CUBO, A Mathematical Journal

Vol. 24, no. 01, pp. 37–51, April 2022

DOI: 10.4067/S0719-06462022000100037

Smooth quotients of abelian surfaces by finite
groups that fix the origin

Robert Auffarth1

Giancarlo Lucchini Arteche1

Pablo Quezada2

1Departamento de Matemáticas,

Facultad de Ciencias, Universidad de Chile,

Las Palmeras 3425, Ñuñoa,

Santiago, Chile.

rfauffar@uchile.cl

luco@uchile.cl

2Facultad de Matemáticas,

Pontificia Universidad Católica,

Vicuña Mackenna 4860, Macul,

Santiago, Chile.

psquezada@uc.cl

ABSTRACT

Let A be an abelian surface and let G be a finite group

of automorphisms of A fixing the origin. Assume that the

analytic representation of G is irreducible. We give a clas-

sification of the pairs (A, G) such that the quotient A/G

is smooth. In particular, we prove that A = E2 with E an

elliptic curve and that A/G ≃ P2 in all cases. Moreover,

for fixed E, there are only finitely many pairs (E2, G) up

to isomorphism. This fills a small gap in the literature and

completes the classification of smooth quotients of abelian

varieties by finite groups fixing the origin started by the

first two authors.

RESUMEN

Sea A una superficie abeliana y sea G un grupo finito de

automorfismos de A fijando el origen. Se asume que la

representación anaĺıtica de G es irreducible. Damos una

clasificación de los pares (A, G) tales que el cociente A/G

es suave. En particular, probamos que A = E2 con E una

curva eĺıptica y que A/G ≃ P2 en todos los casos. Más aún,

para E fija, hay solo una cantidad finita de pares (E2, G),

salvo isomorfismo. Esto llena una pequeña brecha en la

literatura y completa la clasificación de cocientes suaves

de variedades abelianas por grupos finitos fijando el origen

comenzado por los dos primeros autores.

Keywords and Phrases: Abelian surfaces, automorphisms.

2020 AMS Mathematics Subject Classification: 14L30, 14K99.

Accepted: 26 October, 2021

Received: 21 May, 2021

c©2022 R. Auffarth et al. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.4067/S0719-06462022000100037
https://orcid.org/0000-0001-7243-0315
https://orcid.org/0000-0003-3269-1814
mailto:rfauffar@uchile.cl
mailto:luco@uchile.cl
mailto:psquezada@uc.cl


38 R. Auffarth, G. Lucchini Arteche & P. Quezada CUBO
24, 1 (2022)

1 Introduction

The purpose of this paper is to give a complete classification of all smooth quotients of abelian

surfaces by finite groups that fix the origin, and is to be seen as the completion of the classification

given in [2] of smooth quotients of abelian varieties that fix the origin. This kind of quotients

of abelian surfaces has already been studied by Tokunaga and Yoshida in [6], where infinite 2-

dimensional complex reflection groups, which are extensions of a finite complex reflection group

G by a G-invariant lattice, are classified. However, these do not cover all possible G-invariant

lattices and hence not all possible group actions on abelian surfaces. Moreover, there seem to be

some complex reflection groups that the authors missed, as can be seen by looking at Popov’s

classification of the same groups in [3].

The techniques used in this paper are similar, but not exactly the same, to the methods used

in [2]. Indeed, the ideas used in this last paper have been modified in order to apply them to the

two-dimensional case. Moreover our approach is far different from that used in [6].

Our main theorem states the following:

Theorem 1.1. Let A be an abelian surface and let G be a (non-trivial) finite group of automor-

phisms of A that fix the origin. Then the following conditions are equivalent:

(1) A/G is smooth and the analytic representation of G is irreducible.

(2) A/G ≃ P2.

(3) There exists an elliptic curve E such that A ≃ E2 and (A, G) satisfies exactly one of the

following:

(a) G ≃ C2 ⋊ S2 where C is a non-trivial (cyclic) subgroup of automorphisms of E that fix

the origin; here the action of C2 is coordinate-wise and S2 permutes the coordinates.

(b) G ≃ S3 and acts on

A ≃ {(x1, x2, x3) ∈ E
3 : x1 + x2 + x3 = 0},

by permutations.

(c) E = C/Z[i] and G is the order 16 subgroup of GL2(Z[i]) generated by:










−1 1 + i

0 1



 ,





−i i − 1

0 i



 ,





−1 0

i − 1 1











,

acting on A ≃ E2 in the obvious way.

The first two cases found in item (3) of the above theorem were studied in detail in [1] (in arbitrary

dimension), where it was proven that both examples give the projective plane as a quotient.



CUBO
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Smooth quotients of abelian surfaces by finite groups... 39

Throughout the paper we will refer to these two examples as Example (a) and Example (b),

respectively. The equivalent assertion for Example (c) is Proposition 4.1 in this paper. Note that,

aside from Examples (a) and (b) which belong to infinite families, Example (c) is the only new case

of an action of G on an abelian variety satisfying condition (1) from Theorem 1.1, cf. [2, Thm. 1.1].

Remark 1.2. If A/G is smooth and the analytic representation of G is reducible, then the results

in [2] imply that A is isogenous to a product of two elliptic curves. The quotient is then either

P
1 × P1 (in which case A = E1 × E2) or a bielliptic surface.

In [7], Yoshihara introduces the notion of a Galois embedding of a smooth projective variety. If

X is a smooth projective variety of dimension n and D is a very ample divisor that induces an

embedding X →֒ PN, then the embedding is said to be Galois if there exists an (N − n − 1)-

dimensional linear subspace W of PN such that X ∩ W = ∅ and the restriction of the linear

projection πW : P
N

99K P
n to X is Galois. Yoshihara specifically studies when abelian surfaces

have a Galois embedding. He gives a classification of abelian surfaces having a Galois embedding,

along with their Galois groups, and proves that after taking the quotient of the original abelian

variety by the translations of the Galois group, the abelian variety must be isomorphic to the self-

product of an elliptic curve. Unfortunately, his results were incomplete since they depended on a

classification of smooth quotients like the one given in this paper, which Yoshihara attributed to

Tokunaga and Yoshida in [6]. But as stated before, Tokunaga and Yoshida’s results do not imply

such a classification. Nevertheless, we can now safely say, thanks to Theorem 1.1, that Yoshihara’s

results remain correct.

The structure of this paper is as follows: in Section 2 we fix notations and give some preliminary

results that will be needed in the proofs of Theorem 1.1. The implication (2) ⇒ (1) is obvious and

(3) ⇒ (2) was already treated in [1] in the case of Examples (a) and (b). Thus, we are mainly

concerned with (1) ⇒ (3), which we treat in Section 3. Finally, in Section 4 we treat (3) ⇒ (2)

for Example (c), which is a construction of a different nature that only exists in the 2-dimensional

case.

2 Preliminaries on group actions on abelian varieties

We recall here some elementary results that were proved in [2]. Let A be an abelian surface and

let G be a group of automorphisms of A that fix the origin, such that the quotient variety A/G

is smooth. By the Chevalley-Shephard-Todd Theorem, the stabilizer in G of each point in A

must be generated by pseudoreflections; that is, elements that fix pointwise a divisor (i.e. a curve)

containing the point. In particular, G = StabG(0) is generated by pseudoreflections and G acts

on the tangent space at the origin T0(A) (this is the analytic representation). In this context, a

pseudoreflection is an element that fixes a line pointwise. We will often abuse notation and display



40 R. Auffarth, G. Lucchini Arteche & P. Quezada CUBO
24, 1 (2022)

G as either acting on A or T0(A); it will be clear from the context which action we are considering.

In what follows, let L be a fixed G-invariant polarization on A (take the pullback of an ample class

on A/G, for example). For σ a pseudoreflection in G of order r, define

Dσ := im(1 + σ + · · · + σ
r−1), Eσ := im(1 − σ).

These are both abelian subvarieties of A. The following result corresponds to [2, Lem. 2.1].

Lemma 2.1. We have the following:

1. Dσ is the connected component of ker(1 − σ) that contains 0 and Eσ is the complementary

abelian subvariety of Dσ with respect to L. In particular, Dσ and Eσ are elliptic curves.

2. Fσ := Dσ ∩ Eσ consists of 2-torsion points for r = 2, 4, of 3-torsion points for r = 3 and

Dσ ∩ Eσ = 0 for r = 6.

We will consider now a new abelian surface B equipped with a G-equivariant isogeny to A, which

we will call G-isogeny from now on. Let ΛA denote the lattice in C
2 such that A = C2/ΛA. Let

ΛB ⊂ ΛA be a G-invariant sublattice, and let B := C
2/ΛB be the induced abelian surface, along

with the G-isogeny

π : B → A,

whose analytic representation is the identity. Note that this implies that σ ∈ G is a pseudoreflection

of B if and only if it is a pseudoreflection of A. We may then consider the subvarieties Eσ, Dσ, Fσ ⊂

A defined as above, which we will denote by Eσ,A, Dσ,A and Fσ,A. We do similarly for B. Note

that, by definition, π sends Eσ,B to Eσ,A and Dσ,B to Dσ,A, hence Fσ,B to Fσ,A. The following

result was proved in [2, Prop. 2.4].

Proposition 2.2. Let σ ∈ G be a pseudoreflection and let L be the line defining both Eσ,A and

Eσ,B. Assume that the map Fσ,B → Fσ,A is surjective and that ΛA ∩ L = ΛB ∩ L. Then ker(π) is

contained in Dτστ−1,B for every τ ∈ G.

Define ∆ := ker(π). Since π is G-equivariant, G acts on ∆ and hence we may consider the group

∆⋊ G. This group acts on B in the obvious way: ∆ acts by translations and G by automorphisms

that fix the origin. In particular, we see that the quotient B/(∆ ⋊ G) is isomorphic to A/G. We

conclude this section by recalling a result on pseudoreflections in ∆ ⋊ G (cf. [2, Lem. 2.5]).

Lemma 2.3. Let σ ∈ ∆ ⋊ G be a pseudoreflection. Then σ = (t, τ) with τ ∈ G a pseudoreflection

and t ∈ ∆ ∩ Eτ,B.



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Smooth quotients of abelian surfaces by finite groups... 41

3 Proof of (1) ⇒ (3)

Assume (1), that is, we have an abelian surface A with an action of a finite group G that fixes

the origin and such that A/G is smooth and the analytic representation of G is irreducible. Under

these conditions, we see that G is an irreducible finite complex reflection group in the sense of

Shephard-Todd [4]. These groups were completely classified by Shephard and Todd in [4]. In the

particular case of dimension 2, we get that G is either one of 19 sporadic cases or it is isomorphic

to a semidirect product G(m, p) := H(m, p) ⋊ S2, where p|m, m ≥ 2, and

H = H(m, p) = {(ζa1m , ζ
a2
m ) | a1 + a2 ≡ 0 (mod p)} ⊂ µ

2

m,

with ζm denoting a primitive m-th root of unity. The action of S2 on H is the obvious one. The

case G = G(2, 2) is excluded since G is then a Klein group and thus the representation is not

irreducible. The action of G on C2 is given as follows: H acts on C2 coordinate-wise while S2

permutes the coordinates.

Emulating the work done in [2], we wish to describe which of these actions actually appear on

abelian surfaces and give smooth quotients. The sporadic cases were already treated in [2] and

were proven not to give a smooth quotient (cf. [2, §3.3]), so we may and will assume henceforth

that G = G(m, p) as above. This fixes a G-equivariant isomorphism of T0(A) with C
2. We denote

by e1 and e2 the canonical basis of T0(A) thus obtained.

Lemma 3.1. Assume that G acts on A as above. Then m ∈ {2, 3, 4, 6}.

Proof. We have that (ζm, ζ
−1
m ) acts on A, and so the characteristic polynomial of (ζm, ζ

−1
m ) ⊕

(ζm, ζ
−1
m ) must have integer coefficients, and so must be the k-th power of the m-th cyclotomic

polynomial Φm, where k = 2 if m ≥ 2 and k = 4 if m = 2. Looking at the degrees, we get

that 4 = kϕ(m), where ϕ is Euler’s totient function. Therefore, if m 6= 2 then ϕ(m) = 2 and so

m ∈ {3, 4, 6}.

Having proved this result, we see that there is a finite list of cases to be analyzed, that is:

(m, p) ∈ {(2, 1), (2, 2), (3, 1), (4, 1), (6, 1), (3, 3), (4, 2), (4, 4), (6, 2), (6, 3), (6, 6)}.

Recall that we have already eliminated the case (2, 2) since the analytic representation of G(2, 2)

is not irreducible. Moreover, it is well-known that there is an exceptional isomorphism of complex

reflection groups between G(4, 4) and G(2, 1). We will prove then the following:

• If G = G(m, 1) and A/G is smooth, then the pair (A, G) corresponds to Example (a) (Sections

3.1, 3.3, 3.4, 3.5);

• If G = G(3, 3) and A/G is smooth, then the pair (A, G) corresponds to Example (b) (Section

3.8);



42 R. Auffarth, G. Lucchini Arteche & P. Quezada CUBO
24, 1 (2022)

• If G = G(4, 2) and A/G is smooth, then the pair (A, G) corresponds to Example (c) (Section

3.2);

• If G = G(6, p) with p ≥ 2, then A/G cannot be smooth (Sections 3.6, 3.7, 3.9).

In order to do this, we will construct a G-isogeny B → A such that the action of G on B is

“well-known”. Let us concentrate first on the cases where m 6= p. Then we obtain B as follows:

Let Ei be the image of Cei in A via the exponential map. We claim that it corresponds to an elliptic

curve. Indeed, consider the non-trivial element τ = (ζpm, 1) ∈ H. Then a direct computation shows

that im(1 − τ) = Ce1. This tells us that E1 = (1 − τ)(A) and hence it corresponds to an elliptic

curve. The same proof works for E2.

Now, let ΛA be a lattice for A in C
2. Then Cei ∩ ΛA corresponds to the lattice of Ei in C = Cei.

We can thus define the G-stable sublattice of ΛA

ΛB := (Ce1 ∩ ΛA) ⊕ (Ce2 ∩ ΛA).

As in Section 2, this defines a G-isogeny π : B → A. Moreover, we see that B ≃ E1 ×E2 ≃ E
2 and

that π|Ei is an injection. Let ∆ be the kernel of π. We will study the different possible quotients

A/G by studying the possible quotients B/(∆ ⋊ G) and thus by studying the possible ∆’s. Our

first result is the following:

Lemma 3.2. Assume that m 6= p. Then the coordinates of every element in ∆ are invariant by

ζpm, so in particular these elements are

• 2-torsion if (m, p) ∈ {(2, 1), (4, 1), (4, 2), (6, 3)};

• 3-torsion if (m, p) ∈ {(3, 1), (6, 2)};

• trivial if (m, p) = (6, 1).

Proof. Let t̄ = (t1, t2) ∈ ∆. Then, since ∆ is G-stable, we have that, for τ1 = (ζ
p
m, 1) ∈ H,

(1 − τ1)(t̄) = ((1 − ζ
p
m)t1, 0) ∈ ∆.

But, by construction, there are no elements of the form (x, 0) in ∆. We deduce then that t1 is

ζpm-invariant. The same proof works for t2. The assertion on the torsion of t1 and t2 follows

immediately.

Let us study now pseudoreflections in ∆ ⋊ G. Define the elements

ρ := (ζm, ζ
−1
m ) ∈ H ⊂ G; σ := (1 2) ∈ S2 ⊂ G; τ := (ζ

p
m, 1) ∈ H ⊂ G.

Then there are two types of pseudoreflections in G:



CUBO
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Smooth quotients of abelian surfaces by finite groups... 43

• conjugates of ρaσ for 0 ≤ a < m
p
;

• conjugates of powers of τ;

and the corresponding elliptic curves in B are respectively:

Eρaσ = {(x, −ζ
a
mx) | x ∈ E}; Eτ = {(x, 0) | x ∈ E}.

Recall that elements of the form (x, 0) are not in ∆ by construction of the isogeny π : B → A.

Using Lemmas 2.3 and 3.2, we obtain immediately the following result:

Lemma 3.3. Every pseudoreflection in ∆ ⋊ G that is not in G is a conjugate of (t̄, ρaσ), where

0 ≤ a < m
p
, t̄ = (t, −ζamt) ∈ ∆ and t is ζ

p
m-invariant.

With these considerations, we can start a case by case study of the non-trivial ∆’s. We recall that

the main tool will be the Chevalley-Shephard-Todd Theorem, which states that A/G = B/(∆⋊G)

is smooth if and only if the stabilizer in ∆⋊G of each point in B is generated by pseudoreflections.

3.1 The case G = G(2, 1)

By Lemma 3.2, we know that ∆ is 2-torsion. Since we also know that there are no elements of the

form (t, 0) for t ∈ E, we get the following possible options for ∆:

(1) ∆ = {0};

(2) ∆ = 〈(t, t)〉 with t ∈ E[2];

(3) ∆ = {(t, t) | t ∈ E[2]};

(4) ∆ = {(0, 0), (t1, t2), (t2, t1), (t1 + t2, t1 + t2)} with t1, t2 ∈ E[2], t1 6= t2.

Case (1) clearly corresponds to Example (a) (which gives a smooth quotient, cf. [2, Prop. 3.4]).

Case (2) cannot give a smooth quotient and this follows directly from [2, Prop. 3.7].1

In case (3), we claim that the pair (A, G) is isomorphic to the pair (B, G). This will reduce us to

the case with trivial ∆, which was already dealt with. To prove the claim, consider the canonical

basis of T0(A) = T0(B) = C
2. Then the analytic representation of G is given by the following

values in its generators:

ρa((1, −1)) =





1 0

0 −1



 , ρa((1 2)) =





0 1

1 0





1The proof of this proposition only uses two variables and thus it works in dimension 2 as well.



44 R. Auffarth, G. Lucchini Arteche & P. Quezada CUBO
24, 1 (2022)

Now, with this basis and this ∆, we can view the G-isogeny B → A as the morphism E2 → E2

given by the following matrix:

M =





1 1

1 −1



 , (*)

for which one can check that its kernel is precisely the elements in ∆. In order to prove that the

pairs (A, G) and (B, G) are isomorphic, it suffices thus to prove that the image of this representation

of G under conjugation by M is G once again. Direct computations give:

Mρa((1, −1))M
−1 =





0 1

1 0



 = ρa((1 2)), Mρa((1 2))M
−1 =





1 0

0 −1



 = ρa((1, −1)).

And these clearly generate the same group G.

In case (4), consider the element t̄ = (t′
1
, t′

2
) where 2t′i = ti. Note that G cannot fix t̄ as t

′
1
and t′

2

lie in different orbits by the action of µ2. Now, it is easy to see that there is no way the action of

∆ can compensate the action of G except in the case when we add the element (t1, t2). A direct

computation tells us then that the only element fixing t̄ is ((t1, t2), (−1, −1)) ∈ ∆ ⋊ G and since

this stabilizer is not generated by pseudoreflections by Lemma 3.3, we see that A/G is not smooth.

3.2 The case G = G(4, 2)

Since G(4, 2) contains G(2, 1), we may start from the precedent list of possible non-trivial ∆’s.

However, these must also be stable by the new element (i, i) ∈ H(4, 2) (where i = ζ4). Note that

such an element acts on each component E of B by multiplication by i, which implies in particular

that E = C/Z[i]. Defining by t0 the only non-trivial i-invariant element in E, we get the following

possibilities:

(1) ∆ = {0};

(2) ∆ = 〈(t0, t0)〉;

(3) ∆ = {(t, t) | t ∈ E[2]};

(4) ∆ = {(0, 0), (t, t + t0), (t + t0, t), (t0, t0)} with t ∈ E[2], t 6= t0.

Case (1) does not give a smooth quotient A/G, cf. [2, Prop. 3.4]. Case (2) corresponds to Example

(c) (and it actually gives a smooth quotient A/G as we prove in section 4). Indeed, the G-isogeny

B → A corresponds in this case to the morphism E2 → E2 with E = C/Z[i] given by the matrix





1 −1

0 i − 1



 ,



CUBO
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Smooth quotients of abelian surfaces by finite groups... 45

and the generators given in Example (c) correspond to the conjugates by this matrix of the following

respective matrices:










−1 0

0 1



 ,





−i 0

0 i



 ,





0 1

1 0











.

But these are clearly the matrix expressions of the generators (−1, 1), (−i, i) ∈ H and (1 2) ∈ S2

of G = H ⋊ S2.

Remark 3.4. Since the first and third matrices above generate the subgroup G(2, 1) ⊂ G(4, 2), we

see that if we take F to be the subgroup of G spanned by the pseudoreflections




−1 i + 1

0 1



 and





−1 0

i − 1 1



 ,

then F is isomorphic to G(2, 1) and A/F ≃ P2. In particular, the pair (A, F) is isomorphic to

Example (a) with C cyclic of order 2.

In cases (3) and (4), we claim that the pair (A, G) is isomorphic to the pair (B, G). This will

reduce us to the case with trivial ∆, which was already dealt with. To prove the claim, we consider

as for G = G(2, 1) the canonical basis of T0(A) = T0(B) = C
2. Then the analytic representation

of G is given by the following values in its generators:

ρa((i, −i)) =





i 0

0 −i



 , ρa((−1, 1)) =





−1 0

0 1



 , ρa((1 2)) =





0 1

1 0





Now, with this basis and the ∆ from case (2), we already know that B → A looks like E2 → E2

with matrix M from (*). It suffices to check then that the new generator ρa((i, −i)) falls into

ρa(G) after conjugation by M. And indeed we have that Mρa((i, −i))M
−1 = ρa((i, i))ρa((1 2)).

With the ∆ from case (3), the corresponding matrix for B → A is:

N =





1 i

i 1



 .

And once again, direct computations give:

Nρa((i, −i))N
−1 =





0 −1

1 0



 = ρa((−1, 1))ρa((1 2)),

Nρa((−1, 1))N
−1 =





0 −i

i 0



 = ρa((1 2))ρa((i, −i)),

Nρa((1 2))N
−1 =





0 1

1 0



 = ρa((1 2)).

And these clearly generate the same group G.



46 R. Auffarth, G. Lucchini Arteche & P. Quezada CUBO
24, 1 (2022)

3.3 The case G = G(4, 1)

Since G(4, 1) contains G(4, 2), we may start from the precedent list of possible non-trivial ∆’s.

Now, by Lemma 3.2, we know that the coordinates of the elements in ∆ are i-invariant. We get

then that there are only two options for ∆, that is the trivial case and ∆ = 〈(t0, t0)〉.

In the trivial case, we immediately see that (A, G) corresponds to Example (a). Assume then that

∆ is non-trivial and consider the element (s, t) ∈ B with s ∈ E[2], s not i-invariant and 2t = t0.

Since clearly these elements have different order, we see that the orbits of these elements by the

action of 〈t0〉 × µ4 are different. Thus no action of an element in ∆ × H ⊂ ∆ ⋊ G can compensate

the action of (1 2) ∈ G in order to fix (s, t). In other words, the stabilizer of t̄ must be contained

in ∆ × H. It is easy to see then that it corresponds to 〈((t0, t0), (i, −1))〉. By Lemma 3.3, this

stabilizer is not generated by pseudoreflections and hence A/G is not smooth in this case.

3.4 The case G = G(3, 1)

By Lemma 3.2, we know that the coordinates of the elements in ∆ are ζ3-invariant. Now, there

are only two such non-trivial elements that we will denote by s0 and −s0. Since we also know

that there are no elements of the form (t, 0) for t ∈ E, we get the following possible options for a

non-trivial ∆:

(1) ∆ = {0};

(2) ∆ = 〈(s0, s0)〉;

(3) ∆ = 〈(s0, −s0)〉.

We immediately see that the trivial case gives us Example (a). In case (2), Lemma 3.3 tells us that

the only pseudoreflections in ∆ ⋊ G are those coming from G. In particular, in order to prove that

A/G cannot be smooth, it suffices to exhibit an element in B such that its stabilizer in ∆ ⋊ G has

elements that are not in G. Let τ = (ζ3, ζ3) ∈ H ⊂ G, then 1 − τ is surjective. Then there exists

an element z̄ ∈ B such that z̄ − τ(z̄) = (s0, s0). This implies that ((s0, s0), τ) ∈ ∆ ⋊ G stabilizes

z, proving thus that A/G is not smooth in this case.

In case (3), consider the element s̄ = (s, −s) ∈ B with s ∈ E[3] and s not ζ3-invariant. Note that

〈s0〉 × µ3 acts on E[3] and a direct computation tells us that the orbit of s is {s, s + s0, s − s0}. In

particular, we see that s and −s lie in different orbits for this action. The same argument used in

the case of G(4, 1) tells us then that the stabilizer of s̄ must be contained in ∆ × H. It is easy to

see then that, up to changing s̄ by −s̄, it corresponds to 〈((s0, −s0), (ζ3, ζ3))〉. Since this stabilizer

is not generated by pseudoreflections by Lemma 3.3, we see that A/G is not smooth in this case

as well.



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Smooth quotients of abelian surfaces by finite groups... 47

3.5 The case G = G(6, 1)

By Lemma 3.2, we know that the only possibility is a trivial ∆. This clearly corresponds to

Example (a).

3.6 The case G = G(6, 2)

Since G(6, 2) contains G(3, 1), we may start from the possible non-trivial ∆’s for that case. Note

that these are all 3-torsion subgroups. Thus, if x̄ ∈ B denotes a 2-torsion element, we see that its

stabilizer in ∆ ⋊ G can only contain elements in G. Consider then the element t̄ = (t, 0) where t

is a non-trivial 2-torsion element. As it is proven in [2, Prop. 3.4], the stabilizer of this element in

G is not generated by pseudoreflections. This implies that A/G = B/(∆ ⋊ G) cannot be smooth

regardless of the choice of possible ∆.

3.7 The case G = G(6, 3)

Since G(6, 3) contains G(2, 1), we may start from the possible non-trivial ∆’s for that case. Note

that these are all 2-torsion subgroups. Thus, like we noticed in the previous case, if x̄ ∈ B

denotes a 3-torsion element, its stabilizer in ∆ ⋊ G only contains elements in G. Consider then

the element s̄ = (s0, 0) where s0 is a ζ3-invariant element (hence 3-torsion). Once again, as proven

in [2, Prop. 3.4], the stabilizer of this element in G is not generated by pseudoreflections, which

implies that A/G cannot be smooth in any case of ∆.

This finishes the study of the cases where m 6= p. We are left thus with the cases G(3, 3) and G(6, 6).

In these particular cases we forget all the constructions done before and start from scratch.

3.8 The case G = G(3, 3)

The group G(3, 3) is easily seen to be isomorphic as a complex reflection group to S3 acting on

C
2 via the standard representation. As such, it has already been treated by the first two authors

in [2, §3.1] and we know that in that case we get a smooth quotient if and only if we are in

Example (b).

3.9 The case G = G(6, 6)

Note that G(6, 6) is isomorphic to the direct product G(3, 3) × {±1}. Since the actions of S3

and µ2 = {±1} commute, we may follow the approach taken by [2] for S3 and we will prove the

following:



48 R. Auffarth, G. Lucchini Arteche & P. Quezada CUBO
24, 1 (2022)

Proposition 3.5. Let G(6, 6) = S3 × µ2 act on an abelian surface A in such a way that its action

on T0(A) is the standard one for S3 and the obvious one for µ2. Then A/G is not smooth.

Proof. Let σ = (1 2) ∈ S3 and E = Eσ be induced by a line Lσ ⊂ T0(A), and define the lattice

ΛB :=
∑

τ∈S3

τ(Lσ ∩ ΛA).

Since clearly all lattices are µ2-invariant, this gives us a G-invariant sublattice of ΛA. Therefore,

we get a G-equivariant isogeny π : B → A with kernel ∆. Applying this construction to Example

(b), to which we can naturally add the action of µ2 in order to get an action of G, we see that it

gives the whole lattice. We can thus see B as

B = {(x1, x2, x3) ∈ E
3 | x1 + x2 + x3 = 0},

where S3 and µ2 act in their respective natural ways. Using the notations from Section 2, we

see by inspection that Fσ,B = Eσ,B[2] ≃ E[2], hence the map π : Fσ,B → Fσ,A is surjective since

by Lemma 2.1, case 2., we have Fσ,A ⊂ Eσ,A[2] ≃ E[2]. By Proposition 2.2, we have that ∆ is

contained in the fixed locus of all the conjugates of σ, which clearly generate S3. Thus, ∆ consists

of elements of the form (x, x, x) ∈ E3 such that 3x = 0. In particular, ∆ is isomorphic to a

subgroup of E[3] and hence of order 1, 3 or 9.

Assume that ∆ is trivial, that is, that A = B. Then the action of G = S3 × µ2 on B ≃ E
2 induces

an action of µ2 on B/S3 ≃ P
2 (recall that the action of S3 on B is that of Example (b)). We only

need to notice then that any quotient of P2 by a non trivial action of the group µ2 is not smooth.

This is well-known.

Assume now that ∆ has order 3 and let t̄ = (t, t, t) ∈ ∆ be a non-trivial element (thus t ∈ E[3]).

Let x ∈ E[3] be a non-trivial element different from ±t and consider x̄ = (x, x + t, x − t). It is then

easy to see that the element (t̄, (1 2 3)) ∈ ∆ ⋊ G fixes x̄ and that StabG(x̄) = {1}, so that every

pseudoreflection fixing x̄ must lie outside G. Let (s̄, σ) be such a pseudoreflection. Using Lemma

2.3, we see that σ ∈ {−(1 2), −(2 3), −(1 3)}, where −τ denotes (τ, −1) ∈ S3 × µ2 = G. Now, for

any such σ, direct computations tell us that (s̄, σ) fixes x̄ if and only s̄ = (s, s, s) with s = aσx+bσt

for some aσ 6= 0. Since x 6∈ 〈t〉 ⊂ E[3], we see that s̄ 6∈ ∆ and hence these pseudoreflections do not

exist. We get then that Stab∆⋊G(x̄) is not generated by pseudoreflections and hence A/G cannot

be smooth.

Assume finally that ∆ has order 9. We claim that in this case the pair (A, G) is isomorphic to

the pair (B, G). This will reduce us to the case with trivial ∆, which was already dealt with. To

prove the claim, fix the basis {(1, 0, −1), (0, 1, −1)} of T0(B) = T0(A) ⊂ C
3. Then the analytic

representation of G is given by the following values in its generators:

ρa((1 2)) =





0 1

1 0



 , ρa(−1) =





−1 0

0 −1



 , ρa((1 2 3)) =





−1 −1

1 0







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Smooth quotients of abelian surfaces by finite groups... 49

Now, with this basis and this ∆, the analytic representation of B → A is given by the inverse of

the following matrix:

M =





−1 −2

2 1



 .

Indeed, this corresponds to the morphism that sends (x, y, −x − y) ∈ B ⊂ E3 to (−x − 2y, 2x +

y, −x+ y) ∈ A ⊂ E3 and thus its kernel is precisely the elements of the form (x, x, x) ∈ E[3]3 ⊂ B,

that is, ∆. In order to prove that the pairs (A, G) and (B, G) are isomorphic, it suffices thus to

prove that the image of this representation of G under conjugation by M is G once again. Direct

computations give:

Mρa(−1)M
−1 = ρa(−1), Mρa((1 2 3))M

−1 = ρa((1 2 3)),

Mρa((1 2))M
−1 =





0 −1

−1 0



 = ρa((1 2))ρa(−1).

And these clearly generate the same group G.

4 Proof of (3) ⇒ (2)

The only thing left to prove is that Example (c) satisfies property (2) from Theorem 1.1 (the other

two are proved in [1]). Let us then study this example in detail.

Recall that in section 3.2 we proved that the pair (A, G) from Example (c) can be obtained as

follows. Let G = G(4, 2) and let B = E2 with E = C2/Z[i]. Denote by t0 the i-invariant

element in E and denote by q0 the quotient morphism E → E/〈t0〉 ≃ E. Then A = B/∆ with

∆ = 〈(t0, t0)〉 ∈ E
2 = B and the action of G on A is the one induced by B → A.

Note now that G has an index 2 subgroup G1 := G(2, 1) = H1 ⋊ S2, which is thus normal in G

(here, H1 = {±1}
2). Moreover, the pair (B, G1) corresponds to Example (a), so that B/G1 ≃ P

2.

Finally, note that ∆ is an order 2 subgroup of B and thus G acts trivially on it. In particular, the

actions of G and ∆ on B commute and hence we have a commutative diagram of Galois covers

B
∆

//

G1
��

G

��

A

��

G

��

P
2 //

G/G1

��

A/G1

��

B/G // A/G,

where parallel arrows have the same Galois group. Since ∆ and G/G1 have both order 2, we see

then that A/G is a quotient of P2 by the action of a Klein group.



50 R. Auffarth, G. Lucchini Arteche & P. Quezada CUBO
24, 1 (2022)

Proposition 4.1. The quotient A/G is isomorphic to P2.

This proposition finishes the proof of (3) ⇒ (2) in Theorem 1.1.

Remark 4.2. This example was already known to Tokunaga and Yoshida (cf. [6, §5, Table II]).

However, in order to prove that A/G ≃ P2, they cite an article by Švarcman which contains no

proofs (cf. [5]).

Proof. Since A/G is a quotient of P2 by the action of a Klein group K, the only thing we need to

check is that this action gives P2 as a quotient. Note first that the action is faithful since it comes

from the faithful action of G × ∆ on B. Consider then K as a subgroup of PGL3 = Aut(P
2) and

let K1 be its preimage in SL3. This is an order 12 group and hence any 2-Sylow subgroup of K1

gives a lift of K to a subgroup of GL3. This implies that the action lifts to C
3 and it can thus be

seen as a linear representation of K. Since there are exactly four irreducible representations of K,

all of dimension 1, a direct check tells us that any choice of three different representations gives the

same faithful action on P2 up to conjugation, whereas any other choice gives a non-faithful action.

We may assume then that the nontrivial elements xi ∈ K for i = 1, 2, 3 act on P
2 via the diagonal

matrices with 1 on the i-th coordinate and −1 elsewhere. The quotient of P2 by such a group is

the weighted projective space P(2, 2, 2), which is well-known to be isomorphic to P(1, 1, 1) = P2.

This concludes the proof.

Acknowledgments

The first and third authors were partially supported by CONICYT PIA ACT1415. The second

author was partially supported by Fondecyt Grant 11170016 and PAI Grant 79170034.

We would like to thank the anonymous referees for their comments and specially one of them for

showing us a more elegant proof of Lemma 3.1.



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Smooth quotients of abelian surfaces by finite groups... 51

References

[1] R. Auffarth, “A note on Galois embeddings of abelian varieties”, Manuscripta Math., vol. 154,

no. 3–4, pp. 279–284, 2017.

[2] R. Auffarth and G. Lucchini Arteche, “Smooth quotients of abelian varieties by finite groups”,

Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5), vol. 21, pp. 673–694, 2020.

[3] V. Popov. Discrete complex reflection groups, Communications of the Mathematical Institute,

Rijksuniversiteit Utrecht, 15, Netherland: Rijksuniversiteit Utrecht, 1982.

[4] G. C. Shephard and J. A. Todd, “Finite unitary reflection groups”. Canad. J. Math., vol. 6,

pp. 274–304, 1954.

[5] O. V. Švarcman, “A Chevalley theorem for complex crystallographic groups that are generated

by mappings in the affine space C2” (Russian), Uspekhi Mat. Nauk, vol. 34, no.1(205), pp.

249–250, 1979.

[6] S. Tokunaga and M. Yoshida.“Complex crystallographic groups. I.”, J. Math. Soc. Japan, vol.

34, no. 4, pp. 581–593, 1982.

[7] H. Yoshihara, “Galois embedding of algebraic variety and its application to abelian surface”,

Rend. Semin. Mat. Univ. Padova, vol. 117, pp. 69–85, 2007.


	Introduction
	Preliminaries on group actions on abelian varieties
	Proof of (1)(3)
	The case G=G(2,1)
	The case G=G(4,2)
	The case G=G(4,1)
	The case G=G(3,1)
	The case G=G(6,1)
	The case G=G(6,2)
	The case G=G(6,3)
	The case G=G(3,3)
	The case G=G(6,6)

	Proof of (3)(2)