$

of weight $-1$ for the ${\boldsymbol{\mu}}_{4}$-action. So the Gorenstein
index of $\widetilde{X}_{3}/{\frac{1}{4}(1,1,2,3)}=X|_{U_{3}}$ in $Q_{3}$ is
$4$. But $X$ has no singularity of index $4$. We conclude that $c_{0}$ is non-
zero.

Now assume that $\varphi_{1}=0$. Set $X_{4}=X|_{U_{4}}$. We have
$X_{4}=\widetilde{X}_{4}/\frac{1}{7}(1,1,2,4)$, where
$\widetilde{X}_{4}\subset{\mathbb{C}}^{4}$ is the hypersurface defined by the
equation $\\{F_{8}|_{y_{4}=1}=0\\}$. Under our assumption, $F_{8}$ does not
contain $y_{4}$, so $F_{8}|_{y_{4}=1}=F_{8}$ defines the affine cone of an
octic surface in ${\mathbb{P}}(1,1,2,4)$ with coordinates
$y_{0},y_{1},y_{2},y_{3}$. The cone has to be smooth outside of its vertex
$O$, the origin of the affine space ${\mathbb{C}}^{4}$ with the same
coordinates. Indeed, assume the contrary. Then $O$ is a non-isolated
singularity, so the image $Q_{4}$ of $O$ in $X_{4}$ also is a non-isolated
singularity. Hence the unique isolated cyclic quotient singularity $p\in X$ of
type $\frac{1}{7}(1,2,4)$ is different from $Q_{4}$. Moreover, $p\in X$ is not
a hypersurface singularity, but all the singularities of $X\setminus Q_{4}$
are hypersurface, so $p$ cannot be located in the chart $U_{4}$. Then, looking
at the singularities of $X$ in the other charts, we see that all of them are
hypersurface ones, except possibly points on the axis $(y_{2},y_{3})$. By
proving that $c_{0}\neq 0$, we have excluded the possibility that $Q_{3}$
belongs to $X$, so the only non-hypersurface singularity that may occur in a
chart $U_{i}$ with $i\neq 4$ is of the type (hypersurface in
${\mathbb{C}}^{4}$)$/\frac{1}{2}(1,1,0,1)$. The embedding dimension of such a
singularity is at most that of ${\mathbb{C}}^{4}/\frac{1}{2}(1,1,0,1)$, which
is equal to $7$, but the embedding dimension of $p$ is $12$. Hence no point of
a chart $U_{i}$ for $i\neq 4$ can fit the role of $p$, so this case is
impossible, and the singularity of $\widetilde{X}_{4}$ at $O$ is an isolated
quasi-homogeneous singularity.

It remains to see that under this assumption, the quotients
$\\{F_{8}=0\\}/\frac{1}{7}(1,1,2,4)$ and ${\mathbb{C}}^{3}/\frac{1}{7}(1,2,4)$
cannot be isomorphic. This follows from the fact that the first one is non-
canonical, while the second one is canonical. Indeed,
$\widetilde{X}_{4}=\\{F_{8}=0\\}$ is non-canonical by Reid’s criterion of
canonicity (cf. [Rei80, Theorem 4.1]) with monomial valuation $\alpha$ defined
by $(\alpha(y_{i}))=(\frac{1}{8},\frac{1}{8},\frac{1}{4},\frac{1}{2})$; this
implies that $X_{4}=\widetilde{X}_{4}/\mu_{7}$ is non-canonical (cf. [Rei80,
Proposition 1.7]). The canonicity of the second singularity follows from
[Rei80, Theorem 3.1 and Remark 3.2]. We conclude that $\varphi_{1}$ is non-
zero. ∎

By Lemma 6.1, one can assume that $c_{0}\neq 1$ and use the change of
coordinates

$y_{3}\longmapsto
d_{0}y_{3}+d_{1}y_{2}^{2}+y_{2}f_{2}(y_{0},y_{1})+f_{4}(y_{0},y_{1}),$

with $d_{j}\in{\mathbb{C}}$, $d_{0}\in{\mathbb{C}}^{*}$ and the $f_{i}$
homogeneous polynomials of degree $i$, to normalize the value of $c_{0}$ and
to kill the terms containing $y_{3}$ to the power of $1$. We obtain for
$F_{8}$ an expression as in the lemma, but with the constraints

$c_{0}=-1,\quad c_{1}=0,\quad\varphi_{2}=0,\quad\varphi_{4}=0.$

Next, again by the lemma, $\varphi_{1}\neq 0$, and we can change the variables
$y_{0},y_{1}$ in such a way that in new variables, we have
$\varphi_{1}=y_{0}$, so that there is no monomial $y_{1}y_{4}$ in the
expression for $F_{8}$. Then, we use the change of coordinates

$y_{4}\longmapsto
y_{4}+y_{2}^{3}g_{1}(y_{0},y_{1})+y_{2}^{2}g_{3}(y_{0},y_{1})+y_{2}g_{5}(y_{0},y_{1})+g_{7}(y_{0},y_{1}),$

with $g_{i}$ homogeneous of degree $i$, to kill all the monomials divisible by
$y_{0}$. As a result, we obtain an expression of the form

(6.1)
$F_{8}=y_{0}y_{4}-y_{3}^{2}+a_{1}y_{2}^{4}+a_{2}y_{2}^{3}y_{1}^{2}+a_{3}y_{2}^{2}y_{1}^{4}+a_{4}y_{2}y_{1}^{6}+a_{5}y_{1}^{8},\quad
a_{i}\in{\mathbb{C}}.$

Set $\nu=\min\\{i\colon a_{i}\neq 0\\}$. Finally, using changes of coordinates
of the form $y_{1}\mapsto\kappa y_{1}$, $y_{2}\mapsto\lambda y_{2}+\mu
y_{1}^{2}$, $\kappa,\lambda\in{\mathbb{C}}^{*}$, $\mu\in{\mathbb{C}}$, we can
reduce $F_{8}$ to one of the following normal forms:

(6.2)
$\begin{array}[]{lllll}&\nu=\infty\colon&&F_{8}=y_{0}y_{4}-y_{3}^{2},&\\\
&\nu=5\colon&&F_{8}=y_{0}y_{4}-y_{3}^{2}+y_{1}^{8},&\\\
&\nu=4\colon&&F_{8}=y_{0}y_{4}-y_{3}^{2}+y_{2}y_{1}^{6},&\\\
&\nu=3\colon&&F_{8}=y_{0}y_{4}-y_{3}^{2}+(y_{2}^{2}+a_{5}y_{1}^{4})y_{1}^{4},&\\\
&\nu=2\colon&&F_{8}=y_{0}y_{4}-y_{3}^{2}+(y_{2}^{3}+a_{4}y_{2}y_{1}^{4}+a_{5}y_{1}^{6})y_{1}^{2},&\\\
&\nu=1\colon&&F_{8}=y_{0}y_{4}-y_{3}^{2}+y_{2}^{4}+(a_{3}y_{2}^{2}+a_{4}y_{2}y_{1}^{2}+a_{6}y_{1}^{4})y_{1}^{4}.&\end{array}$

######  Proposition 6.2.

Except for the case $\nu=\infty$, the degree $8$ hypersurface $X$ defined by
$F_{8}=0$ in ${\mathbb{P}}(1,1,2,4,7)$, with $F_{8}$ one of the above normal
forms, has a singularity of a type different from the types of singularities
of ${\mathbb{P}}(1,2,4,7)$.

###### Proof.

Let $\nu=5$. Then $X$ passes through the origin of the chart $U_{2}$. We have
$U_{2}={\mathbb{C}}^{4}/\frac{1}{2}(1,1,0,1)$, and $X_{2}:=X\cap U_{2}$ is the
quotient $\widetilde{X}_{2}/\frac{1}{2}(1,1,0,1)$, where
$\widetilde{X}_{2}=\\{y_{0}y_{4}-y_{3}^{2}+y_{1}^{8}=0\\}\subset{\mathbb{C}}^{4}$.
Since $y_{0}y_{4}-y_{3}^{2}+y_{1}^{8}$ is ${\boldsymbol{\mu}}_{2}$-invariant
and the differential $dy_{0}\wedge dy_{1}\wedge dy_{3}\wedge dy_{4}$ is not,
the generator

${\rm res}_{\widetilde{X}_{2}}\left(\dfrac{dy_{0}\wedge dy_{1}\wedge
dy_{3}\wedge dy_{4}}{y_{0}y_{4}-y_{3}^{2}+y_{1}^{8}}\right)$

of $\omega_{\widetilde{X}_{2},O}$, where $O$ is the origin of
${\mathbb{C}}^{4}$, is anti-invariant under ${\boldsymbol{\mu}}_{2}$. So
$Q_{2}$, the image of $O$ in $U_{2}$, is a singular point of Gorenstein index
$2$. But in ${\mathbb{P}}(1,2,4,7)$, the only point of Gorenstein index $2$ is
the origin of the chart $x_{2}=1$, with singularity $\frac{1}{4}(1,2,3)$. The
latter singularity is non-isolated: the whole image of the coordinate axis
with weight $2$ consists of singular points of the quotient. Hence it cannot
be equivalent to the isolated singularity
$\\{y_{0}y_{4}-y_{3}^{2}+y_{1}^{8}=0\\}/{\tiny\frac{1}{2}(1,1,0,1)}$. Thus the
case $\nu=5$ is impossible.

The cases $2\leqslant\nu\leqslant 5$ are all treated in the same way: in all
of them, we find an isolated singularity of Gorenstein index $2$ at the origin
of the chart $U_{2}$, which is impossible.

Let us now look at the case $\nu=1$. In the chart $U_{2}$, we have

$X\cap{U_{2}}=\widetilde{X}_{2}/\tfrac{1}{2}(1,1,0,1),$

where $\widetilde{X}_{2}\subset{\mathbb{C}}^{4}$ is defined by the equation
$\widetilde{F}=0$ with
$\widetilde{F}=y_{0}y_{4}-y_{3}^{2}+1+a_{3}y_{1}^{4}+a_{4}y_{1}^{6}+a_{5}y_{1}^{8}$.
The singular locus of $\widetilde{X}_{2}$ is given by $y_{0}=y_{3}=y_{4}=0$,
$y_{1}^{2}=\gamma$, where $\gamma$ is a double root of
$a_{5}t^{4}+a_{4}t^{3}+a_{3}t^{2}+1=0$. But this polynomial cannot have double
roots. Indeed, otherwise $X$ would have singular points in the chart $U_{1}$,
which are of the type of isolated hypersurface singularities:

$X\cap
U_{1}=\\{y_{0}y_{4}-y_{3}^{2}+y_{2}^{4}+a_{3}y_{2}^{2}+a_{4}y_{2}+a_{5}=0\\}\subset{\mathbb{C}}^{4}.$

This contradicts the fact that the only isolated Gorenstein singularity of
${\mathbb{P}}(1,2,4,7)$ is of type $\frac{1}{7}(1,2,4)$, and this is not a
hypersurface singularity.

Therefore, $\widetilde{X}_{2}$ is smooth, and the singularities of
$\widetilde{X}_{2}/{\boldsymbol{\mu}}_{2}=X\cap{U_{2}}$ can only occur in a
subset of the fixed locus of ${\boldsymbol{\mu}}_{2}$. Thus
$\operatorname{Sing}\nolimits(X)\cap U_{2}$ is the image of the set

$\\{y_{0}=y_{1}=y_{4}=0\\}\cap\widetilde{X}_{2}=\\{y_{0}=y_{1}=y_{4}=-y_{3}^{2}+1=0\\}.$

Hence $X$ has two isolated singular points of type $\frac{1}{2}(1,1,1)$, which
are $(0,0,\pm 1,0)$. But this is impossible. ∎

This brings us to the main result of the paper.

######  Theorem 6.3.

The quotient $X={\pazocal J}/G$ of the Jacobian ${\pazocal J}$ of the Klein
quartic $C$ by its full automorphism group $G$ is isomorphic to the weighted
projective space ${\mathbb{P}}(1,2,4,7)$.

###### Proof.

This is an obvious consequence of Theorem 5.1, Lemma 6.1, Proposition 6.2 and
[MM23, Theorem 4.3]. ∎

######  Remark 6.4.

By arguments similar to those used in the reduction of $F_{8}$ to normal forms
(6.2), one can easily verify that in the $45$-dimensional group of coordinate
changes in ${\mathbb{P}}(1,1,2,4,7)$, the stabilizer of the octic form
$F_{8}^{0}=y_{0}y_{4}-y_{3}^{2}$ is $13$-dimensional and the connected
component of the identity in it is generated by the changes of the form

$y_{0}\mapsto\lambda_{0}y_{0},\ \
y_{4}\mapsto\lambda_{0}^{-1}y_{4},y_{1}\mapsto\mu y_{0}+\lambda_{1}y_{1},\ \
y_{2}\mapsto\lambda_{2}y_{2}+h_{2}(y_{0},y_{1}),\ \ $ $y_{3}\mapsto
y_{3}+y_{0}y_{2}h_{1}(y_{0},y_{1})+y_{0}h_{3}(y_{0},y_{1}),$ $y_{4}\mapsto
y_{4}+2(h_{1}(y_{0},y_{1})y_{2}+h_{3}(y_{0},y_{1}))y_{3}+y_{0}(h_{1}(y_{0},y_{1})y_{2}+h_{3}(y_{0},y_{1}))^{2},$

where $\lambda_{i}\in{\mathbb{C}}^{*}$, $\mu\in{\mathbb{C}}$ and the $h_{i}$
are homogeneous of degree $i$ in $y_{0},y_{1}$. Thus the dimension of the
orbit of $F_{8}^{0}$ is $45-13=32$. As the vector space of octic forms is of
dimension $37$, the orbit of $F_{8}^{0}$ is of codimension $5$. A transversal
slice to the orbit can be given by (6.1). This family of hypersurfaces in
${\mathbb{P}}(1,1,2,4,7)$ represents a versal deformation of the weighted
projective space ${\mathbb{P}}(1,2,4,7)$ inside the space of octics in
${\mathbb{P}}(1,1,2,4,7)$. The computation of the infinitesimal deformation
space $\operatorname{Ext}\nolimits^{1}({}^{1}_{X},{\mathcal{O}}_{X})$ shows
that $\dim T^{1}_{X}=5$, and this implies that (6.1) is a complete versal
deformation of $X$. It is interesting to note that this family contains
partial smoothings of $X$ with only three singular points, of which two are of
type $\frac{1}{2}(1,1,1)$ (the singular points found in the proof of the case
$\nu=1$ of Lemma 6.2) and the third one is $\frac{1}{7}(1,2,4)$. These three
singular points are non-smoothable, and even infinitesimally rigid by the
result of Schlessinger [Sch71].

## 7\. Relation to quotients by the Weyl group for the root systems
$\boldsymbol{B_{3}}$ and $\boldsymbol{C_{3}}$

As we already noticed, $G$ contains the Weyl group $W=W(B_{3})=W(C_{3})$ of
order $48$. We are going to look at the invariants under the action of $W$.
For any $g\in W$, the symplectic matrix
$\gamma_{g}=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)$ is
block diagonal; that is, $b=c=0$, and $a=({d}^{{\sf T}})^{-1}$ is nothing but
the matrix of $g^{-1}$ in the basis given by the columns of $\omega_{2}=\alpha
C$ (we keep the notation from previous sections). It easily follows from the
definitions that $g\cdot\theta_{m,k}$ is just $\theta_{dm,k}$. This implies
that the space of $W$-invariant theta functions of degree $k$ consists of the
functions $\pazocal R_{W}\theta_{m,k}$, $m$ running over $({C}^{{\sf
T}})^{-1}(F)\cap P_{k}$, where $F$ is a fundamental domain for the action of
the (real) affine crystallographic group $M\rtimes W$ on the space
$M_{\mathbb{R}}=M\otimes{\mathbb{R}}$ containing the weight lattice $M$ and
$\pazocal R_{W}$ is the Reynolds operator of taking the average over the
action of $W$. When we say that $F$ is a fundamental domain, we mean that $F$
is the disjoint union of an open convex polyhedron in $M_{\mathbb{R}}$ with
finitely many polyhedra of smaller dimensions contained in its boundary such
that every $W$-orbit in $M_{\mathbb{R}}$ has exactly one representative in
$F$. We are now going to fix the choice of a particular fundamental domain
$F$.

There is a tower of degree $2$ extensions

$Q=Q(C_{3})\subset Q{}^{\scriptscriptstyle\vee}=Q(B_{3})\subset M=Q^{*},$

the three lattices being invariant under $W$, so taking the semi-direct
product with $W$, we obtain a tower of degree $2$ extensions of the
corresponding real affine crystallographic groups:

$\widetilde{W}=Q\rtimes
W\subset\widetilde{W}{}^{\scriptscriptstyle\vee}=Q{}^{\scriptscriptstyle\vee}\rtimes
W\subset\widetilde{W}^{*}=M\rtimes W.$

Both $\widetilde{W}$ and $\widetilde{W}{}^{\scriptscriptstyle\vee}$ are affine
Weyl groups and are generated by affine reflections. By [Bou68, Section
VI.2.2], they have fundamental domains which are closed tetrahedra, called
alcoves. For $\widetilde{W}$, the vertices of the standard alcove
$\widetilde{C}=C_{\widetilde{W}}$ are $f_{0}=0$, $f_{1}=(1,0,0)$,
$f_{2}=\frac{1}{2}(1,1,0)$, $f_{3}=\frac{1}{2}(1,1,1)$. Denoting by
$x_{1},x_{2},x_{3}$ the coordinates in the Euclidean space ${\mathbb{R}}^{3}$,
in which we place our lattices $Q$ and $M$, we obtain an alcove
$\widetilde{C}{}^{\scriptscriptstyle\vee}=C_{\widetilde{W}{}^{\scriptscriptstyle\vee}}$
for $\widetilde{W}{}^{\scriptscriptstyle\vee}$ as one half of $\widetilde{C}$
cut out by the mirror $x_{1}=\frac{1}{2}$ of a reflection contained in
$\widetilde{W}{}^{\scriptscriptstyle\vee}$ but not in $\widetilde{W}$. Thus
$\widetilde{C}{}^{\scriptscriptstyle\vee}$ is the tetrahedron with vertices
$O=f_{0}$, $A=\frac{1}{2}f_{1}$, $E=f_{2}$, $N=f_{3}$. The subgroup of
$\widetilde{W}^{*}$ leaving invariant the tetrahedron $OAEN$ is of order $2$;
besides the identity, it contains an element $R$ of order $2$, the axial
symmetry, or rotation by the angle $\pi$ with axis $KL$, where $K$ and $L$ are
the middles of $AE$ and $ON$, respectively (see the figure below).

Now we can get a fundamental domain $F$ of $\widetilde{W}^{*}$ as follows:
first choose a plane $P$ containing $KL$, and pick up one of the two halves in
which $P$ dissects $\widetilde{C}{}^{\scriptscriptstyle\vee}$, say $C_{1}$.
The intersection $D=P\cap\widetilde{C}{}^{\scriptscriptstyle\vee}$ is a face
of $C_{1}$, and it is dissected by $KL$ in two halves $D_{1}$ and $D_{2}$
symmetric to each other by the action of $R$. Then we obtain the following set
of representatives of the orbits of $R$ acting on
$\widetilde{C}{}^{\scriptscriptstyle\vee}$: the polyhedron $C_{1}$, in which
we include all of its faces, except for the face $D$, from which only the
closed part $D_{1}$ is included. The representatives of the orbits of $R$ in
$\widetilde{C}{}^{\scriptscriptstyle\vee}$ are at the same time the
representatives of the orbits of $\widetilde{W}^{*}$ in $V^{*}_{\mathbb{R}}$,
so the constructible set we have described is a fundamental domain for
$\widetilde{W}^{*}$.

To obtain a particular fundamental domain $F$, we choose in the above
construction the plane $OKN$ for $P$, the tetrahedron $OAKN$ for $C_{1}$, then
the triangle $KON$ for the face $D$, and we keep only one half of it in $F$,
the triangle $D_{1}=OKL$. On the picture, $F$ is shaded; the triangle $OKL$ is
shaded in a darker gray color as it is included in $F$ and the remaining part
of the face $OKN$ is not included. The sides of the three closed triangles of
the boundary of $F$ contained in $F$, as well as the edge $AN$ contained in
$F$, are drawn as thick lines. Only the part $OL$ of the edge $ON$ drawn as a
thick line is contained in $F$, and the edge $KN$, not contained in $F$, is
drawn as a thin line. The black points $O$, $A$, $K$, $L$ are contained in
$F$, while the vertex $N$ is not in $F$ and is represented by a white point.

Looijenga in [Loo76] provides a recipe for the choice of generators of the
algebra of $W$-invariant theta functions for the lattice of the form $\tau
Q+Q$; his generators are the averages of the theta functions
$\theta_{f_{i},k_{i}}$ with characteristics $f_{i}$, the vertices of the
alcove, the degrees $k_{i}$ being the smallest positive integers such that
$k_{i}f_{i}\in M$.

######  Remark 7.1.

In the previous sections, we worked with theta functions $\theta_{m,k}$
defined in Section 2 for a different lattice, $\tau M+Q$, so Looijenga’s
recipe does not apply to $\theta_{m,k}$, but we can use it as a heuristic
principle for the choice of theta functions that would be candidates for
generators of the $G$-invariant subalgebra. We can try the $G$-averages of the
theta functions whose characteristics are the vertices $\mu_{i}$ of a “pseudo-
fundamental domain” $F_{G}\subset F$ for $G$, or a “pseudo-alcove” which is
just a smaller tetrahedron with vertex $O$ contained in $F$, having the
correct lattice volume with respect to $M$, equal to $\frac{1}{336}$, and the
expected weights $k_{i}$. Recall that $b_{1}=f_{1},b_{2}=2f_{2},b_{3}=f_{3}$
are the columns of the matrix $({C}^{{\sf T}})^{-1}$; that is, they form a
basis of $M$ dual to the standard basis of the root system $C_{3}$. This is
the basis in which we write the characteristics of our theta functions. From
this we see, in particular, that the lattice volume of $F$ with respect to $M$
is $\frac{1}{48}$. We can pick up the following ad hoc candidate for the
“pseudo-alcove”: take the tetrahedron $F_{G}$ with vertices

(7.1)
$\mu_{0}=0,\quad\mu_{1}=\tfrac{1}{2}b_{1},\quad\mu_{2}=\tfrac{1}{4}(b_{1}+b_{2}),\quad\mu_{3}=\tfrac{1}{7}(2b_{1}+b_{2}+b_{3}).$

The tetrahedron $F_{G}$ obviously has $M$-volume $\frac{1}{336}$, and the
smallest $k_{i}$ such that $k_{i}\mu_{i}\in M$ coincide with the weights of
the expected weighted projective space: $(k_{0},k_{1},k_{2},k_{3})=(1,2,4,7)$.
We thus may expect that the algebra $S({\mathcal{L}}^{2})^{G}$ is generated by
the $G$-averages of the even-degree theta functions that are obtained as
products of theta functions whose characteristics are the vertices of $F_{G}$:
$\theta_{\mu_{0},1}^{2}$, $\theta_{\mu_{1},2}$, $\theta_{\mu_{2},4}$,
$\theta_{\mu_{0},1}\theta_{\mu_{3},7}$, $\theta_{\mu_{3},7}^{2}$. To present
four algebraically independent $G$-invariant theta functions of degrees
$2,2,4,8$ in Lemma 5.2, we chose another set, for which the computations are
slightly easier: $\theta_{\mu_{0},2}$, $\theta_{\mu_{1},2}$,
$\theta_{\mu_{2},4}$, $\theta_{\frac{7}{8}\mu_{3},8}$.

We now return to the quotient ${\pazocal J}/W$. As we saw above, the theta
functions $\\{\pazocal R_{W}\theta_{m,k}\\}_{m\in({C}^{{\sf T}})^{-1}(F)\cap
P_{k}}$ form a basis of the $W$-invariant theta functions of degree $k$ on
${\pazocal J}$. The fact that ${\pazocal J}/W$ is not a complex
crystallographic reflection quotient manifests itself in that $F$ is not a
closed simplex but a constructible set, which we can describe as follows:

(7.2) $F=OAKN\cup\overline{OAK}\cup\overline{OKL}\cup\overline{OAL}\cup
ANK\cup ANL\cup AN.$

Here we denote by $\overline{A_{1}\ldots A_{n}}$ the convex hull of points
$A_{1},\ldots,A_{n}$ in a real affine space, and by $A_{1}\ldots A_{n}$ its
relative interior. From this we can deduce the Hilbert function of the
invariant algebra $S({\mathcal{L}})^{W}$.

######  Proposition 7.2.

The Hilbert function $h_{\overline{S}}\colon k\mapsto\dim H^{0}({\pazocal
J},{\mathcal{L}}^{k})^{W}$ of the algebra $\overline{S}=S({\mathcal{L}})^{W}$
is the sum of a polynomial $E$ and a function $\ell\colon k\mapsto
d_{0}k+d_{1}$, where $d_{0},d_{1}$ are $4$-periodic functions of the integer
variable $k$,

$E(k)=\tfrac{1}{48}k^{3}+\tfrac{3}{16}k^{2}+\tfrac{2}{3}k+{1},\quad
d_{0}=\left\\{\begin{array}[]{l}0\ \mathrm{if}\ k\equiv 0\ \mathrm{or}\ 2\\\
-\tfrac{3}{16}\ \mathrm{if}\ k\equiv\pm 1\end{array}\right.,\quad
d_{1}=\left\\{\begin{array}[]{l}0\ \mathrm{if}\ k\equiv 0\\\ -\tfrac{11}{16}\
\mathrm{if}\ k\equiv\pm 1\\\ -\tfrac{1}{4}\ \mathrm{if}\ k\equiv
2\end{array}\right.\ \ ({\,\mathrm{mod}\,}4).$

The Hilbert series
$\includegraphics{epiga8-9-figure1.pdf}\\!_{\overline{S}}(t)=\sumop\displaylimits_{k=0}^{\infty}h_{\overline{S}}(k)t^{k}$
of $\overline{S}$ is given by

(7.3)
$\includegraphics{epiga8-9-figure1.pdf}\\!_{\overline{S}}(t)=\frac{1-t+t^{2}}{(1-t)^{2}(1-t^{2})(1-t^{4})}.$

###### Proof.

By Ehrhart’s theorem, the number of integer points $h_{P}(k)$ in the integer
multiples $kP$ of an open or closed polytope $P$ in ${\mathbb{R}}^{n}$ with
rational vertices is a quasi-polynomial in the integer variable $k$ of degree
equal to the dimension of $P$. A quasi-polynomial is a polynomial function
whose coefficients are periodic. The common period of the coefficients of the
Ehrhart quasi-polynomial of $P$ is the smallest positive integer $d$ such that
$dP$ is a lattice polytope; that is, all the coordinates of its vertices are
integers (see, for example, [Sta86, Theorem 4.6.25]). By the above,
$h_{\overline{S}}$ coincides with the function $h_{F}$ counting the number of
points of $M$ in the multiples $kF$ of $F$. By (7.2), $h_{F}(k)$ can be
represented as a linear combination, with coefficients $\pm 1$, of the numbers
$h_{\sigma}(k)$ for $\sigma$ running through a finite set of open or closed
simplices with rational vertices of dimensions between $0$ and $3$:

$h_{F}=h_{OAKN}+h_{\overline{OAK}}+h_{\overline{OKL}}+h_{\overline{OAL}}+h_{ALN}+h_{ANK}-h_{\overline{OA}}-h_{\overline{OK}}-h_{\overline{OL}}+h_{AN}+h_{O}.$

Each of the terms of this linear combination is a quasi-polynomial of degree
at most $3$ with period $d=4$, the least common denominator of the coordinates
of the vertices of the simplices $\sigma_{i}$, so $h_{F}$ also is such a
quasi-polynomial. Moreover, the dominant coefficient of $h_{F}$ is constant
and is equal to the inverse of the lattice volume of $F$, that is,
$\frac{1}{48}$, so it suffices to compute 12 consecutive values of $h_{F}$ in
order to determine the coefficients as solutions to a system of linear
equations. Here is the result of the computation of
$h_{\overline{S}}(k)=h_{F}(k)$ for $0\leqslant k\leqslant 12$ by Macaulay2
[Mac2]:

$1,\ 1,\ 3,\ 4,\ 8,\ 10,\ 16,\ 20,\ 29,\ 35,\ 47,\ 56,\ 72,\ldots.$

These values completely determine $h_{\overline{S}}$, and we thus obtain the
formulas from the statement of the proposition. ∎

######  Proposition 7.3.

Consider the quotient
$X^{\prime}={\mathbb{P}}^{3}/{\boldsymbol{\mu}}_{2}\times{\boldsymbol{\mu}}_{4}$
of the projective space ${\mathbb{P}}^{3}$ by a group of order $8$, where
${\boldsymbol{\mu}}_{n}$ denotes a cyclic group of order $n$ and the
generators of ${\boldsymbol{\mu}}_{2}$, ${\boldsymbol{\mu}}_{4}$ act by
diagonal matrices $\operatorname{diag}\nolimits(1,1,-1,1)$,
$\operatorname{diag}\nolimits(i,1,1,-1)$, respectively. Then $X^{\prime}$ and
${\pazocal J}/W$ have the same Hilbert functions.

###### Proof.

We represent $X^{\prime}$ as a toric variety, the equivariant compactification
$\mathbf{X}_{\Sigma,\mathsf{N}}$ of the $3$-dimensional algebraic torus
${\mathbb{T}}:=({\mathbb{C}}^{*})^{3}$, defined by a fan in the
$3$-dimensional ${\mathbb{R}}$-vector space $\mathsf{N}\otimes{\mathbb{R}}$,
where $\mathsf{N}\simeq{\mathbb{Z}}^{3}$ is the lattice of $1$-parametric
subgroups in ${\mathbb{T}}$ (see e.g. [Ful93] for definitions and basic
properties of toric varieties). Let $y_{0},\ldots,y_{3}$ be the homogeneous
coordinates of ${\mathbb{P}}^{3}$ and
${\mathbb{T}}_{0}=({\mathbb{C}}^{*})^{3}$ the standard torus in the affine
chart ${\mathbb{C}}^{3}$ of ${\mathbb{P}}^{3}$ with affine coordinates
$u_{1}=y_{1}/y_{0},\ u_{2}=y_{2}/y_{0},\ u_{3}=y_{3}/y_{0}$. The simplicial
fan defining ${\mathbb{P}}^{3}$ is determined by its $1$-dimensional cones, or
rays, which are spanned by the four vectors $(1,0,0),\ (0,1,0),\
(0,0,1),(-1,-1,-1)$ of the lattice $\mathsf{N}_{0}={\mathbb{Z}}^{3}$. The dual
to $\mathsf{N}_{0}$ is the lattice $\mathsf{M}_{0}={\mathbb{Z}}^{3}$ of
exponents of monomials in the coordinate algebra
${\mathbb{C}}[\mathsf{M}_{0}]={\mathbb{C}}[u_{1},u_{2},u_{3}]$ of the affine
chart of ${\mathbb{P}}^{3}$ that we have chosen. To get the quotient by
${\boldsymbol{\mu}}_{2}\times{\boldsymbol{\mu}}_{4}$, we replace
$\mathsf{M}_{0}$ with the sublattice
$\mathsf{M}=\mathsf{M}_{0}^{{\boldsymbol{\mu}}_{2}\times{\boldsymbol{\mu}}_{4}}$
of exponents of
$({\boldsymbol{\mu}}_{2}\times{\boldsymbol{\mu}}_{4})$-invariant monomials,
which is given by

$\mathsf{M}=\\{(m_{1},m_{2},m_{3})\in\mathsf{M}_{0}\colon m_{3}\equiv
0{\,\mathrm{mod}\,}2,\ -m_{1}-m_{2}+m_{3}\equiv 0{\,\mathrm{mod}\,}4\\};$

this is the lattice of exponents of the monomials
$\mathbf{u}^{\mathbf{m}}=u_{1}^{m_{1}}u_{2}^{m_{2}}u_{3}^{m_{3}}$ which form a
basis of the regular functions on the quotient torus
${\mathbb{T}}:={\mathbb{T}}_{0}/{\boldsymbol{\mu}}_{2}\times{\boldsymbol{\mu}}_{4}$.
The dual
$\mathsf{N}=\mathsf{M}^{*}=\left(\mathsf{M}_{0}^{{\boldsymbol{\mu}}_{2}\times{\boldsymbol{\mu}}_{4}}\right)^{*}$
is the overlattice of $\mathsf{N}_{0}={\mathbb{Z}}^{3}$ generated by the
vectors $(1,0,0),\frac{1}{2}(0,1,0),\frac{1}{4}(-1,-1,1)$, and
$X^{\prime}=\mathbf{X}_{\Sigma,\mathsf{N}}$ is the equivariant
compactification of ${\mathbb{T}}$ defined by the same fan as the original
${\mathbb{P}}^{3}$, but taken with respect to the lattice $\mathsf{N}$. The
primitive vectors of $\mathsf{N}$ generating the rays of the fan are
$v_{0}=(1,0,0)$, $v_{1}=\frac{1}{2}(0,1,0)$, $v_{2}=-\frac{1}{2}(1,1,1)$,
$v_{3}=(0,0,1)$, and we get four ${\mathbb{T}}$-invariant divisors $D_{i}$ in
$X^{\prime}$, defined by $D_{i}=D_{v_{i}}$, where $D_{v_{i}}$ denotes the
closure in $X^{\prime}$ of the kernel of $v_{i}$ viewed as a $1$-parametric
subgroup of ${\mathbb{T}}$. The Cartier indices of these divisors for
$i=0,1,2,3$ are, respectively, $4,2,2,4$. We omit the routine verification of
the following assertion, which makes more precise the statement we are
proving. ∎

######  Proposition 7.3′.

In the above notation, the Hilbert functions $k\mapsto
h^{0}(X^{\prime},{\mathcal{O}}_{X^{\prime}}(kD_{i}))$ coincide with
$h_{\overline{S}}$ for both divisors $D_{i}$ of Cartier index $4$, that is,
for $i=0$ and $3$.

It is plausible that the two quotients are indeed isomorphic. We observe that
$X^{\prime}$ belongs to the class of projective toric varieties of dimension
$n$ whose fan contains $n+1$ rays. Such varieties are called weak weighted
projective spaces; some authors call them fake weighted projective space (see
[Kas09]), but we prefer the adjective weak because the class of weak weighted
projective spaces contains all the weighted projective spaces. A weak weighted
projective space is a weighted projective space if and only if the primitive
vectors of $\mathsf{N}$ on the rays ${\mathbb{R}}^{*}_{+}v_{i}$ of its fan
generate the whole of $\mathsf{N}$. One easily sees that this is not the case
for $X^{\prime}$. This brings us to the following generalization of the
Bernstein–Schwarzman conjecture.

######  Conjecture 7.4.

If and 1 are commensurable complex crystallographic groups acting on
${\mathbb{C}}^{n}$ such that $\hskip 1.29167pt\mathrm{d}{}_{1}=\hskip
1.29167pt\mathrm{d}\Gamma$, and if is irreducible and generated by
reflections, then the quotient ${\mathbb{C}}^{n}/{}_{1}$ is a weak weighted
projective space. It is a genuine weighted projective space if and only if 1
is also generated by reflections.

The cases when is one of the complex crystallographic reflection groups
$(Q+\tau Q)\rtimes W$ or $(Q+\tau Q{}^{\scriptscriptstyle\vee})\rtimes W$
treated by Looijenga and Bernstein–Schwarzman and ${}_{1}=\Lambda\rtimes W$,
where $\Lambda=Q+\tau M$, are particular cases of this conjecture: while
${\mathbb{C}}^{3}/\Gamma$ is known to be a weighted projective space, we
expect that ${\mathbb{C}}^{3}/{}_{1}$ is the weak weighted projective space
$X^{\prime}$.

## References

  * [BS06] J. Bernstein and O. Schwarzman, Chevalley’s theorem for the complex crystallographic groups, J. Nonlinear Math. Phys. 13 (2006), no. 3, 323–351.
  * [BEW98] B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi sums, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, Inc., New York, 1998.
  * [BL04] C. Birkenhake and H. Lange, Complex abelian varieties, 2nd ed., Grundlehren math. Wiss., vol 302, Springer-Verlag, Berlin, 2004.
  * [Bou68] N. Bourbaki, Éléments de mathématique, Fasc. XXXIV, Groupes et algèbres de Lie, Chapitres IV–VI, Actualités Sci. Industr., No. 1337, Hermann, Paris, 1968.
  * [Coh76] A. M. Cohen, Finite complex reflection groups, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 379–436.
  * [DS23] T. Dedieu and E. Sernesi, Deformations and Extensions of Gorenstein Weighted Projective Spaces, in: The Art of Doing Algebraic Geometry, pp. 119–143, Trends Math., Birkhäuser/Springer, Cham, 2023.
  * [Ful93] W. Fulton, Introduction to toric varieties, Ann. of Math. Stud., vol. 131, William Roever Lectures Geom., Princeton Univ. Press, Princeton, NJ, 1993.
  * [Gun62] R. C. Gunning, Lectures on modular forms (Notes by A. Brumer), Ann. of Math. Stud., vol. 48, Princeton Univ. Press, Princeton, NJ, 1962.
  * [Igu72] J.-i. Igusa, Theta functions, Grundlehren math. Wiss., vol. 194, Springer-Verlag, New York-Heidelberg, 1972.
  * [Kas09] A. M. Kasprzyk, Bounds on fake weighted projective space, Kodai Math. J. 32 (2009), no. 2, 197–208.
  * [Kle78] F. Klein, Ueber die Transformation siebenter Ordnung der elliptischen Functionen, Math. Ann. 14 (1878) , no. 3, 428–471 (1878).
  * [Loo76] E. Looijenga, Root systems and elliptic curves, Invent. Math. 38 (1976/77), no. 1, 17–32.
  * [Mac2] D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
  * [MM23] D. Markushevich and A. Moreau, Action of the Automorphism Group on the Jacobian of Klein’s Quartic Curve, in: Birational Geometry, Kähler–Einstein Metrics and Degenerations, 591–607, Springer Proc. Math. Stat., vol. 409, Springer, Cham, 2023.
  * [Maz86] B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 207–259.
  * [Pop82] V. L. Popov, Discrete complex reflection groups, Commun. Math. Inst. Rijksuniv. Utrecht, vol. 15, Rijksuniv. Utrecht, Math. Inst., Utrecht, 1982.
  * [Rai23] E. M. Rains, Quotients of abelian varieties by reflection groups, preprint arXiv:2303.04786 (2023).
  * [Rei80] M. Reid, Canonical 3-folds, in: Journées de Géométrie Algébrique d’Angers, Juillet 1979, pp. 273–310, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, MD, 1980.
  * [Sch71] M. Schlessinger, Rigidity of quotient singularities, Invent. Math. 14 (1971), 17–26.
  * [ST54] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304.
  * [Sta86] R. P. Stanley, Enumerative combinatorics, Vol. I, Wadsworth & Brooks/Cole Math. Ser., Wadsworth & Brooks/Cole Advanced Books & Software, Montereye , CA, 1986.