arXiv:1612.05737v1 [] 17 Dec 2016

Polynomial invariants and moduli of generic
two-dimensional commutative algebras
M. Rausch de Traubenberg∗a and M. J. Slupinski†b
a Universit

de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France

b Universit

de Strasbourg, CNRS, IRMA UMR 7501, F-67000 Strasbourg, France

January 9, 2018

Abstract
Let V be a two-dimensional vector space over a field F of characteristic not 2 or 3. We
show there is a canonical surjection ν from the set of suitably generic commutative algebra
structures on V modulo the action of GL(V ) onto the plane F2 . In these coordinates,
which are quotients of invariant quartic polynomials, properties such as associativity and
the existence of zero divisors are described by simple algebraic conditions. The map ν is
a bijection over the complement of a degenerate elliptic curve Γ and over Γ we give an
explicit parametrisation of the fibre in terms of Galois extensions of F. Algebras in ν −1 (Γ)
are exactly those which admit non-trivial automorphisms. We show how ν can be lifted to
a map from the SL(V )−moduli space to an algebraic hypersurface Γ′ in a four-dimensional
vector space whose equation is essentially the classical Eisenstein equation for the covariants
of a binary cubic. This map is the restriction of a surjective map from the set of stable
commutative algebras on V modulo the action of SL(V ) onto Γ′ .

Keywords: Commutative algebras, Moduli space, Binary cubics, Galois extensions.

∗
†

e-mail: michel.rausch@iphc.cnr.fr
e-mail: marcus.slupinski@math.unistra.fr

1

Contents
1 Introduction

3

2 Notations and definitions

9

3 Twisted polynomial invariants and the twisted Eisenstein identity
4 Moduli space of generic algebras with respect to the action of GL(V )
4.1 Résumé and comparison with the results of Anan’in and Mironov . . . .
4.2 Fundamental triples and the invariants p2 , p3 . . . . . . . . . . . . . . . .
4.3 Moduli space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Surjectivity of ν . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Moduli over F2 \ ΓCardano . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Moduli over ΓCardano \ {(0, 0)} . . . . . . . . . . . . . . . . . . . .
4.3.4 Moduli over {(0, 0)}: exceptional algebras . . . . . . . . . . . . .

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21
26
26
29
30
32

5 Applications
5.1 Division algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35
35
36

6 Moduli space of generic algebras with respect to the action of SL(V )

38

7 A linear embedding of stable SL(V )−moduli space

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8 Disc(D)-generic algebras modulo SL(V ) × SL(V ) and binary quadratic forms
modulo SL(V )
52
8.1 Isomorphism of moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
8.2 Group structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
A Non-generic commutative algebras
A.1 Stable commutative algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Non-stable commutative algebras . . . . . . . . . . . . . . . . . . . . . . . . . .

2

56
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59

1

Introduction

If F is a field it is well known that the equivalence classes of separable quadratic extensions of F
(including the split extension) are classified by the Galois cohomology group H 1 (F, Z2 ) ∼
= F∗ /F∗2
[1]. It is easy to check that if K/F is a quadratic extension, the associated “twisted” cubic
Q(x) = x2 ∧ x with values in Λ2 (K) has three distinct roots in a splitting field and generically,
the “twisted” cubic of an arbitrary two-dimensional F−algebra will have this property. Such
algebras we will call generic and in this paper we first investigate the moduli spaces with respect
to various group actions of the set of generic two-dimensional commutative F−algebras. A more
general class of algebras is the class of stable algebras which, by definition are algebras for which
some invariant polynomial does not vanish. In the latter part of the paper (Section 7 and the
Appendix) we study moduli spaces of stable two-dimensional commutative F−algebras and their
relation with moduli spaces of generic two-dimensional commutative F−algebras. Of course, different moduli spaces with respect to different group actions reveal different features. In [2] (p.
224) Bhargava shows that there is a natural group structure on the SL(2, Z) × SL(2, Z)−moduli
space of 2 × 2 × 2 cubes with “twofold symmetry” of fixed nonzero discriminant. We will prove
the analogous result in the context of commutative two-dimensional algebras over a field of characteristic not two or three.
It is natural to consider functions invariant under a group action when looking for coordinates
on the corresponding moduli space. If F is algebraically closed of characteristic not two or three,
we show that coordinates on the GL(2)−moduli space of generic, two-dimensional, commutative
algebras are given by two ratios of “twisted” GL(2)−invariant polynomials which are quartic in
the structure constants of the algebra. This gives a natural parametrisation of the GL(2)−moduli
space by F × F. A similar result was obtained by Anan’in and Mironov in [3] for stable twodimensional algebras. However, if F is not algebraically closed, we show that a point in F×F does
not necessarily characterise a class of generic algebras. Another (discrete) invariant, the splitting
field of the associated twisted cubic, may be needed to distinguish non-isomorphic algebras but
we prove that this is only necessary for equivalence classes of algebras with the property that
their moduli lie on a degenerate elliptic curve ΓCardano in F × F whose equation we give explicitly
(cf Eq.[4.26]).
The parametrisation above has the advantage that many intrinsic properties of an algebra
are described by simple algebraic conditions on the corresponding moduli. This is a first step
to understanding the global “geometry” of moduli space. For example, all equivalence classes of
associative algebras correspond to a single point. This is a version of the celebrated result of B.
Peirce (completed by his son C. Peirce) [4] which says that there are, up to equivalence, seven twodimensional real associative algebras (not necessarily commutative). Perhaps more surprisingly,
equivalence classes of algebras which admit non-trivial automorphisms are characterised by the
fact that their moduli lie on the curve ΓCardano already introduced above.
The curve ΓCardano appears naturally in yet another context: if F is algebraically closed, the
SL(2)−moduli space of generic two-dimensional commutative algebras is a principal F∗ −fibration
over its complement in the GL(2)−moduli space of generic two-dimensional commutative algebras. In fact, we will realise this fibration as a projective fibration by defining a bijective map
from the SL(2)−moduli space of stable commutative algebras onto a homogeneous algebraic hypersurface in a four-dimensional vector space. This map is a lift of the projective embedding of
3

the GL(2)−moduli space of stable two-dimensional commutative algebras defined in [3].
There is a considerable literature on the subject of two-dimensional algebras, the vast majority of which [5, 6, 7, 8, 9, 10, 11, 12] is concerned with finding an exhaustive list of normal
forms for algebras satisfying varying constraints (commutative, division etc.). In most of these
articles the base field is R or algebraically closed but the results of [9, 10] are proved for more
general fields. Let us recall that the motivation for [6] was the fact that classifying systems of
two quadratic differential equations in two variables is equivalent to classifying two-dimensional
commutative algebras.
More relevant to this paper is [3] which, to the best of the authors’ knowledge, is the only article to study the structure of moduli space. In this paper Anan’in and Mironov construct explicitly
an embedding Φ of M into the projective space P8 , where M denotes the GL(2)−moduli space
of stable, two-dimensional (not necessarily commutative) algebras over an algebraically closed
field of characteristic not two or three. The nine affine coordinates are given by SL(2)−invariant
polynomials, quartic in the structure constants, and the authors establish that the image of this
embedding is exactly the intersection of six quadrics. If F is algebraically closed there is a link
between this paper and some of our results which we now explain. Commutative algebras which
are generic in our sense are stable in the sense of [3] (the converse is not true) so it makes sense
to restrict Φ to the moduli space of generic commutative algebras. It is easy to see that the
image of this restriction is a P2 \ P1 embedded in P8 and, with respect to a suitable identification
of F × F with P2 \ P1 their coordinates and our coordinates are the same. This allows us to show
that, in the case of generic commutative algebras, the algebro-geometric singular points of M
eCardano
identified by Anan’in and Mironov are precisely the points of the projective completion Γ
of ΓCardano . In other words a generic commutative algebra defines a singular point of moduli
space in the abstract sense if and only if it admits non-trivial automorphisms. This is also true
for any stable commutative algebra as shown in Appendix A, and it would be very interesting to
know whether this property remains true for an arbitrary two-dimensional algebra. If F is not
algebraically closed, let us recall, as mentioned above, “discrete” invariants are needed to give a
complete parametrisation of the GL(2)−moduli space. In Section 6 we lift this projective embedding of the GL(2)−moduli space of generic commutative algebras in P2 to an embedding of
the corresponding SL(2)−moduli space in a four-dimensional vector space as the set of solutions
of an equation of Eisenstein type.
Finally to complete our survey of the literature, let us mention that in the articles
[13, 14, 15, 16, 17, 18] equivalence classes of twisted binary cubics are parametrised by Galois extensions in the case of fields [14] or by cubic rings in the case of Z [15, 16, 17, 18]. This is
relevant to the present paper since, up to equivalence, certain types of algebras are completely
determined by their associated twisted cubic.
We now give a more detailed account of the results in this paper. Let F be a field of
characteristic not two or three and let V be a two-dimensional F−vector space. We denote by
Ac the set of commutative algebra structures on V , i.e., the set of symmetric bilinear maps from
V × V to V . In Sections 2 and 3 we introduce several “twisted” polynomial invariants on Ac
where by “twisted” polynomial invariant (TPI) we mean a GL(V )−equivariant polynomial map

4

from Ac to some power of Λ2 (V ∗ ) (which from now on we denote by L). The decomposition


Ac ∼
= V ∗ ⊕ S 3 (V ∗ ) ⊗ L−1

into GL(V )−irreducible summands has three important consequences. Firstly, there are no TPIs
of degree less than or equal to three. Secondly, the space of “twisted” binary cubics S 3 (V ∗ )⊗L−1
appears naturally and, as has already been observed in the literature [14, 13], the description of
GL(V )−orbits in this space is simpler than the description of GL(V )−orbits in the space S 3 (V ∗ )
of “untwisted” binary cubics. Thirdly it allows us to define analogues of the the four classical
covariants of binary cubics for Ac . Maybe the most important of these is the discriminant
Disc(Qm ) of the fundamental cubic Qm given by Qm (v) = m(v, v) ∧ v for m ∈ Ac , v ∈ V . The
set of generic algebras Ac3 is then defined to be the set of algebras m such that Disc(Qm ) is non
zero, or equivalently, such that Qm has three distinct roots in a splitting field. The other three
invariants p̃2 , p̃3 and Inv are respectively quartic, quartic and sextic in the structure constants.
It essentially follows from the fact that the covariants of a binary cubic satisfy the Eisenstein
identity [19], that these four invariants also satisfy a kind of Eisenstein identity. Furthermore, it
turns out that they also furnish coordinates (Section 4) for the moduli space of generic algebras.
In fact we have the following theorems (see Theorems 3.17, 4.19, 4.21, 4.24, 5.4, 4.12, Lemma
4.20)
Theorem 1.1 Let m ∈ Ac (not necessarily generic). The polynomial invariants p̃3 (m), p̃2 (m),
Disc(Qm ) and Invm satisfy the following identity in L3 :
27Disc(Qm )p̃3 (m)2 + 4p̃2 (m)3 = −24 · 36 Inv2m .
Note that this identity (a fortiori the four TPIs appearing in it) make sense over any unital
commutative ring (e.g. the integers). In the case of generic m we give a second proof of the
identity based on the fact that p̃3 and p̃2 turn out to be symmetric functions of the roots of a
“modular cubic” (see Corollary 4.10).
Theorem 1.2 If m ∈ Ac3 we define p2 (m), p3 (m) ∈ F by
p2 (m) =

p̃2 (m)
,
Disc(Qm )

p3 (m) =

p̃3 (m)
Disc(Qm )

and the map ν : Ac3 /GL(V ) → F × F by
ν(m) = (p3 (m), p2 (m)) .
The Cardano curve is given by
n
o
2
3
ΓCardano = (p3 , p2 ) ∈ F × F s.t. 27p3 + 4p2 = 0 ,
and we denote by FQm a splitting field of Qm .
(i) The map ν is surjective.

5

(1.1)

(ii) Let m1 , m2 ∈ Ac3 be such that ν(m1 ), ν(m2 ) 6∈ ΓCardano . Then there exists g ∈ GL(V ) s.t.
g · m1 = m2 iff
ν(m1 ) = ν(m2 ) .
(iii) Let m1 , m2 ∈ Ac3 be such that ν(m1 ), ν(m2 ) ∈ ΓCardano . Then there exists g ∈ GL(V ) s.t.
g · m1 = m2 iff
ν(m1 ) = ν(m2 ) and FQm ∼
= FQm′ .
(iv) Let m ∈ Ac3 .
(a) If ν(m) ∈ ΓCardano \ {0} then [FQm : F] ≤ 2. Conversely, if (p3 , p2 ) ∈ ΓCardano \ {0}
and K/F is an extension of degree at most two, then there exists m ∈ Ac3 such that
ν(m) = (p3 , p2 ) and FQm ∼
= K.
(b) If ν(m) = (0, 0) then [FQm : F] = 1, 2, 3 or 6. Conversely, if K/F is a Galois extension
of degree at most six, there exists m ∈ Ac3 such that ν(m) = (0, 0) and FQm ∼
= K.
(v) Let m ∈ Ac3 . Then m is associative iff (p3 (m), p2 (m)) = (16, −12).
g
(vi) The “split” automorphism group Aut(m)
(see Corollary 4.12) of m ∈ Ac3 is given by

 {1} if ν(m) 6∈ ΓCardano ,
∼
g
Z2 if ν(m) ∈ ΓCardano and ν(m) 6= (0, 0) ,
Aut(m)
=

S3 if ν(m) = (0, 0) .

The main idea of the proof is to find a normal form for any generic algebra, which we call a
fundamental triple, and prove its uniqueness (see Section 4.2).
Division algebras have always played a special rôle in mathematics and in section 5 we obtain
the following characterisation of generic two-dimensional commutative division algebras in terms
of our moduli.
Theorem 1.3 Let m ∈ Ac3 .

(i) If p3 (m), p2 (m) 6∈ ΓCardano , then m is a division algebra iff

−3(p2 (m) − p3 (m) + 1)(−27p23 (m) − 4p32 (m))

is not a square in F.

√
(ii) If p3 (m), p2 (m) ∈ ΓCardano \ {0}, write FQm = F( a) where a ∈ F. Then m is a division
algebra iff
−3a(p2 (m) − p3 (m) + 1)
is not a square in F.

(iii) If p3 (m), p2 (m) = (0, 0) then m is a division algebra iff −3 is not a square in the splitting
field FQm of Qm .
6

In Section 6 we give moduli for the action of SL(V ) on generic two-dimensional commutative
algebras. For this we introduce functions invariant under SL(V ) but not under GL(V ) which
distinguish algebras with the same GL(V )−moduli. In order to simplify presentation we state
here the results only in the case when the field F is algebraically closed but for the general case,
which is considerably more complicated, see Theorem 6.5.
Theorem 1.4 Let F be an algebraically closed field.
n
o


c
c
c
2
1. Let A3g = m ∈ A3 s.t. ν(m) 6∈ ΓCardano . The map ν̂s : A3g /SL(V ) → F \ ΓCardano ×


L \ {0} given by
!
Invm
ν̂s (m) = ν(m),
Disc(Qm )
is a bijection.
n
o
2. Let Ac3C = m ∈ Ac3 s.t. ν(m) ∈ ΓCardano \ {0} . The map ν̂ˆs : Ac3C /SL(V ) →

 

2
ΓCardano \ {0} × L \ {0} given by
ˆ
ν̂(m)
s = (ν(m), Disc(Qm ))

is a bijection.
n
o
3. Let Ac3 0 = m ∈ Ac3 s.t. ν(m) = (0, 0) . The map ν̂0 : Ac3 0 /SL(V ) → L2 \ {0} given by
ν̂0 (m) = Disc(Qm )

is a bijection.
In Section 7, using Theorem 1.1, we construct a surjective (bijective if F is algebraically closed)
map from Acst /SL(V ) (here Acst is the set of stable two-dimensional commutative algebras) to a
closed hypersurface in a four-dimensional vector space. The hypersurface is the set of solutions
of what is essentially the classical Eisenstein equation considered as an equation in four variables.
The main results can be summarised as follows:
Theorem 1.5 Let F be algebraically closed and let
o
n
2
2
2
3
2
3
4 6 2
ΓEisenstein = (A, B, D, C) ∈ L × L × L × L s.t. 27DA + 4B = −2 3 C .
1. The map ζs : Acst /SL(V ) → ΓEisenstein given by

ζs (m) = (p̃3 (m), p̃2 (m), Disc(Qm ), Invm ) ,
is a bijection.

7

2. Let F∗ act on ΓEisenstein by:
λ · (A, B, D, C) =

1
1
1
1 
A,
B,
D,
C
λ2 λ2 λ2
λ3

λ ∈ F∗ ,

e Eisenstein ) the quotient of ΓEisenstein by this action. Then ζs induces a
and denote by P(Γ
e Eisenstein ).
bijective map ζg : Acst /GL(V ) → P(Γ

The relationship between moduli spaces of generic two-dimensional commutative algebras and
stable two-dimensional commutative algebras can be summarised in the following commutative
cube (Theorem 7.8 and Appendix A).
ζs
/ ΓEisenstein
F
F✍ ✤
✌✌
✌
✍✍✍ ✤
✌
✍ ✤
✌✌
✍
✤
✌
✍
✌
toto
✤
✍✍
✌✌
✍
✌
✍
✌
p✤
π
✍✍
✤
✌✌
✍
✌
✍✍
✤
✌
toto
✍
✌✌
✤
✍
✌
✍
✌
✤
✍✍
✌✌
✤
✍
✌

✍

ζg
✌✌ ❴ ❴ ❴/ e
c
✍✍✍
❴
❴
❴
❴
❴
❴
❴
❴
❴
✌
P(ΓEisenstein )
Ast /GL(V )
✍
✌✌
✌F
✍F
✌
✍✍✍
✌
✌
✌
✍
✍
✌✌
✌✌
✍
 4 ✍✍
 3 ✌✌
✌
ζs
✌
✍
/ Γ∗
Ac3 /SL(V )
✌✌
✍
Eisenstein
✌
✌
✍
✌✌
✍
✌
✍
✌✌
✌
✍
✌
toto
✍
✌✌
✌
✍
✌
p
π
✍
✌✌
✌
✌
toto ✍ ✍
✌✌
✌
✍
✌✌
✍
✌

3✌
 4 ✍
ζg
e ∗
/ P(Γ
)
Ac /GL(V )

Acst /SL(V )

3

Eisenstein

The front face of the cube involves only two-dimensional generic algebras, the back face only
two-dimensional stable algebras and the maps including the former in the latter can be thought
of as “compactifications”.
In the final section of this paper we consider the invariant obtained by associating to a commutative algebra m the quadratic form Dm given by the determinant of (left) multiplication.
The discriminant of Dm is not just SL(V )−invariant but is invariant under the action of a larger
group SL(V ) × SL(V ) in which it is diagonally embedded. We show that this association establishes a bijection between the set of commutative algebra structures of fixed non-vanishing
discriminant ∆ modulo the action of SL(V ) × SL(V ), and the set of quadratic forms of discriminant ∆ modulo the action of SL(V ). By pullback of the Gauss composition law this allows us
8

to endow the set of SL(V ) × SL(V )−orbits of fixed discriminant ∆ with a group law. In the
context of commutative two-dimensional algebras over a field of characteristic not two or three,
this result is analogous to a result of Bhargava [2] who exhibited a natural group structure on
the SL(2, Z) × SL(2, Z) moduli space of 2 × 2 × 2 cubes with “twofold” symmetry.

2

Notations and definitions

Throughout this paper:
- F is a field of characteristic not two or three;
- V is a two-dimensional vector space over F;
- V ∗ is the dual of V ;
- S 2 (V ∗ ) is the set of symmetric bilinear forms on V ;

- L is the set of anti-symmetric bilinear forms on V (this is one-dimensional);
- L−1 is the dual of L;
- Lr ⊗ Ls is always identified with Lr+s for any integers r, s.
Definition 2.1
1. Let Ac = S 2 (V ⋆ ) ⊗ V . We think of this as the set of commutative algebra structures on V ,
that is, the set of bilinear maps m from V × V to V such that for all v, w ∈ V, m(v, w) =
m(w, v).
2. Let F = {m ∈ Ac s.t. m is a field or m is a split field} (by “ m is a split field” we
mean that m has two idempotents e1 , e2 and an identity 1: m(e1 , e1 ) = e1 , m(e2 , e2 ) =
e2 , m(e1 , e2 ) = 0 and 1 = e1 + e2 ).
As usual, the left multiplication operation Lv : V → V is

Lv (m)(w) := m(v, w) ∀v, w ∈ V, m ∈ Ac .

The group GL(V ) acts on V by


g · m(v, w) = g m g −1(v), g −1(w) , ∀v, w ∈ V, m ∈ Ac .

(2.2)

(2.3)

In 1881 Benjamin Peirce introduced the notion of “idempotent” [4] in order to classify twodimensional associative (not necessarily commutative) real algebras. In fact, one can associate
to any two-dimenional algebra a (twisted) binary cubic on V whose roots correspond to its
idempotents (see also for example [11].
Definition 2.2 Let m be in Ac . The fundamental binary cubic of m is the function Qm : V →
L−1 defined for all v ∈ V by
Qm (v) := m(v, v) ∧ v .

(2.4)

An element v ∈ V is an idempotent of m iff
Qm (v) = 0 .
9

(2.5)

3

Twisted polynomial invariants and the twisted Eisenstein identity

In this section, based on the decomposition of Ac into GL(V )−irreducible summands and inspired
by the classical theory of covariants of binary cubics, we introduce four invariant polynomial
maps from Ac to powers of L. These four maps satisfy an identity analogous to the classical
Eisenstein identify satisfied by the four classical covariants of a binary cubic. Furthermore, as we
shall see in Section 6, they also define an embedding (at least if F is algebraically closed) of the
SL(V )−moduli space of stable two-dimensional commutative algebras onto a hypersurface in a
four-dimensional vector space. This embedding turns out to be a lift of the projective embedding
of the GL(V )−moduli space of stable two-dimensional commutative algebras defined in [3].
It has been known for a long time that one can associate three covariants to a homogeneous
cubic polynomial in two variables [19]. These are polynomial functions of the cubic of which
perhaps the most important is the discriminant, and we will now generalise them to the case of
“twisted” cubics.
We begin with the discriminant. Recall that the classical discriminant of P (x, y) = Ax3 +
2
Bx y + Cxy 2 + Dy 3 is given by
P 7→ 18ABCD + B 2 C 2 − 4AC 3 − 4B 3 D − 27A2 D 2 .
This formula defines a quartic function of P which is invariant under special linear transformations but not under a general linear transformation. In order to define a GL(V )−invariant
“discriminant” we need to consider quartic functions of the cubic P with values in powers of the
one dimensional vector space L. Since λ Id acts on cubic polynomials by 1/λ3 and on L−1 by
λ2 , it follows that the formula
Discc (P ) = (18ABCD + B 2 C 2 − 4AC 3 − 4B 3 D − 27A2 D 2 )(ǫ1 ∧ ǫ2 )⊗

6

defines a GL(V )−equivariant map Discc : S 3 (V ∗ ) → L6 , where {ǫ1 , ǫ2 } is the dual basis of
{e1 , e2 }. Motivated by this definition of the classical discriminant of a cubic we define Disc:
S 3 (V ∗ ) ⊗ L−1 → L2 by


Disc (Ax3 + Bx2 y + Cxy 2 + Dy 3)(e1 ∧ e2 ) =

18ABCD + B 2 C 2 − 4AC 3 − 4B 3 D − 27A2 D 2 (ǫ1 ∧ ǫ2 )2 .
If m ∈ Ac , we now give the explicit formulæ for Qm and Disc(Qm ) in terms of the structure
constants of m. Let (e1 , e2 ) be a basis of V with dual basis (ε1 , ε2) and m : V × V → V be given
by:
m(e1 , e2 ) = ae1 + be2 , m(e2 , e2 ) = ce1 + de2 ,
m(e1 , e2 ) = ee1 + f e2 , m(e2 , e1 ) = ee1 + f e2 .

10

(3.6)

Then Qm and Disc(Qm ) are given by


Qm =
− bx3 + (a − 2f )x2 y + (2e − d)xy 2 + cy 3 (e1 ∧ e2 ) ,
(3.7)

Disc(Qm ) =
− 36bcf d + 72bcf e + 18bcad − 36bcae + 4f 2d2 − 16f 2 de + 16f 2 e2 − 4f ad2
+16f ade − 16f ae2 + a2 d2 − 4a2 de + 4a2 e2 − 4bd3 + 24bd2 e − 48bde2 + 32be3

+32cf 3 − 48cf 2 a + 24cf a2 − 4ca3 − 27b2 c2 (ε1 ∧ ε2 ) ⊗ (ε1 ∧ ε2 ) .

It follows from this that given any binary cubic G there exists m in Ac such that Qm = Ge1 ∧ e2 .
Recall also that G is irreducible iff its splitting field is of degree three or six, and in this case
the degree is three iff Disc(G) is a square.
Remark 3.1 In [3] Anan’in and Mironov (p 4483) attach to any two-dimensional algebra nine
“projective coordinates”:
[x1 , x2 , x3 , y1 , y2 , y3, z0 , z1 , z2 ] .
These are all quartic functions of the structure constants and if the algebra is commutative only
x1 , x2 , z0 do not vanish identically. One can check that the link between our invariant Disc(Q)
and z0 is Disc(Q) = 81z0 (ǫ1 ∧ ǫ2 )2 where (ǫ1 , ǫ2 ) is the basis dual to the basis (e1 , e2 ) given by
their Eq.[6] (p 4483).
The normalisation of the discriminant above has been chosen so that if m is a split field then
Disc(Qm ) ∈ L2 is a perfect square. In fact using Disc, we have a natural parametrisation of field
structures on V up to SL(V )−equivalence which extends the well-known natural parametrisation
by F∗ /F∗2 of field structures on V up to GL(V )−equivalence:
Proposition 3.2 Recall that F = {m ∈ Ac s.t. m is a field or m is a split field}. Let:
• s : L2 \ {0} → F∗ /F∗2 be defined by


s λ(ω0 ⊗ ω0 ) = [λ] ,

for ω0 any in L. This is clearly independent of the choice of ω0 .
• δ : F /GL(V ) → F∗ /F∗2 be defined by
δ([m]) = [z 2 ] ,
where z is any (nonzero) purely imaginary element of m.
• p : F /SL(V ) → F /GL(V ) be the natural projection.
Then the diagram
F /SL(V )

Disc

/

L2 \ {0}

p

s



F /GL(V )

δ

/



F∗ /F∗2

is commutative and the maps Disc and δ are bijections.
11

Proof. The fact that δ is a bijection is well known (see e.g. [1]) and one can check easily that
the diagram is commutative.
To prove the surjectivity of Disc, let x ∈ L2 \ {0}. Then by the surjectivity of δ there exists
m ∈ F such that δ([m]GL(V ) ) = [x]. Hence by the definition of s there exists λ ∈ F∗ such that
Disc([m]SL(V ) ) = λ2 x
and from this it follows that
1
Disc( Id · m) = x .
λ
This prove that the map Disc is surjective.
To prove the injectivity of Disc, let m1 , m2 ∈ F be such that
Disc([m1 ]SL(V ) ) = Disc([m2 ]SL(V ) ) .
Then
δ([m1 ]GL(V ) ) = δ([m2 ]GL(V ) ) ,
and by the injectivity of δ there exists g ∈ GL(V ) such that m1 = g · m2 . Thus
Disc(m1 ) =

1
Disc(m2 )
det(g)2

from which it follows that det(g) = ±1. However, we can always assume that det(g) = 1 since if
not, we compose with the non-trivial element of the Galois group which is always of determinant
equal to minus one. This prove injectivity.
QED

Remark 3.3 Our convention is the following: if (e1 , e2 ) is a basis of V with dual basis (ε1 , ε2 )
and vi = xi e1 + yi e2 are vectors in V then
(ε1 ∧ ε2 )(v1 , v2 ) = (x1 y2 − x2 y1 ) .
We now give the decomposition of Ac into its GL(V )−irreducible components and for this
we need the following
Definition 3.4 Let m ∈ Ac and let Lv (m) be left multiplication as in (2.2). Then Tm ∈ V ∗ and
Dm ∈ S 2 (V ∗ ) are defined by
Tm (v) = Tr(Lv (m)) , Dm (v) = det(Lv (m)) , ∀v ∈ V .

(3.8)

In terms of the structure constants of m (see (3.6)) this gives the formuæ,
Tm (xe1 + ye2 ) = (a + f )x + (e + d)y ,
Dm (xe1 + ye2 ) = (af − be)x2 + (ad − bc)xy + (ed − cf )y 2 .
12

(3.9)

Proposition 3.5


∗
3
∗
−1
∼
1. As GL(V )−representations we have: A = V ⊕ S (V ) ⊗ L
.
c

2. For m ∈ Ac define m′ ∈ Ac by

m′ (v, w) =
Then


1
Tm (v)w + Tm (w)v .
3

m = m′ + (m − m′ ) ,
is the irreducible decomposition of m.
Proof.
(1): Let K = {m ∈ Ac
GL(V )−representations:

s.t.

Qm ≡ 0} and consider the exact sequence of

{0} −→ K −→ Ac −→ S 3 (V ∗ ) ⊗ L−1 −→ {0} .
Since the map Q is not identically zero and S 3 (V ∗ )⊗L−1 is irreducible, it follows that Q is surjective and that K is a two-dimensional representation of GL(V ). Define the GL(V )−equivariant
map k : V ∗ → Ac by
kα (v, w) = α(v)w + α(w)v , ∀α ∈ V ∗ , ∀u, v ∈ V .
Since Qkα (v) = 2α(v)v ∧ v = 0, we have Im(kα ) ⊂ K and, since V ∗ is irreducible of the same
dimension as K, the map k is an isomorphism of GL(V )−representations.
(2): For all m ∈ Ac , it is evident that m′ = k 1 Tm and easily checked that Tm′ = Tm . QED
3

Remark 3.6 The space of twisted binary cubics B = S 3 (V ∗ ) ⊗ L−1 appearing in the above
decomposition has already been studied in the literature. In both [13, 14] it was explicitly pointed
out that the crucial advantage of B over the space of untwisted binary cubics S 3 (V ∗ ) is that
λ·IdV acts by λ1 IdB instead of by λ13 IdS 3 (V ∗ ) (recall that in [13] the authors are working over Z
whereas in [14] the author is working over over a number field).
Remark 3.7 Let q ∈ S 3 (V ∗ ) ⊗ L−1 be a “twisted” binary cubic. It follows from the proposition
that there exists a unique algebra m such that Qm = q and Tm = 0. Explicitly, if (e1 , e2 ) is a
basis of V and q(xe1 + ye2) = (αx3 + βx2 y + γxy 2 + δy 3 )e1 ∧ e2 in this basis, it is easy to see
that:
1
1
1
1
m(e1 , e1 ) = βe1 − αe2 , m(e2 , e2 ) = δe1 − γe2 , m(e1 , e2 ) = γe1 − βe2 .
3
3
3
3
In the rest of this paper an important rôle will be played by twisted polynomial invariants
(TPI) on Ac . A basic example of a TPI is the quartic polynomial Disc: Ac → L2 and this
motivates the following definition:
13

Definition 3.8 A twisted polynomial invariant (TPI) of degree (2k, k) on Ac is a degree 2k
polynomial map P : Ac 7→ Lk which is GL(V )−equivariant, i.e., P is an element of S 2k (A∗c )⊗Lk
which is fixed by the action of GL(V ).
Using Proposition 3.5 we can adapt the classical theory of polynomial invariants of binary
cubics to construct TPIs as follows. Setting B = S 3 (V ∗ ) ⊗ L−1 , the decomposition Ac = V ∗ ⊕ B
implies the GL(V )−equivariant decomposition of degree (2k, k) polynomials on Ac :
M
S 2k (Ac∗ ) ⊗ Lk =
S m (V ) ⊗ S n (B ∗ ) ⊗ Lk .
m+n=2k

Recall that there is an GL(V )−equivariant isomorphism
V ∼
= V ∗ ⊗ L−1
given by:
iα b o

✤

α⊗b

∀α ∈ V ∗ , b ∈ L−1 ,

which induces an GL(V )−equivariant isomorphisms
S m (V ) ⊗ S n (B ∗ ) ⊗ Lk ∼
= S m (V ∗ ) ⊗ Lk−m ⊗ S n (B ∗ ) .
and
S m (V ) ⊗ S n (B ∗ ) ⊗ Lp ∼
= S m (V ∗ ) ⊗ Lp−m ⊗ S n (B ∗ ) .
Hence TPIs of degree (2k, k) are described by the GL(V )−equivariant isomorphism:
M
S 2k (Ac∗) ⊗ Lk =
S m (V ∗ ) ⊗ S n (B ∗ ) ⊗ Lk−m .

(3.10)

m+n=2k

Now we recall that the algebra of classical covariants of the space of binary cubics S 3 (V ∗ )
is the algebra of GL(V )−equivariant polynomial maps f : S 3 (V ∗ ) → S(V ∗ ), i.e., the
GL(V )−invariant elements of

S (S 3 (V ∗ ))∗ ⊗ S(V ∗ ) = S(B ∗ ) ⊗ S(L−1 ) ⊗ S(V ∗ ) .

Hence by (3.10) each classical covariant suitably adapted to our situation will give rise to a TPI.
For this we need the following definition:

Definition 3.9 Let E be a vector space, let α ∈ V ∗ and let P ∈ S k (V ∗ ) ⊗ E. Then we denote
by P (α) the element of Lk ⊗ E defined by:
P (α) = P (v) ⊗ ω k ,

(3.11)

where v ∈ V and ω ∈ L are chosen so that α = iv ω. One easily checks that this definition is
independent of the choice of such pair (v, ω).
14

Let us illustrate this with: α = Tm , P = f (Qm ) where m ∈ Ac and f : B 7→ S k (V ∗ ) ⊗ Lp is
a polynomial function. In the basis of (3.6) we have
Tm (xe1 + ye2 ) = (a + f )x + (e + d)y ,
in other words,
Tm = (a + f )ǫ1 + (e + d)ǫ2 .
Hence
Tm = i

(e+d)e1 −(a+f )e2

and

 ǫ1 ∧ ǫ2 ,


f (Qm )(Tm ) = f (Qm ) (e + d)e1 − (a + f )e2 ⊗ (ǫ1 ∧ ǫ2 )deg(f ) .

(3.12)

Note that the function f (Qm )(Tm ) is of bidegree (k, deg(f )) (see (3.10)). We now give explicit
formulæ for the TPIs on Ac which are obtained by taking for f in (3.12) the classical covariants
of a binary cubic [19].
The simplest example is obtained by taking for f the covariant Disc: B → L2 (corresponding
to n = 4, m = 0, k = 2 in (3.10)) which gives rise to Disc(Qm )(Tm ) : Ac → L2 of bidegree (4, 2)
(see (3.7) for the explicit formula).
The second example is obtained by taking for f the covariant Id: B → S 3 (V ∗ ) ⊗ L−1 (corresponding to m = 3, n = 1, k = 2 in (3.10)):
Definition 3.10 If f : B → S 3 (V ∗ )⊗L−1 is the identity then we define p̃3 : Ac → L2 of bidegree
(4, 2) by
p̃3 (m) = −8Qm (Tm ) .
In the basis (3.6) we have

p̃3 (m) =
− 32f ade + 24bde2 + 24bd2 e + 24cf a2 + 24cf 2 a + 8a2 de
−40f ae2 + 8f ad2 − 40f 2 de − 8f 2 d2 − 32f 2 e2 + 16a2 d2

−8a2 e2 + 8bd3 + 8be3 + 8cf 3 + 8ca3 (ǫ1 ∧ ǫ2 )2 .

(3.13)

Remark 3.11 In terms of the coordinates of [3] we have p̃3 = 81x2 (ǫ1 ∧ ǫ2 )2 where (ǫ1 , ǫ2 ) is the
basis dual to the basis (e1 , e2 ) given by their Eq.[6] (p 4483).
We now recall the definition of the the two remaining classical covariants of a twisted binary
cubic Q ∈ B.

15

Definition 3.12 Let (e1 , e2 ) be a basis of V , let x, y be the coordinates of v ∈ V in this basis
and let Q = Q1 e1 ∧ e2 ∈ B.
1. The map µ : B → S 2 (V ∗ ) is defined by taking the Hessian of Q1 :
∂ 2 Q1 ∂ 2 Q1  ∂ 2 Q1 2
µ(Q) =
−
.
∂x2 ∂y 2
∂x∂y

One checks that the quadratic form µ(Q) does not depend on the choice of (e1 , e2 ). In the
basis of (3.6) this gives
µ(Q)(xe1 + ye2 ) =

(−16f 2 + 16f a − 4a2 + 12bd − 24eb)x2
+(−36bc − 8f d + 4ad − 8ae + 16f e)yx
+(16de − 16e2 + 12ac − 24cf − 4d2 )y 2 ,

(3.14)

2. The map G : B → S 3 (V ∗ ) is defined by taking the Poisson bracket of Q1 and µ:
G(Q) =

∂Q1 ∂µ ∂Q1 ∂µ
−
.
∂x ∂y
∂y ∂x

One checks that the binary cubic G(Q) does not depend on the choice of (e1 , e2 ). In the
basis of (3.6) this gives
G(Q)(xe1 + ye2 ) =


2
2
2
3
3
− 144bf e + 72bae + 96f a + 72bf d − 36bad − 48f a + 108b c − 64f + 8a x3

+ 48f ad − 288bde + 96f 2 e + 288be2 − 12a2 d + 24a2 e

(3.15)
−108bac + 216bcf − 48f 2 d − 96f ae + 72bd2 yx2

+ 72a2 c − 12ad2 + 48ade − 48ae2 + 96f e2 − 96f de

−288f ac + 288cf 2 + 24f d2 − 108bcd + 216bce y 2 x


+ 72dcf + 72eac − 144ecf − 36dac + 96de2 + 108c2b + 8d3 − 64e3 − 48d2e y 3
Note that these covariants are invariant under the action of GL(V ) whereas the classical
covariants of (untwisted) binary cubics are only invariant under the action of SL(V ). This means
that that we have to slightly adapt the Eisenstein identity satisfied by the classical covariants to
our situation.
Proposition 3.13 The values of the covariants Disc, Id, µ, G at any v in V satisfy the Eisenstein identity
16 × 27Disc(Q)Q(v)2 + µ(Q)(v)3 = −G(Q)(v)2 .
Proof. A straightforward consequence of the classical Eiseinstein equation [19].
Taking for f in (3.12) the covariants µ and G we obtain two new TPIs.
16

QED

Definition 3.14
1. If f : B → S 2 (V ∗ ) is the covariant µ we define p̃2 : Ac → S 2 (V ∗ ) of bidegree (4, 2) by
p̃2 (m) = µ(Qm )(Tm ) .
In the basis of (3.6) this gives

p̃2 (m) = 36bcf d + 36bcf e + 36bcad + 36bcae + 60f ade − 36bde2

−36cf 2 a + 12a2 de − 24f ae2 + 12f ad2 − 24f 2 de − 12f 2 d2

(3.16)

−48f 2 e2 − 12a2 d2 − 12a2 e2 + 12bd3 − 24be3 − 24cf 3 + 12ca3 (ǫ1 ∧ ǫ2 )2 .

2. If f : B → S 6 (V ∗ ) is the covariant G we define Inv: Ac → S 3 (V ∗ ) of bidegree (6, 3) by
Invm =

1
G(Qm )(Tm ) .
54

In the basis of (3.6) this gives

Invm =
− 2a3 cdf − 4a3 cef − 6a2 bc2 f + 6b2 cde2 − 6a2 de2 f

+4bd3 ef − 12bde3 f + 12acef 3 − 8de2 f 3 + 6a2 bce2
+2acdf 3 + 4a2 e3 f − 2abd4 − 4abe4 + 2ad3 f 2 + 8ae3 f 2
+2b2 cd3 + 2b2 ce3 − 2bc2 f 3 − 8be4 f + 4cdf 4 + 8cef 4
−4d2 ef 3 − 2abde3 + 6abd2 e2 − 2a2 d3 f − 2a3 de2 + 2a4 cd
−2a3 bc2 + 2a3 d2 e + 6b2 cd2 e + 2abd3 e − 6bcd2 f 2 + 6ad2 ef 2



(3.17)

−6abc2 f 2 − 6bcdef 2 + 6abce2 f − 6abcd2 f + 6a2 bcde − 6a2 cdf 2 × (ǫ1 ∧ ǫ2 )3 .
Remark 3.15 In terms of the coordinates of [3] we have p̃2 = 81x1 (ǫ1 ∧ ǫ2 )2 where (ǫ1 , ǫ2 ) is the
basis dual to the basis (e1 , e2 ) given by their Eq.[6] (p 4483).
Remark 3.16 In contrast to Disc(Q), p̃3 and p̃2 which are quartic in the structure constants
and correspond to coordinates in [3], the invariant Inv is sextic and is not one of the coordinates
in [3]. It turns out that these four invariants satisfy a twisted Eisentein identity (see below)
which is essential to our parametrisation of the SL(V )−moduli space of commutative algebras
in Section 6.
We now prove the twisted Eisenstein identity mentioned above. This will be a consequence
of Proposition 3.13.
Theorem 3.17 Let m ∈ Ac . Then as elements of Λ2 (V ∗ )6 we have the identity:
27Disc(Qm )p̃3 (m)2 + 4p̃2 (m)3 = −(4 × 27)2 Inv2m .
17

(3.18)

Proof. Since
p̃3 (m) = −8Qm (Tm ) ,
p̃2 (m) = µ(Qm )(Tm ) ,
1
Invm =
G(Qm )(Tm ) ,
54
the theorem follows from Proposition 3.13.
QED
To conclude this section we introduce one more TPI on Ac which, although it can be expressed
in terms of the polynomial invariants above (Proposition 3.19), is of interest in its own right and
will be useful later.
Definition 3.18 Let m ∈ Ac and let Lv be the left multiplication by v ∈ V . Define the quartic
twisted polynomial invariant Disc◦D : Ac → L2 by
Disc ◦ Dm (v) = Disc(det(Lv )) .
In the basis of (3.6) we have


Disc(Dm ) =
a2 d2 − 2bcad + b2 c2 − 4f ade + 4cf 2 a + 4bde2 − 4bcf e (ǫ1 ∧ ǫ2 )2 (3.19)
The relationship between Disc(D) and the TPIs appearing in the twisted Eisenstein identity
is given by the following proposition for which we omit the proof.
Proposition 3.19 Let m ∈ Ac . Then as elements of L2 we have the identity
27Disc(Dm ) = p̃3 (m) − p̃2 (m) − Disc(Qm ) .
Remark 3.20 All of the TPIs appearing in the identities of Theorem 3.17 and Proposition 3.19
are defined over Z and hence the identities themselves are valid over Z.
Finally let us point out that one can always generate “new” TPIs from “old” TPIs by applying
to them natural operations on A = V ∗ ⊗ V ∗ ⊗ V , i.e., elements of GL(A) which commute with
the action of GL(V ). For example in this way, the invariant D will generate three (in general)
linearly independent invariants, say D, D ′ , D ′′ . These are essentially the three quadratic forms
introduced by Bhargava in [2]. If the multiplication is commutative it turns out there are only
three possiblities: (i) D = D ′ = D ′′ ; (ii) D = D ′ = −D ′′ ; (iii) D = D ′ and D ′′ independent of
D. Explicitly, in the basis of (3.6) with v = xe1 + ye2, we have the following formulæ
Dm (v) = (af − be)x2 + (ad − bc)xy + (ed − cf )y 2 ,
′′
Dm
(v) = (−f 2 + bd)x2 + (2ef − ad − bc)xy + (−e2 + ac)y 2 .
′′
Notice that if Tm = 0 a + f = d + e = 0 and hence Dm = Dm
. Bhargava observed an analogous
property for cubes with “twofold symmetry” or “threefold symmetry” [2], p. 225.
In a forthcoming paper we will investigate invariants of arbitrary two-dimensional algebras
in more detail.

18

4

Moduli space of generic algebras with respect to the
action of GL(V )

4.1

Résumé and comparison with the results of Anan’in and Mironov

In this section we consider the set of generic commutative algebras
n
o
Ac3 = m ∈ Ac s.t. Disc(Qm ) 6= 0 ,

the set of stable commutative algebras [3]
n
o
Acst = m ∈ Ac s.t. ∃ SL(V ) − invariant polynomial F 6≡ 0 on Ac s.t. F (m) 6= 0 (4.20)
and especially their moduli spaces

Mc3 = Ac3 /GL(V ) , Mcst = Acst /GL(V ) .
Commutative algebras which are generic are stable but the converse is not true (Appendix A).
Recall that p̃2 (m), p̃3 (m) and Disc(Qm ) are obtained by evaluation of the three classical covariants
of the fundamental cubic Qm of m at the “trace” Tm (see Definition 3.4) and hence the functions
p2 , p3
p3 (m) =

p̃3 (m)
p̃2 (m)
, p2 (m) =
,
Disc(Qm )
Disc(Qm )

(4.21)

are well defined on the moduli space Mc3 and take their values in F. For m ∈ Ac3 we introduce
the following notations:
1. ν̂(m) = (p3 (m), p2 (m));
2. FQm is the splitting field of Qm (this is well defined up to equivalence of field extensions);
n
o
3. ΓCardano = (a, b) ∈ F × F s.t. 27a2 + 4b3 = 0 ;

4. Ek denotes the set of equivalences classes of extensions of degree at most k which are
isomorphic to splitting fields of cubic polynomials.
The essential result of this section can now be summarised in the
Theorem 4.1 Let m, m′ be in Ac3 .
(i) There exists a non-trivial automorphism of m iff ν̂(m) ∈ ΓCardano .
(ii) Suppose ν̂(m), ν̂(m′ ) 6∈ ΓCardano . Then there exists g ∈ GL(V ) such that m′ = g · m iff
ν̂(m) = ν̂(m′ ) and ν̂ establishes a bijection of Mc3 \ ν̂ −1 (ΓCardano ) with F2 \ ΓCardano .
(iii) Suppose ν̂(m), ν̂(m′ ) ∈ ΓCardano . Then exists g ∈ GL(V ) such that m′ = g · m iff ν̂(m) =
ν̂(m′ ) and FQm ∼
= FQ′m .
19


(iii.1) The map (ν̂, FQ• ) establishes a bijection between ν̂ −1 ΓCardano \ {0} and ΓCardano \
{0} × E2 ;
(iii.2) The map FQ• establishes a bijection between ν̂ −1 {0} and E6 ;

Remark 4.2 Note that 27p3 (m)2 + 4p2 (m)3 is minus the discrimant of the cubic x3 + p2 (m)x −
p3 (m). By Theorem 3.17 this equals −24 · 36 · Inv2m /Disc(Qm )3 .
If F is an algebraically closed field, Anan’in and Mironov [3] constructed an embedding Φ :
M → P8 of M, the GL(V )−moduli space of stable two-dimensional algebras, in the projective
space P8 . By Remarks 3.1, 3.11 and 3.15 the restriction of Φ to Mc3 is given by
Φ(m) = [p̃2 (m), p̃3 (m), 0, 0, 0, Disc(Qm ), 0, 0]

(4.22)

which clearly defines an embedding (also denoted Φ) of Mc3 in P2 :
Φ(m) = [p̃2 (m), p̃3 (m), Disc(Qm )] .

(4.23)

It follows from the definition of Ac3 that

 n
o
c
2
Φ M3 = [a, b, c] ∈ P s.t. c 6= 0 .

Hence we can think of the map ν̂ : Mc3 → F × F of Theorem 4.1 given by
!
p̃2 (m)
p̃3 (m)
= (p3 (m), p2 (m)) ,
,
ν̂(m) =
Disc(Qm ) Disc(Qm )
as providing two affine coordinates on the moduli space Mc3 .
In order to prove the “projective” versions Theorem 4.1 we need to consider the homogenisation of ΓCardano :
(
)
eCardano = [a, b, c] ∈ P2 s.t. 27ca2 + 4b3 = 0 .
Γ

e Cardano then Φ(m) uniquely determines the
We will show in Appendix A that if Φ(m) 6∈ Γ
equivalence class of the stable commutative algebra m. What is more, we will give a complete set of discrete invariants which distinguish stable commutative algebras m, m′ such that
e Cardano .
Φ(m) = Φ(m′ ) ∈ Γ
Finally to complete this section let us point out that, as we shall see in Sections 6 and 7, the
map ζ : Acst /SL(V ) → W (here W is a suitable four-dimensional vector space) given by
ζ(m) = (p̃3 (m), p̃2 (m), Disc(Qm ), Invm ) ,
defines an embedding (at least if F is algebraically closed) of the SL(V )−moduli space of stable,
commutative, two-dimensional algebras onto the Eisenstein hypersurface
(
)
ΓEisenstein =

(a, b, c, d) ∈ W s.t. 27ca2 + 4b3 + 24 36 d2 = 0
20

.

Since (p̃3 (m), p̃2 (m), Disc(Qm )) 6= (0, 0, 0), one can define the map π : ΓEisenstein → P2 given by


π p̃3 (m), p̃2 (m), Disc(Qm ), Invm = [p̃3 (m), p̃2 (m), Disc(Qm )] ,

(some care must be taken to make sense of this) and then
(
)


eCardano = (a, b, c, d) ∈ ΓEisenstein s.t. d = 0 .
π −1 Γ

4.2

Fundamental triples and the invariants p2 , p3

The main tool in the proof of Theorem 4.1 will be the fact that one can associate to m in Ac3
a unique triple of vectors in VFQm (i.e. V tensored by the splitting field FQm of Qm ) together
with a unique triple of elements of FQm . Very surprisingly, it turns out that the two polynomial
invariants p2 and p3 in (4.21) are essentially the elementary symmetric functions of these three
numbers. Recall that since Disc(Qm ) 6= 0, the polynomial Qm has three distinct roots in its
splitting field.
Remark 4.3 Since Qm is a cubic, the degree of a splitting field is either one, two, three or six.
Remark 4.4 If FQm is a splitting field of Qm we can write Qm = F1 F2 F3 ⊗ b with b ∈ Λ2 (VFQm )
are pairwise independent. The Galois group Gal(FQm /F)
and Fi ∈ (VFQm )∗ such that (F1 , F2 , F3 ) 
permutes the elements of the set R = Ker(F1 ), Ker(F2 ), Ker(F3 ) and this gives a homomorphism to the permutation group S3 (R). Note that although the factorisation of Qm is not unique,
the set R is uniquely defined by Qm .
Theorem 4.5 Let m ∈ Ac3 , let Qm be its fundamental cubic and let VFQm = V ⊗F FQm , where
FQm is the splitting field of Qm .
(i) There exists a triple ((f1 , γ1 ), (f2 , γ2 ), (f3 , γ3 )) ∈ (VFQm × FQm )3 such that
a)
b)
c)
d)

(f1 , f2 , f3 ) are pairwise linearly independent;
f1 + f2 + f3 = 0 ,
fi2 = γi fi ,
γ1 + γ2 + γ3 = −1 .

(ii) The triple of a) is unique up to permutation.
Proof. By hypothesis there exist three non-zero linear forms F1 , F2 , F3 ∈ VF∗Qm which are pairwise
independent and b ∈ Λ2 (VFQm ) such Qm (v) = F1 (v)F2 (v)F3 (v)b for all v in V . The vector space
n
o
B = (f1 , f2 , f3 ) ∈ Ker(F1 ) × Ker(F2 ) × Ker(F3 ) : f1 + f2 + f3 = 0 ,

is one-dimensional. Since Qm (v) = m(v, v) ∧ v and Qm (fi ) = 0, there exist three linear forms
γi : Ker(Fi ) → FQm such that
m(fi , fi ) = γi (fi )fi .
21

We define γ : B → FQm by
γ(f1 , f2 , f3 ) = γ1 (f1 ) + γ2 (f2 ) + γ3 (f3 ) .

Lemma 4.6 The map γ is non-zero.
Proof. Suppose for contradiction that γ ≡ 0. Then

1
m(f1 + f2 , f1 + f2 ) − m(f1 , f1 ) − m(f2 , f2 )
2
 1

1
=
m(f3 , f3 ) − m(f1 , f1 ) − m(f2 , f2 ) = γ3 (f3 )f3 − γ1 (f1 )f1 − γ2 (f2 )f2
2
2
1
1
γ2 (f2 )f1 + γ1 (f1 )f2 .
=
2
2

m(f1 , f2 ) =

But now for any vector v = xf1 + yf2 we have
m(v, v) = x2 γ1 (f1 )f1 + y 2 γ2 (f2 ) + xy(γ2 (f2 )f1 + γ1 (f1 )f2 ) = (xγ1 (f1 ) + yγ2(f2 )v .
From this Qm (v) = 0 for all v ∈ VFQm which is a contradiction.

QED

By Lemma 4.6 there exists a unique (f1 , f2 , f3 ) ∈ B such that γ(f1 , f2 , f3 ) = −1 and this
completes the proof of the first part of the theorem.
To prove unicity assume now that one can find another triple ((f1′ , γ1′ ), (f2′ , γ2′ ), (f3′ , γ3′ )) ∈
(VFQm × FQm )3 satisfying
i)
ii)
iii)
iv)

(f1′ , f2′ , f3′ ) pairwise linearly independent,
f1′ + f2′ + f3′ = 0 ,
fi′ 2 = γi′ fi′ ,
γ1′ + γ2′ + γ3′ = −1 .

Since the fi are idempotent and pairwise linearly independent, renumbering if necessary, there
exist non-zero λ1 , λ2 , λ3 ∈ K such that
fi′ = λi fi .
Substituting in ii) gives
λ1 f1 + λ2 f2 + λ3 f3 = 0 ,
and since f3 = −f1 − f2 and (f1 , f2 ) is a basis of VFQm , we get λ1 = λ2 = λ3 = λ.
Substituting fi′ = λfi in iii) gives γi′ = λγi and finally, substituting in iv), gives λ = 1. QED
Remark 4.7 The Galois group of the extension FQm /F acts on VFQm by automorphism and hence
by Theorem 4.5 (ii), each element induces a permutation of the triple ((f1 , γ1 ), (f2 , γ2), (f3 , γ3)).
22

As an application of Theorem 4.5 we now characterise the possible automorphism groups of
m ∈ Ac3 . For this we need the notion “idemvalue” which is well defined by Theorem 4.5
Definition 4.8 The fundamental triple of m ∈ Acc is the triple ((f1 , γ1), (f2 , γ2 ), (f3 , γ3 )) associated to m by Theorem 4.5. The idemvalues of m are γ1 , γ2 , γ3 and the reduced idemvalues of
m are ∆i = 3γi + 1. Note that
∆1 + ∆2 + ∆3 = 0 .
Theorem 4.9 Let m ∈ Ac3 and let ∆1 , ∆2 , ∆3 be the reduced idemvalues. Then the polynomial
invariants p2 (m), p3 (m) are given by
p2 (m) = ∆1 ∆2 + ∆2 ∆3 + ∆3 ∆1 ,
p3 (m) = ∆1 ∆2 ∆3 .
Proof. In the basis (f1 , f2 ) the multiplication table is given by
∆1 − 1
∆2 − 1
2 + ∆2
2 + ∆1
f1 , m(f2 , f2 ) =
f2 , m(f1 , f2 ) =
f1 +
f2 . (4.24)
3
3
6
6
A straightforward but long computation gives (see (3.7), (3.19) and (3.17))
m(f1 , f1 ) =

Disc(Qm )(v1 , v2 , v3 , v4 ) = v12 v34 ,
1
Disc(Dm )(v1 , v2 , v3 , v4 ) =
(∆1 − 1)(∆2 − 1)(∆3 − 1)v12 v34 ,
27
1
(∆1 − ∆2 )(∆2 − ∆3 )(∆3 − ∆1 )v12 v34 v56 ,
Invm (v1 , v2 , v3 , v4 , v5 , v6 ) = −
108
where vij = xi yj − xj yi and hence (see (3.13) and (3.16))

(4.25)

p2 (m) = ∆1 ∆2 + ∆2 ∆3 + ∆3 ∆1 ,
p3 (m) = ∆1 ∆2 ∆3 .
QED
Corollary 4.10 (Alternative proof of Theorem 3.17 for a generic algebra.) Let m be in Ac3 .
Then as elements of L6 we have the identity
27Disc(Qm )p̃3 (m)2 + 4p̃2 (m)3 = −24 · 36 Inv2m .
Proof. Since Disc(Qm ) 6= 0, the identity above in L6 is equivalent to the identity in F
27p3 (m)2 + 4p2 (m)3 = −24 · 36

Inv2m
,
Disc(Qm )3

which, using (4.25), reduces to

2
27p3 (m)2 + 4p2 (m)3 = − (∆1 − ∆2 )(∆2 − ∆3 )(∆3 − ∆1 ) .

This is the classical formula for the discriminant of the cubic polynomial x3 + p2 (m)x − p3 (m)
in terms of its roots.
QED

23

Remark 4.11 It is important to emphasise that the bivectors f1 ∧ f2 , f2 ∧ f3 , f3 ∧ f1 although
a priori elements of Λ2 (VFQm ) are in fact elements of L−1 . This means that in order to deduce
the expression for the invariants above in a basis (e1 , e2 ) of V from their expressions in the basis
(f1 , f2 ) of VFQm , one has to express f1 ∧ f2 in terms of e1 ∧ e2 . When the ∆i ’s are distinct this
is the formula (4.28) below.
A simple characterisation of the automorphism group of a generic algebra can be given in terms
of the invariants Invm and p2 (m), p3 (m). (For alternative calculations of the automorphism group
of real division algebras see [11, 20], Proposition 4/Proposition 2.8.)

Corollary 4.12
g
= {g ∈ GL(V ⊗ FQm ) such
1. Let m ∈ Ac3 , let FQm be a splitting field of Qm and let Aut(m)
that g · m = m}. Then,

 {1} if Invm 6= 0 ,
∼
g
Z2 if Invm = 0 and (p2 (m), p3 (m)) 6= (0, 0) ,
Aut(m)
=

S3 if p2 (m) = p3 (m) = 0 ,

where S3 is the group of permutations of three objects. Recall that Invm = 0 iff 27p3 (m)2 +
4p2 (m)3 = 0.

2. Let m ∈ Ac3 , and let Aut(m) = {g ∈ GL(V ) such that g · m = m}

{1} if Invm 6= 0 ,




Z2 if Invm = 0 and (p2 (m), p3 (m)) 6= (0, 0) ,



S3 if p2 (m) = p3 (m) = 0 and [FQm : F] = 1,
Aut(m) ∼
=
Z2 if p2 (m) = p3 (m) = 0 and [FQm : F] = 2,




Z if p2 (m) = p3 (m) = 0 and [FQm : F] = 3,


 3
{1} if p2 (m) = p3 (m) = 0 and [FQm : F] = 6 .

Proof. (1): From the formulæ 4.25 and ∆1 + ∆2 + ∆3 = 0 it follows that
Invm 6= 0


Invm = 0
(p2 (m), p3 (m)) 6= (0, 0)
p2 (m) = p3 (m) = 0

if m has three distinct reduced idemvalues,
if m has two distinct reduced idemvalues,
if m has one reduced idemvalue.

The result follows from this and the fact that by Theorem 4.5 any automorphism is equivalent
to a permutation of the triple ((f1 , ∆1 ), (f2 , ∆2 ), (f3 , ∆3 )).
(2): This is a consequence of Propositions 2.1 and 2.3 in [14].
QED

24

Corollary 4.13 Let m ∈ Ac3 be such that Invm = 0 (or, equivalently, m ∈ ΓCardano in the
g
such
terminology of Eq.[4.26]). Then there exists g in Aut(m)
1. g · m = m;

2. det(g) = −1.
Proof. First note that −IdV ⊗FQm cannot be an automorphism of m in Ac3 . If so, we would have
m(−v, −w) = −m(v, w) , ∀v, w ∈ V ,
which is impossible. Secondly, by Corollary 4.12(1), the group of automorphisms of m is isomorphic to either Z2 or S3 . Hence m has a non-trivial automorphism g of square one which cannot
be −IdV ⊗FQm by the above remark. Thus g 6=IdV ⊗FQm , g 6= −IdV ⊗FQm and det(g) = −1. QED
Remark 4.14 In [3] the authors identify explicitly the image of the singular locus Sing(M)
under the embedding Φ : M → P8 . Furthermore, they show that the image of Sing(Sing(M)) is
a single point which they also identify. Restricting to generic commutative algebras their results
imply


c
ν̂ Sing(M) ∩ M3 = ΓCardano ,


ν̂ Sing(Sing(M)) ∩ Mc3 = (0, 0) ∈ ΓCardano ,
and restricting to stable commutative algebras


c
eCardano ,
Φ Sing(M) ∩ Mst = Γ


eCardano .
Φ Sing(Sing(M ∩ Mcst ) = [0, 0, 1] ∈ Γ

The significance of the Corollary above is that it gives a “geometric” interpretation of the
singular locus and of the singular-singular locus. Namely, if m is a generic commutative twodimensional algebra then:
 
m ∈ Sing(M)

iff

 
m ∈ Sing(M) \ Sing(Sing(M)) iff
 
m ∈ Sing(Sing(M))

iff

Aut(m) 6= {Id} ;
g
Aut(m)
= Z2 ;
g
Aut(m)
= S3 ;

We will give the proof of the “stable” version of this result in the Appendix.

25

4.3

Moduli space

In this section we give a simple parametrisation of the GL(V )−moduli space of generic commutative algebras. When the field F is algebraically closed the parameter space turns out to be
F2 . If F is not algebraically closed, a point in the plane F2 does not necessarily characterise a
class of algebras, and another (discrete) invariant may be needed to distinguish non-isomorphic
algebras which become isomorphic after tensoring with an extension of F.
In order to ease the notation, from now on we write (p3 , p2 ) instead of (p3 (m), p2 (m)) whenever
the choice of m is clear from the context.
Our strategy will be to consider the map ν : Ac3 → F2 defined by
ν(m) = (p3 , p2 ) .
This factors to a map ν̂ : Mc3 → F2 (c.f. (4.21)). Note that in terms of the reduced idemvalues
of m, we have ν(m) = (∆1 ∆2 ∆3 , ∆1 ∆2 + ∆2 ∆3 + ∆3 ∆1 ) and so the degenerate elliptic curve
n
o
2
2
3
ΓCardano = (p3 , p2 ) ∈ F : −27p3 − 4p2 = 0 ,
(4.26)

defined by the vanishing of the discriminant of the cubic Qmod (x) = x3 + p2 x − p3 will play a
special rôle. This is because the roots of Qmod are ∆1 , ∆2 , ∆3 and we will call Qmod the modular
cubic of the algebra Ac . Note that by Corollary 4.12: ν(m) ∈ ΓCardano iff the algebra Ac has a
non-trivial automorphism.
We first show that over F2 \ ΓCardano the map ν̂ is a bijection. Then we will show that over
ΓCardano \ {(0, 0)} there is a point in the moduli space for every class of extensions of F of degree
at most two and and finally that over (0, 0) there is a point in the moduli space for every class
of Galois extensions of F of degree one, two, three and six.
4.3.1

Surjectivity of ν

Theorem 4.15
(i) ν(g · m) = ν(m), ∀m ∈ Ac3 , ∀g ∈ GL(V ).
(ii) ν : Ac3 → F2 is surjective.
Proof. Part (i) is a consequence of (4.21).
To prove (ii), let (p3 , p2 ) ∈ F2 . First we suppose that (p3 , p2 ) 6∈ ΓCardano , i.e., −27p23 −4p32 6= 0.
Since −27p23 − 4p32 is the discriminant of the cubic P (x) = x3 + p2 x − p3 , this means P has three
distinct roots ∆1 , ∆2 and ∆3 in a splitting field K1 . Since the characteristic of F is zero or greater
than three and P is of degree three, K1 /F is a Galois extension (see e.g. Artin [21]).
Lemma 4.16 Let VK1 = V ⊗F K1 . Then there exists a triplet f1 , f2 , f3 ∈ VK1 such that
(a) (f1 , f2 , f3 ) are pairwise linearly independent;
(b) f1 + f2 + f3 = 0;

26

(c)

n
o
f1 , f2 , f3 is stable under the action of the Galois group Gal(K1 /F);

(d) There exists a group homomorphism γ : Gal(K1 /F) → S3 such that for all g ∈ Gal(K1 /F)
and for i = 1, 2, 3,
g · (fi , ∆i ) = (fγ(g)·i , ∆γ(g)·i ) .
Proof. Let e1 , e2 be a basis of V and define the following elements of VK1 :
p2
)e2 ,
3
p2
= ∆2 e1 + (∆3 ∆1 − )e2 ,
3
p2
= ∆3 e1 + (∆1 ∆2 − )e2 .
3

f1 = ∆1 e1 + (∆2 ∆3 −
f2
f3

(4.27)

If K1 = F, (c) and (d) are evident. If K1 6= F the Galois group acts by permutations on the
indices of the roots and so (c) and (d) are obviously satisfied. Property (b) follows since ∆1 , ∆2
and ∆3 are the roots of the polynomial P (x) = x3 + p2 x − p3 .
We now prove (a). This follows from

1
p2 
e1 ∧ e2 = (∆1 − ∆2 )(∆2 − ∆3 )(∆3 − ∆1 )e1 ∧ e2(4.28)
fi ∧ fj = (∆j − ∆i ) (∆i + ∆j )2 +
3
3

and the fact that the ∆i are distinct by hypothesis. Hence, we have proved (a) for (f1 , f2 ,
f3 ).
QED
Since f1 , f2 ∈ VK1 is a basis we can define a multiplication m on VK1 by,
m(f1 , f1 ) =

∆1 − 1
∆2 − 1
2 + ∆2
2 + ∆1
f1 , m(f2 , f2 ) =
f2 , m(f1 , f2 ) =
f1 +
f2 . (4.29)
3
3
6
6

Then a direct calculation shows that in fact we have for all i, j = 1, 2, 3 such that i 6= j,
m(fi , fi ) =

2 + ∆j
2 + ∆i
∆i − 1
fi , m(fi , fj ) =
fi +
fj .
3
6
6

(4.30)

Since for any g ∈ Gal(K1 /F) we have
g · fi = fγ(g)·i , g · ∆i = ∆γ(g)·i ,
it follows from (4.30) that
g · m(fi , fj ) = m(g · fi , g · fj )
and so
g · m(v, w) = m(g · v, g · w) , ∀v, w ∈ VK1 .
In particular, since V ⊆ VK1 is the fixed point set of the Galois group and since K1 /F is Galois,
this implies that m(v, w) ∈ V if v, w ∈ V .
27

It remains to prove that the fundamental cubic of m has three distinct roots. This follows
since
∆i − 1
fi ,
m(fi , fi ) =
3
so by (2.5) there are three pairwise independent idempotents and the fundamental cubic (not to
be confused with P (x)!) indeed has three distinct roots.
To recap we have now shown that if (p3 , p2 ) 6∈ ΓCardano then there exists m ∈ Ac3 such that
ν(m) = (p3 , p2 ). To complete the proof of (ii) we now consider the case (p3 , p2 ) ∈ ΓCardano . This
means the cubic P (x) has a multiple root in F and we have

3
3 2
) (x + 3p
) if p2 6= 0
(x − 3p
3
2p2
p2
P (x) = x + p2 x − p3 =
3
x if p2 = 0 .
Let (∆, ∆, −2∆) be the roots of P (x). Choose (e1 , e2 ) a basis of V and set,
f1 = e1 − 2e2 , f2 = −2e1 + e2 , f3 = e1 + e2 .

It is clear that (f1 , f2 , f3 ) are pairwise linearly independent and that f1 + f2 + f3 = 0. Define
the multiplication m : V × V → V by
∆−1
2+∆
∆−1
f1 + f2 ) .
f1 , m(f2 , f2 ) =
f2 , m(f1 , f2 ) =
m(f1 , f1 ) =
3
3
6
One checks that
2 − 2∆
2+∆
−2∆ − 1
f3 , m(fi , f3 ) =
fi +
f3 , i = 1, 2 .
m(f3 , f3 ) =
3
6
6
Thus m has three idempotents f1 , f2 , f3 and by definition,
ν(m) = (−2∆3 , −3∆2 ) = (p3 , p2 ) .

This complete the proof of (ii).

QED

Remark 4.17 In the proof we showed that the product (4.29) defined a priori on VK1 in fact
defines a product on V because it is compatible with the action of the Galois group Gal(K1 /F).
In order to get explicit formulæ for the products of the ei , inverting the system (4.27) gives


3
1
e1 =
(3∆3 ∆1 − p2 )f1 + (p2 − 3∆2 ∆3 )f2 , e2 = − ∆2 f1 − ∆1 f2 ,
d
d
where d = (∆1 − ∆2 )(∆2 − ∆3 )(∆3 − ∆1 ) and then (tedious) calculation shows that

 i
1 h
3
2
3
2
m(e1 , e1 ) =
−
4p
−
27p
−
9p
p
e
+
2p
+
27p
2
3
1
2
3
2
3 e2 ,
3d2
i
1h
(4.31)
m(e2 , e2 ) = 2 9p3 e1 + 2p22 e2 ,
d
 i
3h 2
2
3
m(e1 , e2 ) = 2 − p22 e1 + − p32 − p23 + p2 p3 e2 .
d
3
9
2
It should be noted that
d2 = −27p23 − 4p32 ,

so that all of the structure constants are rational functions in the moduli (p3 , p2 ). Note further
that these formulæ are valid in characteristic not two or three but not over Z.
28

4.3.2

Moduli over F2 \ ΓCardano

We saw in Theorem 4.15 that ν(m) ∈ F2 is an invariant of the equivalence classe of m. We now
show that if ν(m) is not on a ΓCardano it completely determines the class of m.
Lemma 4.18 Let m ∈ Ac3 with reduced idemvalues (∆1 , ∆3 , ∆3 ) and let K be a splitting field of
Qmod (x) = (x − ∆1 )(x − ∆2 )(x − ∆3 ). If the idemvalues are distinct, then K is also a splitting
field of the fundamental binary cubic Qm (cf 2.2).
Proof. Let FQm be a splitting field of Qm . By definition, it is always true that the polynomial
Qmod splits over any splitting field of Qm so without loss of generality we can suppose K ⊂ FQm .
We now show that if the idemvalues are distinct the converse inclusion is also true.
Let G = Gal(FQm /K) be the Galois group of the extension FQm /K. By definition, there exist
three idempotents f1 , f2 , f3 ∈ V ⊗F FQm and three idemvalues ∆1 , ∆2 , ∆3 ∈ K such that
m(fi , fi ) =

∆i − 1
fi .
3

The Galois group G acts by automorphism on V ⊗F FQm and maps idempotents to idempotents
but fixes the idemvalues (∆1 , ∆2 , ∆3 ). Since the idemvalues are distinct, the Galois group must
fix each idempotent because it act by automorphisms. However, the extension FQm /K is Galois
(being a splitting field of the separable polynomial Qm ), so the fundamental theorem of Galois
theory implies that the fixed point set of G in V ⊗F FQm is exactly V ⊗F K. Hence f1 , f2 , f3 ∈ V ⊗K
QED
and by (2.5), Qm (f1 ) = Qm (f2 ) = Qm (f3 ) = 0 and Qm splits over K.
Perhaps surprisingly, the next theorem shows that just as in the case when F is algebraically
closed [3], the invariant ν(m) completely determines the GL(V )−equivalence classe of m even
when F is not algebraically closed.
n
o
Theorem 4.19 Let ΓCardano = (p3 , p2 ) ∈ F2 : 27p23 + 4p32 = 0 , and let Ac3 g = {m ∈ Ac3 :
ν(m) 6∈ ΓCardano }. If m, m′ ∈ Ac3 g satisfy ν(m) = ν(m′ ) then there exists g ∈ GL(V ) such that
m′ = g · m.
Proof. Suppose ν(m) = ν(m′ ) ∈ F2 \ ΓCardano with m, m′ ∈ Ac3 . Let (∆1 , ∆2 , ∆3 ) and
(∆′1 , ∆′2 , ∆′3 ) be the idemvalues of m and m′ respectively. Since ν(m) = ν(m′ ) we have
Qmod (x) = (x − ∆1 )(x − ∆2 )(x − ∆3 ) = (x − ∆′1 )(x − ∆′2 )(x − ∆′3 ) ,
and therefore Qm and Qm′ split over FQmod by lemma 4.18. Without loss of generality we can
suppose ∆i = ∆′i . By Theorem 4.5 one can find fundamental triples (f1 , f2 , f3 ) and (f1′ , f2′ , f3′ )
in V ⊗F FQmod , such that
m(fi , fi ) =

∆i − 1 ′
∆i − 1
fi , m′ (fi′ , fi′ ) =
fi .
3
3

The map h : VFQmod → VFQmod given by h(fi ) = fi′ clearly defines a FQmod −algebra isomorphism.
By Remark 4.7, if g ∈ Gal(FQmod /F) and g · ∆i = ∆j then g · fi = fj and g · fi′ = fj′ . Hence
h ◦ g = g ◦ h and h maps V to V .
QED
29

4.3.3

Moduli over ΓCardano \ {(0, 0)}

Theorem 4.19 gives a complete set of invariants for isomorphism classes of algebras m such
that ν(m) 6∈ ΓCardano . As we remarked above, this set of invariants is the same whether F is
algebraically closed or not. This is not the case if ν(m) ∈ ΓCardano .
Suppose now that ν(m) ∈ ΓCardano . In this case the idemvalues of Qmod are not distinct
and hence are in F. Consequently, Lemma 4.18 does not apply, i.e., the splitting field of the
fundamental cubic Qm is not determined by the splitting field of the cubic polynomial Qmod .
Given a binary cubic Q we denote by FQ a splitting field of Q.
Lemma 4.20 Let m ∈ Ac3 .

(i) If ν(m) ∈ ΓCardano \ (0, 0) then [FQm : F] = 1 or 2;
(ii) If ν(m) = (0, 0) then [FQm : F] = 1, 2, 3 or 6.

Proof. (i): Let FQm be a splitting field of the fundamental cubic, and let f1 , f2 , f3 ∈ V ⊗F FQm
be such
∆−1
∆−1
−2∆ − 1
f1 , m(f2 , f2 ) =
f2 , m(f3 , f3 ) =
f3 .
3
3
3


If g ∈ Gal(FQm /F) then g permutes (f1 , −2∆), (f2 , ∆), (f3 , ∆) by Remark 4.7. Since −2∆ 6= ∆
we must have g · f1 = f1 and either g · f2 = f2 , g · f3 = f3 or g · f2 = f3 , g · f3 = f2 . This
means that Gal(FQm /F) has at most two elements and hence [FQm : F] ≤ 2. Part two follows
from Remark 4.3 since the only possible degree for a splitting field of Qm is 1, 2, 3 or 6. QED
m(f1 , f1 ) =

Given a field F and an integer k ∈ N∗ , we denote by Ek (F) the set of equivalence classes of
extensions of F of degree at most k which are isomorphic to a splitting field of a degree three
polynomial with distinct roots. It is well known that there is a bijection
E2 (F) ∼
= F∗ /F∗2 ,

(4.32)

given by
√
F( a) 7→ [a] .
(Under this bijection the field F itself corresponds to the identity element.)




Theorem 4.21 Let Ac3 C = ν −1 ΓCardano \ (0, 0) . Define ν̃ : Ac3 C → ΓCardano \ (0, 0) ×
E2 (F) by
ν̃(m) = (ν(m), [FQm ]), ∀m ∈ Ac3 C .
(i) Let m, m′ be in Ac3C . Then
ν̃(m) = ν̃(m′ ) ⇒ ∃g ∈ GL(V ) s.t. m′ = g · m .
(ii) The map ν̃ is surjective.
30

Proof. (i): Since ν̃(m) = ν̃(m′ ) we have ν(m) = ν(m′ ) and FQm ∼
= FQm′ . Hence m and
′
m have the same idemvalues and without loss of generality we can suppose FQm = FQm′ . Let
((f1 , −2∆), (f2 , ∆), (f3 , ∆)) and ((f1′ , −2∆′ ), (f2′ , ∆′ ), (f3′ , ∆′ )) be two fundamental triples in V ⊗F
FQm s.t.
m(f1 , f1 ) =
m′ (f1′ , f1′ ) =

−2∆−1
f1
3
−2∆−1 ′
f1
3



,

m(f2 , f2 ) =

, m′ (f2′ , f2′ ) =

∆−1
f2
3
∆−1 ′
f2
3

,

m(f3 , f3 ) =

, m′ (f3′ , f3′ ) =

∆−1
f3
3
∆−1 ′
f3
3

;
.

Define h ∈ GL V ⊗F FQm by h(fi ) = fi′ then h is an isomorphism of FQm −algebras. There
are two possibilities: either FQm = F or FQm is a quadratic extension of F (see Lemma 4.20
(i)). In the first case h is an isomorphism of F−algebras and we are finished. In the second
case to conclude, we have to show that h commutes with the action of Gal(FQm /F) ∼
= Z2 . The
Galois group has to fix f1 and f1′ and has to permute f2 , f3 and f2′ , f3′ . This says exactly that h
commutes with the action of the Galois group.
(ii): We now show the surjectivity of ν̃. Let (p3 , p2 ) ∈ ΓCaradano \ {(0, 0)} and let K/F be an
extension of degree at most two. By hypothesis there exists ∆ ∈ F⋆ s.t.
x3 + p2 x − p3 = (x + 2∆)(x − ∆)2 .

Since K is of degree at most two there exists√a ∈ F \ {0} s.t. K is a splitting field of x2 − a. Let
{e1 , e2 } be a basis of V , pick a square root a in K and define the following elements of VK :
√
1
ae1 + ae2 ,
f1 =
3
√
1
(4.33)
f2 = − ae1 + ae2 ,
3
2
f3 = − e2 .
3
Define the multiplication m : VK × VK → VK by
∆−1
∆−1
2+∆
f1 + f2 ) .
m(f1 , f1 ) =
f1 , m(f2 , f2 ) =
f2 , m(f1 , f2 ) =
3
3
6
One then checks that
1
2∆ + 1
∆−1
m(e1 , e1 ) = − e2 , m(e2 , e2 ) =
e2 , m(e1 , e2 ) =
e1 .
(4.34)
6
2a
2a
From this it follows that ν(m) = (−2∆3 , −3∆2 ) = (p3 , p2 ). It remains to show that the splitting
field of the fundamental cubic Qm is K. Up to a constant, Qm = F1 F2 F3 ⊗ e1 ∧ e2 where
√
√
1
1
2
F1 (xe1 + ye2) = ax − ay , F2 (xe1 + ye2 ) = ax + ay , F3 (xe1 + ye2 ) = − ax (4.35)
3
3
3
since Fi (fi ) = 0, i = 1, 2, 3. A short calculation then gives
1
2
Qm (xe1 + ye2 ) = − a3 x(x2 − y 2) ⊗ e1 ∧ e2 ,
27
a
2
which shows that x − a and Qm have isomorphic splitting fields.
Hence ν̃(m) =
3
2
((−2∆ , −3∆ ), K) = ((p3 , p2 ), K).
QED

31

4.3.4

Moduli over {(0, 0)}: exceptional algebras

Theorems 4.19 and 4.21 give a complete set of invariants for isomorphism classes of algebras
m such that ν(m) 6= (0, 0). We now consider the final case which corresponds to the case of
algebras m satisfying ν(m) = (0, 0) which we call exceptional. Recall that, by definition, E6 (F)
is the set of equivalence classes of extensions of F of degree at most six and which are isomorphic
to splitting fields of cubic polynomials over F. Recall also that the splitting field of a cubic
polynomial is of degree three or six iff it is irreducible.
Exceptional algebras have a very simple characterisation in terms of the representation theory
of GL(V ). In fact, recall that as a GL(V ) representation there is an isomorphism (see Proposition
3.5)
Ac ∼
=V∗⊕B
(here B = S 3 (V ∗ ) ⊗ L−1 ) and we now show that m ∈ Ac3 is exceptional iff its component along
V ∗ vanishes.
Lemma 4.22 Let m be in Ac3 and recall that Tm : V → F is defined by Tm (v) = Tr(Lv ) for all
v ∈ V . Then Tm (v) = 0, ∀v ∈ V iff m is an exceptional algebra.
Proof. Let ((f1 , ∆1 ), (f2 , ∆2 ), (f3 , ∆3 )) ∈ (VFQm ⊗F FQm )3 ) be a reduced fundamental triple:
m(fi , fi ) =

∆j + 2
∆i + 2
∆i − 1
fi , m(fi , fj ) =
fi +
fj , i 6= j .
3
6
6

Without loss of generality, we can suppose that FQm = F since tr(Lv ⊗F IdFQm ) = tr(Lv ). Then,
Lxf1 +yf2 =

 ∆1 −1
3

x+

∆2 +2
y
6

∆1 +2
y
6

∆2 +2
x
6
∆2 −1
∆1 +2
y+ 6 x
3



and Tm (xf1 + yf2 ) = 21 (∆1 x + ∆2 y). Since by defintion m is exceptional iff ν(m) = 0 iff
∆1 = ∆2 = ∆3 = 0, this proves the result.
QED

Corollary 4.23
1. Let m1 , m2 ∈ Ac3 be exceptional and let g ∈ GL(V ). Then g · m1 = m2 iff g · Qm1 = Qm2 .
2. If Q ∈ B is non-zero, there exists a unique m ∈ Ac3 such that m is exceptional and Qm = Q.
Proof. This is an immediate consequence of the lemma above and Proposition 3.5.

Theorem 4.24 Let Ac3 0 = ν −1 (0, 0) . Define ν0 : Ac3 0 → E6 (F) by

ν0 (m) = [FQm ] , ∀m ∈ ν −1 (0, 0) .
32

QED

(i) Let m, m′ ∈ Ac30 . Then
ν0 (m) = ν0 (m′ ) ⇒ ∃g ∈ GL(V ) s.t. m′ = g · m .
(ii) The map ν0 is surjective.
Proof. (i): By Proposition 2.1(iv) in [14], if Q1 , Q2 ∈ B are non-degenerate then they are in the
same GL(V )−orbit iff their splitting fields are isomorphic. Combining this with the previous
corollary it follows that two generic exceptional algebras m1 , m2 are in the same GL(V )−orbit
iff the splitting fields of Qm1 and Qm2 are isomorphic.
(ii): By definition E6 (F) is the set of equivalence classes of extensions of F of degree at most six
which are isomorphic to splitting fields of cubic polynomials over F. Hence by Corollary 4.23 (2)
ν0 is surjective.
QED
Example 4.25 Let K/F be an extension of F such that [K] ∈ E6 (F). We now give explicitly the
multiplication table of an exceptional algebra m such that ν0 (m) = [K]. Without loss of generality
we can assume that K is the splitting field of x3 + d2 x − d3 ∈ F[X] and that the roots of this
polynomial are distinct. If
x3 + d2 x − d3 = (x − ρ1 )(x − ρ2 )(x − ρ3 ) in K,

(4.36)

we define f1 , f2 , f3 ∈ VK by
1
f1 = ρ1 e1 + (ρ2 ρ3 − d2 )e2 ,
3
1
f2 = ρ2 e1 + (ρ3 ρ1 − d2 )e2 ,
3
1
f3 = ρ3 e1 + (ρ1 ρ2 − d2 )e2 .
3

(4.37)

Let m be the unique multiplication on VK s.t.
1
1
1
m(fi , fi ) = − fi , m(fi , fj ) = fi + fj , for i 6= j .
3
3
3

(4.38)

One can check, after lengthy but straightforward calculation, that the product of two elements in
V is also in V and in the basis e1 , e2 we have

1 
m(e1 , e1 ) = − 2 9d3 d2 e1 − (27d23 + 2d32 )e2 ,
3D
2 2 
9 
d
e
+
d e2 ,
(4.39)
m(e2 , e2 ) =
3 1
D2
9 2

1 
2
−
2d
e
+
3d
d
e
m(e1 , e2 ) =
3 2 2 ,
2 1
D2
where

D = (ρ1 − ρ2 )(ρ2 − ρ3 )(ρ3 − ρ1 ) .
33

(4.40)

Notice that D 2 = −27d23 − 4d32 is the discriminant of the cubic x3 + d2 x − d3 . It is clear that
ν(m) = (0, 0) from (4.38) and to see that ν0 (m) = K we have to show that K is a splitting field
of Qm . Computation gives
Qm = −

9
F1 F2 F3 ⊗ e1 ∧ e2
D2

(4.41)

where
Fi (xe1 + ye2 ) = (ρj ρk − d2 /3)x − ρi y , {i, j, k} = {1, 2, 3} .

(4.42)

Since ρi is a root of x3 + d2 x − d3 whose splitting field is K, the splitting field of Qm is also K.
After some (tedious) calculation, we get


2
2
Qm (xe1 + ye2 ) = (d23 + d32 )x3 + d3 d2 x2 y + d22 xy 2 − d3 y 3 ⊗ e1 ∧ e2 .
27
3
Note that it follows from (4.39) that

m(m(xe1 + ye2 , xe1 + ye2 ), xe1 + ye2 ) =

1
(d2 x2 − 9d3xy − 3d2 y 2)(xe1 + ye2 ) ,
3D 4 2

in other words the cube of an element in an exceptional algebra is proportional to itself.
√
Note also that equations (4.34) define a algebra m such that ν̃(m) = ((−2∆3 , −3∆2 ), F( a).
If we set ∆ = 0, m becomes the algebra (4.39) with d3 = 0, d2 = −a and D 2 = 4a3 . This means
that in some sense exceptional algebras corresponding to extensions of degree at most two are
limits of non-exceptional algebras over the Cardano curve.
Remark 4.26 If K/F is an extension of degree six we can describe the action of the Galois group
on the six dimensional representation K as follows. Without loss of generality we can suppose
that K is the splitting field of x3 + d2 x − N. Then we can find ρ, ∆ ∈ K such that K = F(ρ, ∆)
where ρ, ∆ are defined by
x3 + d2 x − N = (x − ρ)(x2 + ρx + Nρ )
= (x − ρ)(x − −ρ+∆
)(x −
2
= (x − ρ)(x − ρ̄)(x − ρ̄¯) .

−ρ−∆
)
2

(in F(ρ))
(in K)

From ρρ̄ + ρ̄ρ̄¯ + ρ̄¯ρ = d2 it follows that ∆2 = −3ρ2 − 4d2 . Since ∆ = ρ̄ − ρ it follows immediately
¯ = 1/2(3ρ − ∆). The action of the Galois group S3 on K is then given by
that ∆



ρ → 21 − ρ − ∆
ρ → ρ
−
,
= (123) −→
(23) −→
,
∆ → −∆
∆ → 21 3ρ − ∆

and the decomposition of K into irreducible F−representations by

1
1
1
K = N ⊕ D ⊕ ρ, ρ̄, ρ̄¯ ⊕ ρρ̄ − d2 , ρ̄ρ̄¯ − d2 , ρ̄¯ρ − d2 ,
3
3
3
where D = (ρ − ρ̄)(ρ̄ − ρ̄¯)(ρ̄¯ − ρ). These representations as representations of S3 are respectively
isomorphic to: 1, 1ǫ , 2 and 2.
34

5

Applications

5.1

Division algebras

In this section we give a criterion for division algebras in terms of the invariants defined above.
Recall that
Definition 5.1 The algebra m ∈ Ac3 is a division algebra iff for all v ∈ V \ {0}, Lv is invertible.
Lemma 5.2 (If F = R see [22].) An algebra m ∈ Ac3 is a division algebra iff Dm is an
anisotropic quadratic form.
Proof. Let v ∈ V \ {0} then v is Dm −isotropic iff Dm (v) = 0 iff det(Lv (m)) = 0. This is true
iff there exists a non-zero u ∈ V such Lv (m)(u) = 0, i.e., such that m(u, v) = 0. Hence Dm is
anisotropic iff m defines a division algebra.
QED
We now show that the moduli p3 , p2 enable us to detect when an algebra is a division algebra.
Theorem 5.3 Let m ∈ Ac3 .
(i) If ν(m) = (0, 0) then m is a division algebra iff −3 is not a square in the splitting field
FQm of Qm .
(ii) If ν(m) ∈ ΓCardano \ {(0, 0)}, let ν̃(m) = (p3 , p2 , [FQm ]). Then m is a division algebra iff
∼
−3(p2 − p3 + 1) is not a square in F when
√ FQm = F or iff −3a(p2 − p3 + 1) is not a square
∼
in F when [FQm : F] = 2 and FQm = F( a).
(iii) If ν(m) 6∈ ΓCardano , let ν(m) = (p3 , p2 ). Then m is a division algebra iff −3(p2 − p3 +
1)(−27p23 − 4p32 ) is not a square in F.
Proof. (i): Since ν(m) = (0, 0) the equivalence class of m is completely determined by ν0 (m) =
K. Let x3 + d2 x − d3 ∈ F[X] be a polynomial with three distinct roots ρ1 , ρ2 , ρ3 whose splitting
field is isomorphic to FQm . By (4.39) there exists a basis {e1 , e2 } of V such that the matrix of
the left multiplication by xe1 + ye2 is given by


− D3x2 d3 d2 − D2y2 d22
− D2x2 d22 + D9y2 d3
Lxe1 +ye2 = x
(9d23 + 23 d32 ) + D3y2 d3 d2 D3x2 d3 d2 + D2y2 d22
D2
where D 2 = (ρ1 − ρ2 )2 (ρ2 − ρ3 )2 (ρ3 − ρ1 )2 = −27d23 − 4d32 . By Lemma 5.2, m defines a division algebra iff Dm (x, y) = det(Lxe1 +ye2 ) is an anisotropic quadratic form. Computing the determinant
we get

27d23 + 4d32  2 2
2
Dm (x, y) =
d
x
−
9d
xy
−
3d
y
3
2
2
3D 4
1
and the discriminant reduces to −3 (3D)
2 . Since D ∈ FQm \ F this proves (i).
(ii-a): When FQm ∼
= F there exists a reduced fundamental triple ((f1 , ∆1 = ∆), (f2 , ∆2 =
∆), (f3 , ∆3 = −2∆)) ∈ (V × F)3 . The fundamental quadratic Dm (xf1 + yf2 ) is given by

1
1
1
1
1
Dm (xf1 + yf2 ) = x2 (∆1 − 1)(1 + ∆1 ) + xy(∆1 − 1)(∆2 − 1) + y 2 (∆2 − 1)(1 + ∆2 ) ,
9
2
9
9
2
35

1
(p3 − p2 − 1). This proves (ii-a).
and its discriminant by 27
(ii-b). By (4.34) there exists a basis {e1 , e2 } of V such that the matrix of left multiplication
by xe1 + ye2 is given by
 ∆−1

y ∆−1
x
2a
2a
Lxe1 +ye2 =
.
− 16 x 2∆+1
y
2a

By Lemma 5.2 m defines a division algebra iff det(Lxe1 +ye2 ) is a anisotropic quadratic form. A
short calculation shows that
det(Lxe1 +ye2 ) =

p3 − p2 − 1
,
12a3

and this proves (ii-b).
(iii): If ν(m) = (p3 , p2 ) then by (4.31) there is a basis {e1 , e2 } of V such that the matrix of the
left multiplication by xe1 + ye2 is given by




9y
2y 2
2x 2
x
3
2
p
−
p
+
p
−
4p
−
27p
−
9p
p
−
2 3
2
3
3d2
d2 2
d2 2
d2 3






Lxe1 +ye2 =  
3y
2y 2
2 3
3 2
2 3
3 2
2x
x
3
2
+
2p
+
27p
−
p
−
p
+
p
p
−
p
−
p
+
p
p
+
p
2 3
2 3
2
3
3d2
d2
9 2
2 3
d2
9 2
2 3
d2 2
with d2 = (∆1 − ∆2 )2 (∆2 − ∆3 )2 (∆3 − ∆2 )2 = −(27p23 + 4p32 ). By Lemma 5.2, m defines a
division algebra iff Dm (x, y) = det(Lxe1 +ye2 ) is an anisotropic quadratic form. Computing the
determinant we get
)
(





4p32 + 27p23  2
Dm (x, y) =
27p3 − 9p2 p3 + 6p22 + 4p32 x2 − 6 9p3 + 2p22 xy + 9 3p3 − 2p2 y 2 ,
18d4

and then the discriminant of Dm (x, y) reduces to − 91 × 3(p2 − p3 + 1)(27p23 + 4p32 ) . This proves
(iii).
QED
Part one of this Theorem implies that there is (up to isomorphism) only one two-dimensional
real commutative division algebra whose automorphism group is S3 . In fact one can show that
any real division algebra whose automorphism group is S3 must be commutative [11].

5.2

Associativity

In this section we show that generic commutative associative algebras correspond to a single
point in the Cardano plane.
Theorem 5.4 Let m ∈ Ac3 . Then m is associative iff ν(m) = (16, −12). Note that ν(m) ∈
ΓCardano .
Proof. Let FQm be a splitting field of Qm and let f1 , f2 , f3 ∈ VK be the fundamental triple:
m(f1 , f1 ) =

∆2 − 1
2 + ∆2
2 + ∆1
∆1 − 1
f1 , m(f2 , f2 ) =
f2 , m(f1 , f2 ) =
f1 +
f2 .
3
3
6
6
36

Among the eight associators A(fi , fj , fk ), i, j, k = 1, 2 one checks that only two do not trivially
vanish:
m(m(f2 , f1 ), f1 ) − m(f2 , m(f1 , f1 )) , m(m(f1 , f2 ), f2 ) − m(f1 , m(f2 , f2 )) .

(5.43)

Calculation shows that


2 + ∆1 
(2 + ∆2 )f1 + (4 − ∆1 )f2 ,
36

2 + ∆2 
m(m(f1 , f2 ), f2 ) − m(f1 , m(f2 , f2 )) =
(4 − ∆2 )f1 + (2 + ∆1 )f2 .
36
It follows from this that m is associative iff (∆1 , ∆2 ) = (−2, −2), (−2, 4) or (4, −2) and hence iff
ν(m) = (16, −12).
QED
m(m(f2 , f1 ), f1 ) − m(f2 , m(f1 , f1 )) =

We have already seen in Theorem 4.21 that equivalence classes of algebras such that ν(m) =
(16, −12) are completely determined by the class of the field extension FQm /F. Moreover, since
(16, −12) ∈ ΓCardano \ {(0, 0)}, FQm can be any extension of F of degree at most two.

Corollary 5.5 Let m ∈ Ac3 be associative and let ν̃(m) = ((16, −12), [K]) where K/F is an
extension of degree one or two.
(i) If K ∼
= F, then the F−algebra (V, m) is isomorphic to F × F.
(ii) If K is a quadratic extension, then the F−algebra (V, m) is isomorphic to K.
Proof. (i): If K ∼
= F, the splitting field of Qm is F. Hence, as we saw in the proof of Theorem
5.4 one can suppose ∆1 = ∆2 = −2, ∆3 = 4 and hence there exist f1 , f2 ∈ V such that
m(f1 , f1 ) = −f1 , m(f2 , f2 ) = −f2 , m(f1 , f2 ) = 0 ,

and the algebra (V, m) is the product of the ideals Vect(f1 ) × Vect(f2 ). The identity element is
f3 = −f1 − f2 .
(ii): The splitting field FQm of Qm is an extension of degree two so Gal(FQm /F) ∼
= Z2 and
−1
∗
∗
there exist α ∈ V , β ∈ VFQm and b ∈ L such that Qm = αβ β̄ ⊗ b. One can choose f ∈ Ker(β)
such that ((f, −2), (f¯, −2), (−f − f¯, 4)) is a reduced fundamental triple (see Theorem 4.5), thus
m(f, f ) = −f , m(f¯, f¯) = −f¯ , m(f, f¯) = 0 ,
(5.44)
∗
(F∗ )2 (i.e., a is not a square in F∗ )
Since FQm is a quadratic
√ extension of F there exists ¯a ∈ F \√
such that FQm = F( a). In the basis e1 = −(f + f ), e2 = a(f − f¯) of V the multiplication
defined in (5.44) reduces to

m(e1 , e1 ) = e1 , m(e2 , e2 ) = ae1 , m(e1 , e2 ) = e2 .
√
Here the map e1 → 1, e2 → a defines an F−algebra isomorphism of (V, m) with FQm .

QED

Remark 5.6 In case (ii) of the corollary above, the algebra (V, m) is isomorphic to a field and
in particular is a division algebra. This means that the criterion of Theorem 5.3 (ii) must be
satisfied which is true since
− 3a(p2 − p3 + 1) = 81a
is not a square in F.
37

6

Moduli space of generic algebras with respect to the
action of SL(V )

In this section we consider the set of generic algebras
n
o
c
c
A3 = m ∈ A , s.t. Disc(Qm ) 6= 0 ,

modulo the action of SL(V ). Since SL(V ) is a subgroup of GL(V ) there is a natural map from
the SL(V )−moduli space Ac3 /SL(V ) to the GL(V )−moduli space Mc3 . It turns out that the
description of the fibre of this map is quite different depending on whether or not we are working
Invm
over the Cardano curve. We will show that the invariant Disc(Q
distinguishes SL(V )−moduli
m)
corresponding to the same GL(V )−moduli if the latter do not belong to the Cardano curve.
If they do belong to the Cardano curve, given by the vanishing of Inv (see Theorem 3.17), we
will show that the invariant Disc(Qm ) distinguishes the corresponding SL(V )−moduli unless
Disc(Qm ) is irreducible (see (3.2) of Proposition 6.1).
n
o
3
′
c
2
2
Proposition 6.1 Let m, m ∈ A3 and let ΓCardano = (p3 , p2 ) ∈ F : 27p3 + 4p2 = 0 .

1. If ν(m), ν(m′ ) 6∈ ΓCardano and ν(m) = ν(m′ ), then there exists g ∈ SL(V ) such that
Inv
Invm
m′ = g · m iff Disc(Q
= Disc(Qm′ ′ ) .
m)
m

2. If ν̃(m), ν̃(m′ ) ∈ ΓCardano \ {0} and ν̃(m) = ν̃(m′ ), then there exists g ∈ SL(V ) such that
m′ = g · m iff Disc(Qm ) = Disc(Qm′ ).
3. If ν(m) = ν(m′ ) = (0, 0) and ν0 (m) = ν0 (m′ ),
3.1 if Qm is reducible (equivalently if [FQm : F] = 1 or 2) then there exists g ∈ SL(V )
such that m′ = g · m iff Disc(Qm ) = Disc(Qm′ ),

3.2 if Qm is irreducible (equivalentlyf if [FQm : F] = 3 or 6) then Disc(Qm) determines
the SL(V )−classes up to a Z2 −factor.

Proof. The implications “⇒” are immediate for (1),(2),(3), so we now prove the three converse
implications.
(1): Suppose m, m′ ∈ Ac3 are such that ν(m), ν(m′ ) 6∈ ΓCardano and
Invm′
Invm
, ν(m) = ν(m′ ) .
=
Disc(Qm )
Disc(Qm′ )
By Theorem 4.19 there exists g in GL(V ) such that m′ = g · m. Hence
Invm′
Invm
Invm
= (det g)−1
=
,
Disc(Qm′ )
Disc(Qm )
Disc(Qm )
which means that g is in SL(V ).

38

(2): Suppose m, m′ ∈ Ac are such that ν̃(m), ν̃(m′ ) ∈ ΓCardano \ {0} and
Disc(Qm ) = Disc(Qm′ ) , ν̃(m) = ν̃(m′ ) .
By Theorems 4.21 and 4.24 there exists g in GL(V ) such that m′ = g · m. Hence
Disc(Qm′ ) = (det g)−2Disc(Qm ) = Disc(Qm ) ,
whence det(g) = ±1. Without loss of generality we can assume det(g) = 1 since by Corollary
4.12 there exists an automorphism of m′ of determinant minus one.
The proof of (3.1) is the same as the proof of (2) (see corollary 4.12).
(3.2): By Theorem 4.24, m and m′ are GL(V )−equivalent iff the twisted binary cubics Qm
and Qm′ are GL(V )−equivalent. It follows that m and m′ are SL(V )−equivalent iff Qm and
Qm′ are SL(V )−equivalent. As an SL(V )−representation the space of twisted binary cubics and
the space of binary cubics are isomorphic so we are led to the problem of classifying irreducible
(untwisted) binary cubics under the action of SL(V ).
Let us first consider the particular case where in some basis {e1 , e2 } of V we can write
qm (x, y) = ax3 + by 3 , qm′ (x, y) = a′ x3 + b′ y 3 ,
with Qm = qm e1 ∧ e2 , Qm′ = qm′ e1 ∧ e2 . Then one can show that


Disc(Qm ) = −3 3ab ⊗ (ǫ1 ∧ ǫ2 )2 , Disc(Qm′ ) = −3 3a′ b′ ⊗ (ǫ1 ∧ ǫ2 )2 ,

where {ǫ1 , ǫ2 } is the basis dual to {e1 , e2 }. It is shown in [23] that qm and qm′ are in the same
SL(V )−orbit iff
(  
(  
 a′ 
 ′
a
a
=
= ab′ F∗ /F∗3
∗ /F∗3
b F∗ /F∗3
b′ F∗ /F∗3
b
F
or
.
ab
=
a′ b′
ab
=
−a′ b′
Now the fact that Qm and Qm′ are in the same GL(V )−orbit implies that there exists λ in F∗
such that λ(ab) = a′ b′ , and the fact that Disc(Qm )=Disc(Qm′ ) implies that ab = ±a′ b′ , i.e.,
λ2 = 1. If λ = 1 we must have
(  
(  
 a′ 
 b′ 
a
a
=
=
′
∗
∗3
∗
∗3
∗
∗3
b F /F
b F /F
b F /F
a′ F∗ /F∗3
or
,
′ ′
ab
=
ab
ab
=
−a′ b′
and if λ = −1 we must have
(  
 ′
a
= ab′ F∗ /F∗3
b F∗ /F∗3
ab
=
−a′ b′

or

(  
a

b F∗ /F∗3

ab

=
=

 b′ 

a′ F∗ /F∗3
′ ′

ab

.

Hence either Qm′ and Qm are in the same orbit or Qm′ is in the SL(V )−orbit corresponding
to the parameters ([a/b], −ab). This proves (3.2) in the special case where Qm and Qm′ can be
written as above. In fact, as shown in [23], this is the general situation, and this completes the
proof of part (3.2).
QED

39

Remark 6.2 Since
Disc(Dm ) =

1
(p3 − p2 − 1)Disc(Qm ) ,
27

the invariant Disc(D) also distinguishes SL(V )−modules with the same GL(V )−moduli on the
Cardano curve if p3 − p2 − 1 6= 0 if [FQm /F] ≤ 2. Note that the only point on the Cardano curve
such that p3 − p2 − 1 = 0 is the “associative” point (16, −12) (see Theorem 5.4).
Before stating the next theorem we need the
Lemma 6.3 The natural action of the group F∗2 on L2 \ {0} defines a principal F∗2 −fibration
s : L2 \ {0} → F∗ /F∗2 .
Proof. It is clear that
ω ⊗ ω 7→ (λω) ⊗ (λω) , ∀λ ∈ F∗ , ω ∈ L ,
is a well defined principal F∗2 −action, i.e., the only element of F∗2 with a fixed point is the
identity. The point of the Lemma is to identify the quotient space with F∗ /F∗2 . For this pick
{ω0 } of L and define
s:

L2 \ {0}
→ F∗ /F∗2
(λω0 ) ⊗ (λω0 ) 7→ [λ2 ] .

One easily checks that s is independent of the choice of basis {ω0 } and that s(ω1 ⊗ω1 ) = s(ω2 ⊗ω2 )
iff there exists λ ∈ F∗ such that ω2 ⊗ ω2 = λ2 ω1 ⊗ ω1 . This proves the lemma.
QED
This lemma enables us to associate a half-line (more precisely a F∗2 −orbit) in L2 \ {0} to any
quadratic extension of F since F∗ /F∗2 is naturally identified with the set of at most quadratic
extensions of F.
Definition 6.4 Let [a] be an element of F∗ /F∗2 . We define the half-line L[a] ⊂ L2 \ {0} by
L[a] = s−1 ([a]) .
In part (3) of the following theorem we use the notation:

•
•
•

Ei is the set of equivalence classes of Galois extensions of F of degree equal to i.
n
o
i c
c
A
=
m
∈
A
s.t.
ν
(m)
∈
E
.
0
0
0
i
3
3
n
o
∆6,2 = (F̃6 , ω) ∈ E6 × (L2 \ {0}) s.t. s(ω) ⊂ F̃6 ,

where i takes the values 1, 2, 3 or 6.

40

Theorem 6.5


1. The map ν̂s : Ac3g /SL(V ) → F2 \ ΓCardano × (L \ {0}) given by
Inv
p3 , p2 ,
Disc(Q)

ν̂s (m) =

!

is a bijection.


c
ˆ
2. The map ν̂s : A3C /SL(V ) → ΓCardano \ {0} × (L2 \ {0}) given by
ν̂ˆs (m) = (p3 , p2 , Disc(Q))

is a bijection.
3. If ν(m) = (0, 0) there are three cases:

.


3.1 The map 1 ν̂0 : 1Ac3 0 ∪ 3Ac3 0 SL(V ) → E1 ∪ E3 × L[1] given by
1 ν̂0 (m)

= (ν0 (m), Disc(Qm )) ,

is a bijection.
3.2 The map 2 ν̂0 : 2Ac3 0 /SL(V ) → (L2 \ {0}) \ L[1] given by
2 ν̂0 (m)

= Disc(Qm ) ,

is a bijection.
3.3 The map 6 ν̂0 : 6Ac3 0 /SL(V ) → ∆6,2 given by
6 ν̂0 (m)

= (ν0 (m), Disc(Qm )) .

Proof. (1): By (1) of Proposition 6.1, the map is injective so it remains to prove surjectivity.
Taking e = f = b = 0 and d = 1 in the formulæ (3.7) and (3.17), we get
2a2 c
Invm
=
(ǫ1 ∧ ǫ2 ) .
Disc(Qm )
1 − 4ac
Let X ∈ F\{0}. Then one checks immediately that if we take the multiplication m corresponding
to
a = 4X , c =

1
,
48X

then Invm /Disc(Qm ) = Xǫ1 ∧ ǫ2 and this proves surjectivity.
(2): To prove surjectivity in this case, we start with two lemmas.
Since (p3 , p2 ) ∈ ΓCardano \ {0} we know that the splitting field FQm of Qm is of degree at most
two (see Lemma 4.20).
41

√
Lemma 6.6 Write FQm = F( a) where a ∈ F is a square in F if [FQm : F] = 1 and not a square
in F if [FQm : F] = 2. Then
s(Disc(Qm )) = [a] .
Proof. Since ν̃(m) = (p3 , p2 , [FQm ]), as we showed in (4.33), there is a basis of idempotents
{f1 , f3 } of V ⊗ FQm such that
√
2
1
f2 = − ae1 + ae2 , f3 = − e2 .
3
3
The dual basis is

√
3
1
a
ǫ1 − ǫ2 ,
ϕ2 = − √ ǫ1 , ϕ3 = −
2
2
a

and hence from (4.25)
Disc(Qm ) = (f2 ∧ f3 )2 =

9
(ǫ1 ∧ ǫ2 )2 .
4a

By definition
h9i
= [a]
s(Disc(Qm )) =
4a

and this proves the lemma.

QED

Recall now (see Theorem 4.21) that ν̃ : Ac3C /GL(V ) 7→ ΓCardano \ {0} × E2 (F) is a bijection.
In the following Lemma, by abuse of notation, we have identified E2 (F) with F∗ /F∗2 as in (4.32)
Lemma 6.7 The following diagram is commutative


ν̂ˆs
c
/
A3C /SL(V )
ΓCardano \ {0} × (L2 \ {0})
p

Id×s



Ac3C /GL(V )

ν̃

 

/
ΓCardano \ {0} × F∗ /F∗2

Proof. Let m ∈ Ac3C . We denote by [m]SL and [m]GL respectively, the corresponding equivalence
classes with respect to the actions of SL(V ) and GL(V ). By definition,
ν̃ [m]GL ) = (p3 , p2 , FQm )
where FQm is the splitting field of Qm . Identifying E2 (F) with F∗ /F∗2 (see (4.32)) we can write
this as
ν̃ [m]GL ) = (p3 , p2 , FQm ) = (p3 , p2 , [a]) ,
√
if FQm = F( a). By Lemma 6.6 this is equivalent to
ν̃ [m]GL ) = (p3 , p2 , s(Disc(Qm ))
42

and by definition the RHS of this equation is equal to
(Id × s) ◦ ν̂ˆs [m]SL ) .
This shows that the diagram is commutative.

QED

We now show that ν̂ˆs is surjective. Let I = Im(ν̂ˆs ) ⊂ (ΓCardano \ {0}) × (L2 \ {0}). On the one
hand, Id×s(I) is equal to (ΓCardano \ {0}) × F∗ /F∗2 since ν̃ is surjective and the above diagram
commutes. On the other hand, I is stable under the action of F∗2 since

ν̂ˆs [m]SL = (p3 , p2 , Disc(Qm ))

ν̂ˆs [λId · m]SL = (p3 , p2 , λ2 Disc(Qm )) .

for any λ ∈ F∗ . It follows that I = (ΓCardano \ {0}) × (L2 \ {0}) and ν̂ˆs is surjective.
(3): We will need the following lemma.

Lemma 6.8 Let m be an element of Ac3 0 , let FQm be a splitting field of Qm and let Disc(Qm )
be the discriminant of Qm . Then


1
or
3
⇒
s
Disc(Q
)

m
 = [1] ,


2
⇒ s Disc(Qm ) = [FQm ] ,
[FQm : F] =
6
⇒
s Disc(Qm ) = [F̃] , where F̃ is the unique



quadratic extension of F contained in FQm .

Proof. We can always find a polynomial of the form P (x) = x3 + d2 x − d3 in F[X] such that FQm
is the splitting field of P . By Theorem 4.24 there exist ρ1 , ρ2 , ρ3 ∈ FQm and f1 , f2 , f3 in V ⊗ FQm
such that the multiplication table of m in the basis {f1 , f2 } is given by (4.37). In particular by
(4.41) and (4.42)

−1 
Qm =
( 2ρ1 ρ2 − ρ2 ρ3 − ρ3 ρ1 )x − ρ3 y
3D 2 

(−ρ1 ρ2 + 2ρ2 ρ3 − ρ3 ρ1 )x − ρ1 y


(−ρ1 ρ2 − ρ2 ρ3 + 2ρ3 ρ1 )x − ρ2 y (f1 ∧ f2 ) ,
where D = (ρ1 − ρ2 )(ρ2 − ρ3 )(ρ3 − ρ1 ). Note that D 2 is in F. By (4.25) we know that
2

Disc(Qm ) = (ϕ1 ∧ ϕ2 )⊗ ,
with {ϕ1 , ϕ2 } the dual basis of f1 , f2 . On the other hand it follows from (4.37) that
(ϕ1 ∧ ϕ2 ) =

3
(ǫ1 ∧ ǫ2 ) .
(ρ1 − ρ2 )(ρ2 − ρ3 )(ρ3 − ρ1 )

Hence we have the formula
Disc(Qm ) =

(ρ1 −

ρ2 )2 (ρ2

9
2
(ǫ1 ∧ ǫ2 )⊗ .
2
2
− ρ3 ) (ρ3 − ρ1 )
43


If [FQm : F] = 1 then ρ1 , ρ2 , ρ3 ∈ F and hence D ∈ F which means that s Disc(Qm ) = [1]. If
[FQm : F] = 3 then ρ1 , ρ2 , ρ3 ∈ FQm \ F but D, being invariant under the Galois group Z3 , is an
element of F. Hence s Disc(Qm ) = [1].
If [FQm : F] = 2 then Gal(FQm /F) = Z2 and the roots of P (x) are ρ1 ∈ F, ρ2 ∈ FQm \ F and
ρ3 = ρ̄2 ∈ FQm \ F. Hence
D = (ρ1 − ρ2 )(ρ2 − ρ̄2 )(ρ̄2 − ρ1 )
which is not invariant under the Galois group. Thus F(D) is a non-trivial quadratic extension
of F contained in FQm which means that s Disc(Qm ) = [FQm ]. Finally if [FQm : F] = 6, then
Gal(FQm /F) ∼
= S3 which acts on ρ1 , ρ2 , ρ3 ∈ FQm \ F by permutation. In particular D is not
invariant under odd permutations and hence D 6∈ F. Thus F(D) is a non-trivial quadratic
extension of F contained in FQm . By the fundamental theorem of Galois theory there is only
one such quadratic
extension since the group S3 has only one subgroup of index two. Hence

s Disc(Qm ) must be this quadratic extension.
QED
We now prove (3.1) − (3.3) case by case.

(3.1): If [FQm : F] = 1 or 3 it follows from Lemma 6.8 that Disc(Qm ) ∈ L[1] and since
Disc(QλId·m ) = λ2 Disc(Qm ), it is clear that 1 ν̂0 is surjective.
(3.2): If [FQm : F] = 2 it follows from Lemma 6.8 that Disc(Qm ) = [FQm ]. As in the proof
of (3.1), Disc(QλId·m ) = λ2 Disc(Qm ) and hence all points of the half-line L[FQm ] are in the image
of 2 ν0 (m). Since FQm can be an arbitrary non-trivial quadratic extension of F, this implies that
the image of 2 ν0 is (L2 \ {0}) \ L[1] .
(3.3): If [FQm : F] = 6 it follows from Lemma 6.8 that Disc(Qm ) = [F̃m ] where F̃m is
the unique quadratic extension of F such that F ⊂ F̃m ⊂ FQm . As in the proof of (3.1),
Disc(QλId·m ) = λ2 Disc(Qm ) and hence all points in the half-line L[F̃m ] are in the image of 6 ν̂0 .
By the very definition of ∆6,2 this is enough to prove that 6 ν0 is surjective.
QED
Remark 6.9 We do not know how to characterise quadratic extensions which are contained in
Galois extensions of order six.

7

A linear embedding of stable SL(V )−moduli space

In the two previous sections we gave a parametrisation of the moduli spaces of generic twodimensional commutative algebras under the action of GL(V ) and SL(V ). In this section, we F
is algebraically closed, we will construct an embedding ζs of the SL(V )−moduli space of stable
commutative two-dimensional algebras onto a hypersurface ΓEisenstein in a four-dimensional vector
space. The equation defining this hypersurface is essentially the classical Eisenstein identity for
the covariants of a binary cubic. Furthermore, the “projectivisation” of ζs gives exactly the
embedding of the stable GL(V )−moduli space onto P2 of [3].
With this in mind we make the following definitions. Recall that Acst denotes the set of all
stable commutative algebra structures on V and L = Λ2 (V ∗ ).
44

Definition 7.1
n
o
1. Let ΓEisenstein = (A, B, D, C) ∈ L2 × L2 × L2 × L3 \ {0} s.t. 27DA2 + 4B 3 + 24 36 C 2 = 0 .
2. Let ζ : Acst → L2 × L2 × L2 × L3 be defined by

ζ(m) = (p̃3 (m), p̃2 (m), Disc(Qm ), Invm ) , ∀m ∈ Acst .

Note it follows from Theorem 3.17 and the definition of Acst (cf. (4.20)) that Im(ζ) ⊂
ΓEisenstein .
3. The group F∗ acts on L2 × L2 × L2 × L3 by
1
1
1
1 
A,
B,
D,
C .
λ · (A, B, D, C) =
λ2 λ2 λ2
λ3
We set
e 2 × L2 × L2 × L3 ) = (L2 × L2 × L2 × L3 )/F∗ .
P(L

e Eisenstein ) =
Note that ΓEisenstein is stable under this action and we will write P(Γ

e Eisenstein ) s.t. D 6= 0 , we define the canonical
ΓEisenstein /F∗ . If U3 = [A, B, D, C] ∈ P(Γ
e Eisenstein ) → P2 by
affine chart ψ3 : U3 → F2 and the canonical projective chart ψ̃3 : P(Γ
A B
Ψ3 ([A, B, D, C]) =
, Ψ̃3 ([A, B, D, C]) = [A, B, D] .
,
D D

Remark 7.2 Note that the action of F∗ on the set {(A, B, D, C) ∈ ΓEisenstein s.t. C 6= 0} is
principal. It follows from the definition of ζ that ζ(m) is in this set iff ν(m) 6∈ ΓCardano .
Consider the natural map p : Acst /SL(V ) → Acst /GL(V ). There is a GL(V )/SL(V ) ∼
= F∗ −action
on Acst /SL(V ) given by
λ · [m]SL(V ) = [g · m] where g ∈ GL(V ) is s.t. det(g) = λ

which satisfies







(7.45)



p λ · [m]SL(V ) = p [m]SL(V ) .
Since the coordinates of ζ are GL(V )−equivariant “twisted” polynomials, the map ζ factors
to define a map ζs : Acst /SL(V ) → ΓEisenstein . It follows that ζs is F∗ −equivariant with respect
to the natural F∗ −actions defined above (c.f. Definition 7.1 (3) and Equation 7.45) and factors
e Eisenstein ). We have the commutative diagram
to define a map ζg : Acst /GL(V ) → P(Γ
Ac3


Ac3 /SL(V )

Acst PP

⊂
⊂

Acst

PPP
PPPζ
PPP
PPP

(
/ ΓEisenstein
/SL(V )
ζs

p



Ac3/GL(V )

⊂

π





Acst /GL(V )

ζg

/

e Eisenstein )
P(Γ

⊃

and now determine the isotropy of the two F∗ −actions defined above.
45

U3

Ψ3

/

F2

Proposition 7.3
n
1. Let (A, B, D, C) ∈ L2 × L2 × L2 × L3 and I(A,B,D,C) = λ ∈ F∗ s.t. λ · (A, B, D, C) =
o
(A, B, D, C) . Then

{1}
if C 6= 0
I(A,B,D,C) =
{1, −1} if C = 0 .
n
o
2. Let [m]SL(V ) ∈ Acst /SL(V ) and I[m]SL(V ) = λ ∈ F∗ s.t. λ · [m]SL(V ) = [m]SL(V ) . Then
I[m]SL(V )



= {1}
if Invm 6= 0
⊂ {1, −1} if Invm = 0 .

Proof. (1) is immediate and (2) follows from the fact that I[m]SL(V ) ⊂ Iζs ([m]) since ζs is
F∗ −equivariant.
QED
The following theorem shows that many of the solutions of the Eisenstein equation are in the
image of ζ. In fact, as we will show in Corollary 7.7, it turns out that all solutions are in the
image of ζ.
Theorem 7.4 The image of ζs : Ac3 /SL(V ) → ΓEisenstein is given by
n
o
Im(ζs )|Ac /SL(V ) = Γ∗Eisenstein = (A, B, D, C) ∈ ΓEisenstein s.t. D 6= 0 .
3

Proof.
We split the proof into two lemmas corresponding to whether C = 0 or C 6= 0.
Lemma 7.5 Let (A, B, D, 0) ∈ Γ∗Eisenstein . Then there exists m ∈ Ac3 such that ζs ([m]SL(V ) ) =
(A, B, D, 0).
Proof. By Theorem 6.5 (2), (3.1) and (3.2) there exists m ∈ Ac3 such that
p̃3
A
p̃2
B
=
,
=
, Disc(Qm ) = D .
Disc(Qm )
D
Disc(Qm )
D

It follows that ζs ([m]SL(V ) ) = (p̃3 , p̃2 , Disc(Qm ), 0) = (A, B, D, 0).

QED

Lemma 7.6 Let (A, B, D, C) ∈ Γ∗Eisenstein be s.t. C 6= 0. Then there exists m ∈ Ac3 such that
ζs ([m]SL(V ) ) = (A, B, D, C).
Proof. By Theorem 6.5 (1) the exists m ∈ Ac3 such that
A
p̃2
B
Invm
C
p̃3
=
,
=
,
=
.
Disc(Qm )
D
Disc(Qm )
D
Disc(Qm )
D

46

By Theorem 3.17,
27Disc(Qm )p̃23 + 4p̃32 = −24 36 Inv2m
and hence
27Disc(Qm )

A

D

Disc(Qm )

2

+4

Dividing by Disc(Qm )2 this reduces to
Disc(Qm ) 27

B

 A 2
D

D

Disc(Qm )

+4

 B 3
D

!

3

= −24 36

4 6

= −2 3

C

D

Disc(Qm )

2

.

 C 2
D

but since by hypothesis
27DA2 + 4B 3 = −24 36 C 2
and C 6= 0, we get
Disc(Qm ) = D .
It follows that ζs ([m]SL(V ) ) = (p̃3 , p̃2 , Disc(Qm ), Invm ) = (A, B, D, C).

QED
QED

Corollary 7.7 The map ζs : A3st /SL(V ) → ΓEisenstein is surjective.
Proof. By Theorem 7.4 it remains to show that the image of ζs contains
n
o
∗
ΓEisenstein \ ΓEisenstein = (A, B, 0, C) ∈ ΓEisenstein .

First note that ζ(m) = (A, B, 0, C) implies that the fundamental cubic Qm has a multiple root
and hence m admits at least one v ∈ V such that m(v, v) is proportional to v. Motivated by
this observation, let us consider m ∈ Acst of the form
m(e1 , e1 ) = ae1 , m(e2 , e2 ) = de2 , m(m1 , e2 ) = ee1 + f e2 ,
where a, d, e, f ∈ F. Using (3.7) the condition Disc(Qm ) = 0 is equivalent to (d−2e)2 (a−2f )2 = 0.
Choosing d = 2 and e = 1 leads to the multiplication table
m(e1 , e1 ) = ae1 , m(e2 , e2 ) = 2e2 , m(m1 , e2 ) = e1 + f e2 ,
and we now show that if (A, B, 0, C) ∈ ΓEisenstein satisfies BC 6= 0 then there exists m of this form
such that ζ(m) = (A, B, 0, C). From (3.13), (3.16) and (3.17) the equation ζ(m) = (A, B, 0, C)
is equivalent to
p̃3 = 72(a + f )(a − 2f )(ǫ1 ∧ ǫ2 )2 = A ,
p̃2 = −36(a − 2f )2(ǫ1 ∧ ǫ2 )2 = B ,
Invm = 4(a − 2f )3 (ǫ1 ∧ ǫ2 )3 = C ,
47

where B and C satisfy the Eisenstein constraint
4(−B)3 = (4 × 27)2 C 2 .
One easily checks that if (a, f ) ∈ F2 is any solution of the linear system
C
,
B
AB
,
= −
9 × 72C

(a − 2f )ǫ1 ∧ ǫ2 = −9
(a + f )ǫ1 ∧ ǫ2

the corresponding multiplication m satisfies ζ(m) = (A, B, 0, C).
To complete the proof of the corollary, it only remains to show that the image of ζs contains
points of ΓEisenstein which are of the form: (A, 0, 0, 0). (Recall that B = 0 is equivalent to C = 0
by the Eisenstein condition.) For this, consider m ∈ Acst of the form
1
m(e1 , e1 ) = e1 , m(e2 , e2 ) = ce1 , m(e1 , e2 ) = e2 ,
2
c ∈ F. Using (3.13), (3.16) and (3.17) we obtain
p̃3 = 27c(ǫ1 ∧ ǫ2 )2 , p̃2 = Disc(Qm ) = 0 , Invm = 0,
and hence taking
c=

A
,
27

the corresponding multiplication m satisfies ζ(m) = (A, 0, 0, 0).

QED

The following theorem is the main theorem of the paper.Assuming F is algebraically closed we
show that ζs is an embedding of the SL(V )−moduli space of stable commutative two-dimensional
algebras onto the hypersurface ΓEisenstein in a four-dimensional vector space. Furthermore, the
“projectivisation” of ζs gives exactly the embedding of the GL(V )−moduli space onto P2 of [3].
The theorem also shows that moduli spaces − both for the action of GL(V ) and SL(V )−
of stable two-dimensional commutative algebras can be thought of as compactifications of moduli spaces of generic two-dimensional commutative algebras. For instance ζs (Acst/SL(V )) =
Γ
of ζs (Ac3 /SL(V )) where “the points at infinity” are given by
nEisenstein is a “compactification”
o
(A, B, 0, C) ∈ ΓEisenstein .

48

Theorem 7.8 Suppose F is algebraically closed and consider the commutative diagram:

Acst /SL(V )

ζs

/ ΓEisenstein
F
F ✤
✍
✌✌
✍ ✤
✌
✍
✌
✍ ✤
✌✌
✍✍ ✤
✌
✍
✌
toto
✍
✤
✌✌
✍✍✍
✌
p✤
π
✍
✌✌
✤
✌
✍✍✍
✌
✤
✌
✍
toto
✌✌
✤
✍✍
Φ|Sing(M)
✌
✍
✤ ❝ ❜ ❜ ❛ ❛ ❵ ❵ ❴ ❫ ❫ ❪ ✌✌❪ ❭ ❭ ❬
✍
❬ ❩ ❩
✍✍✍ ❢ ❡ ❡ ❞ ❝✤
❨ ❳ ❳

✌✌
❢
✌
+
ζg
✍
✌ ❴ ❴ ❴/ e
✍❴
c
eCardano
❴
❴
/
❴
❴
❴
❴
❴
❴
❴
❴
❴
✌
Γ
Sing(M)✍✍
P(ΓEisenstein
)
Ast /GL(V
)❲
✌
j G
F
F
✌
❲ ❲ ❲
♣
✍✍
✌
✍
✌
♣ ♣ J ✎✎
❲ ❲Φ
♣
✌
✌
♣
✎
✍
❲
✍
= Ψ̃3
✍
✌✌∼
♣♣♣
✌❲✌ ❲ ❲ ❲
✎✎✎
✍
 4 ✍✍
 3 ✌✌
❲ ❲ ❲✌✌  ♣♣♣♣
✎
✌ ❲+ 2 x♣
ζs
✍
/ Γ∗
PF
Ac3 /SL(V )
✌✌
✎✎
✍
Eisenstein
✌
✎
✌
✌
✎
✍
✌✌
✌✌
✎✎
✍
✌
✌
✎
✌
✎
✍
✌✌
✌✌
✌
✎✎
✍
✌
✌
toto
✎
✌
✌
✎
✍
✌
✌✌
✎✎
✍
✌✌
✌
p
π
✎
✌
✌
✌
✍
✎
✌✌
✌✌
✍
✎✎✎
✌
✌
toto ✍
✌
✌
✎
✌✌
✌✌
✍
✎✎
✌
✌
✎
✌
✌
✍
✎
  3 ✌✌
 4 ✎✎
✌✌
 4 ✍
✌
ζ
g
✌
e ∗
/ P(Γ
Ac3 /GL(V ) ❲❲
ΓCardano
Eisenstein
✌✌ )
❲❲❲❲❲
✌
♥♥ I i
♥
❲❲❲❲❲ν̂
✌
♥
♥
❲❲❲❲❲
Ψ3 ∼
= ✌✌
♥♥♥
❲❲❲❲❲
 3 ✌✌ ♥♥♥♥♥
❲❲❲❲❲
❲❲+ 2 v♥♥

F

Then
1. ζs : Ac3 /SL(V ) → Γ∗Eisenstein is a bijection;
2. ζg : Ac3 /GL(V ) → P(Γ∗Eisenstein) is a bijection;
3. ζs : Acst /SL(V ) → ΓEisenstein is a bijection;
e Eisenstein ) is a bijection;
4. ζg : Acst /GL(V ) → P(Γ
2
e ∗
5. Ψ3 : P(Γ
Eisenstein ) → F is a bijection;

e Eisenstein) → P2 is a bijection;
6. Ψ̃3 : P(Γ

Proof. Let us remark that the front face of the diagram involves only generic commutative
algebras and the back face only stable commutative algebras.
We first prove part 1 and part 2. Surjectivity for both maps was proved in Theorem 7.4 over
an arbitrary field.

49

To prove the injectivity of ζs let [m]SL(V ) , [m′ ]SL(V ) ∈ Ac3 /SL(V ) be such that ζs ([m]SL(V ) ) =
ζs ([m′ ]SL(V ) ). Then by definition,
(p̃3 (m), p̃2 (m), Disc(Qm ), Invm ) = (p̃3 (m′ ), p̃2 (m′ ), Disc(Qm′ ), Invm′ )

(7.46)

and since F has no non-trivial extensions, it follows from Theorem 6.5 (1), (2) and (3) that
[m]SL(V ) = [m′ ]SL(V ) .
e ∗
To prove injectivity of ζg , first observe that by definition, P(Γ
Eisenstein ) = U3 and ν̂ = Ψ3 ◦ ζg .
′
′
Hence if ζg ([m]GL(V ) ) = ζg ([m ]GL(V ) ) then ν̂([m]) = ν̂([m )] and, since F has no non-trivial extensions, it follows from Theorems 4.19, 4.21 and 4.24 that [m]GL(V ) = [m′ ]GL(V ) .
We now prove part 3 and part 4. Surjectivity for both maps was proved in Corollary 7.7 over
an arbitrary field.
e Eisenstein ) recall that, as already observed in
To prove the injectivity of ζg : Acst /GL(V ) → P(Γ
(4.23), we have
Φ = Ψ̃3 ◦ ζg ,

where Φ : Acst /GL(V ) → P2 is given by (cf [3])
Φ([m]) = [p̃3 (m), p̃2 (m), Disc(Qm )] .
It was shown in [3] that Φ is injective and hence it follows that both ζg and Ψ̃3 are injective (as
we have already shown that ζg is surjective). This proves part (3) of the theorem.
To prove the injectivity of ζs : Acst /SL(V ) → ΓEisenstein let m, m′ ∈ Acst be such that
ζs ([m]SL(V ) ) = ζs ([m′ ]SL(V ) ) .
Then
ζg ([m]GL(V ) ) = ζg ([m′ ]GL(V ) )
and so Φ([m]) = Φ([m′ ]). By [3] it follows that there exists g ∈ GL(V ) such that m′ = g · m.
On the other hand ζs ([m]SL(V ) = ζs ([m′ ]SL(V ) is equivalent to
(p̃2 (m), p̃3 (m), Disc(Qm ), Invm ) = (p̃2 (m′ ), p̃3 (m′ ), Disc(Qm′ ), Invm′ ) .

(7.47)

If Disc(Qm ) = Disc(Qm′ ) 6= 0 it follows from Theorem 6.5 (1), (2) and (3) that [m]SL(V ) =
[m′ ]SL(V ) .
If Disc(Qm ) = Disc(Qm′ ) = 0 it follows (see Appendix A) that m and m′ are GL(V )−equivalent
to one of the algebras in Table 1 (two types) or Table 2 (one type). The condition (7.47) implies
that m and m′ must be of the same type. In fact, as we show in the following lemma, for such
algebras GL(V )−equivalence implies SL(V )−equivalence.
Lemma 7.9 Let (e1 , e2 ) and (e′1 , e′2 ) be bases of V . We denote by (ǫ1 , ǫ2 ) and (ǫ′1 , ǫ′2 ) the corresponding dual bases. Define α ∈ F∗ by e′1 ∧ e′e = αe1 ∧ e2 .
50

(i) Let ν, ν ′ ∈ F \ { 12 } and consider the two commutative algebras m, m′ given by
m(e1 , e1 ) = e1 , m(e2 , e2 ) = e2 , m(e1 , e2 ) = 21 e1 + νe2 ,
m′ (e′1 , e′1 ) = e′1 , m(e′2 , e′2 ) = e′2 , m(e′1 , e′2 ) = 21 e′1 + ν ′ e′2 .
If m and m′ are GL(V )−equivalent and satisfy (7.46) then they are SL(V )−equivalent.
(ii) Consider the two commutative algebras m, m′ given by
m(e1 , e1 ) = 0 , m(e2 , e2 ) = e2 , m(e1 , e2 ) = 21 e1 + e2 ,
m′ (e′1 , e′1 ) = 0 , m(e′2 , e′2 ) = e′2 , m(e′1 , e′2 ) = 21 e′1 + e′2 ,
If m and m′ are GL(V )−equivalent and satisfy (7.46) then they are SL(V )−equivalent.
(iii) Let λ, λ′ ∈ F∗ and consider the two commutative algebras m, m′ given by
m(e1 , e1 ) = e1 , m(e2 , e2 ) = λe1 , m(e1 , e2 ) = 21 e2 ,
m′ (e′1 , e′1 ) = e′1 , m(e′2 , e′2 ) = λe′1 , m(e′1 , e′2 ) = 21 e′2 ,
If m and m′ are GL(V )−equivalent and satisfy (7.46) then they are SL(V )−equivalent.
Proof. (i) : Since m and m′ are GL(V )−equivalent we must have ν = ν ′ (see Theorem A.1).
Since by (7.46) we have
p̃3 (m) = −18(2ν − 1)(ν + 1)(ǫ1 ∧ ǫ2 )2 = p̃3 (m′ ) = −18(2ν − 1)(ν + 1)(ǫ′1 ∧ ǫ′2 )2
p̃2 (m) = −9(2ν − 1)2 (ǫ1 ∧ ǫ2 )2
= p̃2 (m′ ) = −9(2ν − 1)2 (ǫ′1 ∧ ǫ′2 )2 .
Substituting ǫ′1 ∧ ǫ′2 = 1/αǫ1 ∧ ǫ and solving the two equations above gives α2 = 1. Since
Invm = −1/2(2ν − 1)3 (ǫ1 ∧ ǫ2 )3 = Inv′m = −1/2(2ν − 1)3 (ǫ′1 ∧ ǫ′2 )3 then α3 = 1. This implies
that α = 1 and the linear mapping g : V → V given by g(e1 ) = e′1 , g(e2 ) = e′2 is an algebra
isomorphism of determinant one.
(ii): Just as above the equality (7.46) implies that α3 = α2 = 1 and hence the linear mapping
g : V → V given by g(e1) = e′1 , g(e2 ) = e′2 is an algebra isomorphism of determinant one.
(iii): Since p̃3 (m) = −27λ(ǫ1 ∧ ǫ2 )2 = p̃3 (m′ ) = −27λ′ (ǫ′1 ∧ ǫ′2 )2 , we have λ′ = α2 λ. Let µ ∈ F∗
be the unique square root of λ/λ′ such that µα = 1. It is straightforward to check that the linear
map g : V → V given
g(e1 ) = e′1 , g(e2 ) = µe′2 ,
is an algebra isomorphism of determinant one.
This completes the proof of part (3) of the Theorem.

QED

Finally, observe that part (6) of the theorem implies part (5) of the theorem and that part
(6) is a consequence of the fact that
Ψ̃3 = Φ ◦ ζg−1
is the composition of the bijections ζg−1 (cf part (4) above) and Φ (cf [3]).
QED

51

Disc(D)-generic algebras modulo SL(V ) × SL(V ) and binary quadratic forms modulo SL(V )

8

Recall that if m ∈ Ac , by definition Disc(Dm ) is the discriminant of the quadratic form det Lm
(cf Definition 3.18). In this section we consider the set of Disc(D)-generic algebras
n
o
AcD = m ∈ Ac s.t. Disc(Dm ) 6= 0 ,

modulo the left action of SL(V ) × SL(V ) given by
(g1 ,g2 )

m(u, v) = g1 m(g2−1 u, g2−1v) .

Restricting this action to the diagonally embedded SL(V )
g 7→ (g, g)
we recover the action of SL(V ) of (2.3). It turns out that Disc(D) is not just SL(V ) invariant
but in fact SL(V ) × SL(V ) invariant. We first show that the map which associates to m in AcD
the binary quadratic form Dm induces a bijection D̂ of the moduli space AcD /(SL(V ) × SL(V ))
with the moduli space of non-degenerate quadratic forms S 2 (V ∗ )n.d. /SL(V ). As an application
we define a group structure on the elements of AcD /SL(V ) × SL(V ) of fixed discriminant ∆, and
show that D̂ induces a group isomorphism between this group and the group of SL(V )−orbits
of quadratic forms of discriminant ∆ (with Gauss composition as the group law).

8.1

Isomorphism of moduli spaces

We first show that Disc(Dm ) is an SL(V ) × SL(V ) invariant.
Proposition 8.1 Let m ∈ Ac and (g1 , g2 ) ∈ SL(V ) × SL(V ). Then,
(i) D(g1 ,g2 ) m = g2 · Dm .
(ii) Disc(D(g1 ,g2 ) m ) = Disc(Dm ) .
Proof. We set m̃ = (g1 ,g2 ) m. By definition for all u, v in V ,
m̃(u, v) = g1 m(g2−1 u, g2−1v)
and thus
m
−1
Lm̃
u (v) = g1 Lg −1 u (g2 v) .
2

Hence
m
−1
Lm̃
u = g1 ◦ Lg −1 u ◦ g2 ,
2

and taking determinants
m
det(Lm̃
u ) = det(Lg −1 u ) .
2

52

This implies (see (3.8))
Dm̃ (u) = Dm (g2−1u) = g2 · Dm (u) , ∀u ∈ V .
This proves part (i). Part (ii) follows from this and the fact that, as is well known, two binary
quadratic forms in the same SL(V )−orbit have the same discriminant.
QED
The following corollary is a simple consequence of the proof of the proposition and the fact that
Dm is non-degenerate if m ∈ AcD .
Corollary 8.2 The map D : AcD → S 2 (V ∗ )n.d. factors to a map D̂ : AcD /SL(V ) × SL(V ) →
S 2 (V ∗ )n.d. /SL(V ). If m ∈ AcD we write D̂[m] = [Dm ] for the image of the equivalence class [m]
by D̂.
Proposition 8.3 The map D̂ : AcD /SL(V ) × SL(V ) → S 2 (V ∗ )n.d. /SL(V ) defined in Corollary
8.2 is a bijection.
Proof. To show that D̂ : AcD /SL(V ) × SL(V ) → S 2 (V ∗ )n.d. /SL(V ) is surjective it is sufficient
to show that D : Ac → S 2 (V ∗ ) is surjective. Let (e1 , e2 ) be a basis of V with dual basis (ε1 , ε2 )
and m : V × V → V be given by:
m(e1 , e2 ) = ae1 + be2 , m(e2 , e2 ) = ce1 + de2 , m(e1 , e2 ) = ee1 + f e2 , m(e2 , e1 ) = ee1 + f e2 ,
then by (3.9)
Dm (xe1 + ye2) = (af − be)x2 + (ad − bc)xy + (ed − cf )y 2 .
If q = αx2 + 2βxy + γy 2 then Dm = q iff
af − be = α ; ad − bc = 2β ; ed − cf = γ .

(8.48)

If q = 0 then m ≡ 0 satisfies D̂(m) = q. If q 6= 0 then without loss of generality we can suppose
that β 6= 0. Then the vectors v1 = ae1 + be2 and v2 = ce1 + de2 form a basis of V and there exist
constants A, B s.t.
ee1 + f e2 = Av1 + Bv2 .
Substituting in the system above leads to
A=

α
γ
; B=
; ad − bc = 2β .
2β
2β

In other words our system has a solution iff ad − bc = 2β has a solution but this is obvious (e.g.
a = 2, d = β, b = c = 0, e = βγ , f = α2 ). This completes the proof of the surjectivity of D̂.
To show that D̂ is injective we first prove the following lemma:
Lemma 8.4 Let q be in S 2 (V ∗ )n.d. and suppose that D(m1 ) = D(m2 ) = q for m1 , m2 in AcD .
Then there exists (g, 1) ∈ SL(V ) × SL(V ) such that (g,1) m1 = m2 .
53

Proof. It follows from (8.48) that Dm = q iff
af − be = α , ad − bc = 2β , ed − cf = γ .
In particular, the multiplication m0 given by
a0 = 2 , d0 = β , b0 = c0 = 0 , e0 =

α
γ
, f0 =
β
2

satisfies Dm0 = q. It is easy to check that if we set
a c 
g = 2b βd ,
2

β

then det(g) = 1 and
(g,1)

m0 = m .

Thus the group SL(V ) × {1} acts transitively on the set of solutions of Dm = q which proves
the lemma.
QED
We now prove D̂ is injective. Suppose that
D̂[m1 ] = D̂[m2 ] .
Thus by definition
[Dm1 ] = [Dm2 ] ,
from which it follows that there exist h in SL(V ) s.t.
h · Dm1 = Dm2 .
By Proposition 8.1(i) this means that
D(1,h) m1 = Dm2 ,
and therefore by Lemma 8.4 there exists g in SL(V ) such that
(g,1)

((1,h) m1 ) = m2 .

Hence
(g,h)

m1 = m2 ,

in other words
[m1 ] = [m2 ]
and D̂ is injective.

QED

54

8.2

Group structure

The Gauss composition law is a non-trivial group law on the set of SL(V )−orbits of nondegenerate quadratic forms with fixed non-zero discriminant. There is therefore a unique group
structure on the set of SL(V ) × SL(V )−orbits of Disc(D)−generic algebras with fixed discriminant, which makes D̂ (cf Proposition 8.3) into a group isomorphism. We will now give an
intrinsic description of this group structure.
Let now [m], [m′ ] be two elements of AcD /SL(V ) × SL(V ) such that Disc(D̂[m] ) =Disc(D̂[m′ ] )
is not a square in L2 . We can always find a basis {e1 , e2 } of V which is neither orthogonal for
D̂[m] nor for D̂[m′ ] . In this basis we write
D̂[m] (xe1 + ye2 ) = α x2 + 2β xy + γ y 2 ,
D̂[m′ ] (xe1 + ye2 ) = α′ x2 + 2β ′xy + γ ′ y 2 ,
2

Disc(D̂[m] ) = Disc(D̂[m′ ] ) = ∆(ǫ1 ∧ ǫ2 )⊗ ,
where {ǫ1 , ǫ2 } is the dual basis of {e1 , e2 }, ∆ is an element of F∗ which is not a square in F∗ and
ββ ′ 6= 0. Note that ∆ = 4β 2 − 4αγ = 4β ′2 − 4α′γ ′ and hence it follows that αγα′γ ′ 6= 0. By
Lemma 8.4 the multiplications m0 and m′0 given by
γ
α
, f0 =
β
2
′
γ
α′
a′0 = 2 , d′0 = β ′ , b′0 = c′0 = 0 , e′0 = ′ , f0′ =
β
2

a0 = 2 , d0 = β , b0 = c0 = 0 , e0 =

satisfy D̂[m0 ] = D̂[m] and D[m′0 ] = D̂[m′ ] . The discriminants of m0 , m′0 are both equal to ∆(ǫ1 ∧ǫ2 )⊗
so we can write
f0 =

2

d′ 2 − ∆
d20 − ∆4
, f0′ = 0 ′ ′ 4
2e0 d0
2e0 d0

and this suggests that we define the “product” of [m0 ] and [m′0 ] to be the class of the multiplication m′′0 given in the basis {e1 , e2 } by
a′′0 =

(d0 d′0 )2 − ∆4
a0 a′0
.
, d′′0 = d0 d′0 , b′′0 = b0 b′0 , c′′0 = c0 c′0 , e′′0 = e0 e′0 , f0′′ =
2
2e0 d0 e′0 d′0

By Lemma 8.4 one gets
D̂

[m′′
0]

(ββ ′)2 −
(xe1 + ye2 ) =
γγ ′

∆
4

x2 + 2ββ ′xy + γγ ′ y 2
2

from which it follows that the discriminant of m′′0 is equal to ∆(ǫ1 ∧ ǫ2 )⊗ .
Proposition 8.5 Let Q∆ = S 2 (V ∗ )∆ /SL(V ) denote the set of SL(V )−equivalent classes of
quadratic forms of discriminant ∆ and let ⋆ : Q∆ × Q∆ → Q∆ be Gauss composition. Then
D̂[m] ⋆ D̂[m′ ] = D̂[m′′ ] .
55

Proof. Set
n
F∗∆ = a2 − ∆b2 ∈ F∗

and

s.t. a, b ∈ F

o

G∆ = F∗ /F∗∆ .
It is well know that Q∆ with Gauss composition is isomorphic to G∆ under the map [q] 7→ [q(v)]
where q ∈ S 2 (V ∗ )∆ and q(v) is its value at any non-isotropic v ∈ V . Hence to prove the
proposition it is sufficient to check that
D̂[m′′ ] (e2 ) = D̂[m] (e2 )D̂[m′ ] (e2 ) ,
which is immediate since
D̂[m0 ] (e2 ) = γ , D̂[m′0 ] (e2 ) = γ ′ , D̂[m′′0 ] (e2 ) = γ ′′ = γγ ′ .

(8.49)
QED

Remark 8.6 The essential point of our construction was the choice of the basis {e1 , e2 } with
respect to which neither D̂[m] nor D̂[m′ ] was diagonal. Once this choice has been made m0 , m′0 and
m′′0 are uniquely defined but there are many bases of V with the above property. Our proposition
shows that in fact the SL(V ) × SL(V ) equivalence class of m′′ is independent of this choice.
Remark 8.7 Proposition 8.5 is an analogue of the isomorphism
Cl(Z2 ⊗ Sym2 Z2 ; ∆) → Cl((Sym2 Z2 )∗ ; ∆) ,
of Bhargava [2] (p. 225).

A
A.1

Non-generic commutative algebras
Stable commutative algebras

As we pointed out in Section 4.1 the GL(V )−moduli space Mcst of stable two-dimensional commutative algebras is larger than the GL(V )−moduli space of generic two-dimensional commutative
algebras Mc3 . In fact there is a stratification
Mcst = Mc3 ∪ Mc2 ∪ Mc1 ,
where
Mc3
Mc2
Mc1

n
= m ∈ M s.t.
n
= m ∈ M s.t.
n
= m ∈ M s.t.

o

Disc(Qm ) 6= 0 ;

o
Disc(Qm ) = 0 and p̃2 (m) 6= 0 ;

o
Disc(Qm ) = p̃2 (m) = 0 and p̃3 (m) 6= 0 .
56

By definition the open stratum is the moduli space of generic algebras Mc3 and is of dimension
two. The next stratum Mc2 is of dimension one and corresponds to algebras whose fundamental
cubic has exactly two distinct roots, and the last stratum Mc1 is a single point and corresponds
to algebras whose fundamental cubic has exactly one root (see below).
In Section 4 we showed that for [m] ∈ M3 :
• if ν̂([m]) 6∈ ΓCardano then ν̂([m]) completely determines the equivalence class of m (even if
F is not algebraically closed);
• if ν̂([m]) ∈ ΓCardano \{(0, 0)} then ν̂([m]) together with the equivalence class of the splitting
field (at most quadratic) of the fundamental cubic Qm completely determine the equivalence
class of m;
• if ν̂([m]) = (0, 0) then ν̂([m]) together with the equivalence class of the splitting field of
the fundamental cubic Qm completely determine the equivalence class of m;


• ν̂([m]) ∈ ΓCardano \ {(0, 0)} iff [m] ∈ Sing(Mcst )\ Sing(Sing(Mcst )) ∩ Mc3 ;
• ν̂([m]) = (0, 0) iff [m] ∈ Sing(Sing(Mcst )) ∩ Mc3 ;

• ν̂([m]) ∈ ΓCardano iff m admits a non-trivial automorphism.
In this appendix we will extend all of the above results to Mcst . In order to do this we will give
normal forms for algebras in Mc2 and Mc1 based on the fact that algebras in Mc2 have a basis
of elements satisfying m(x, x) = λx, and algebras in Mc1 have a one-dimensional subspace of
elements satisfying m(x, x) = λx.
Theorem A.1 In Table 1 (resp. Table 2) we give the normal forms, continuous moduli, discrete
moduli, the automorphisms, the values of the coordinates p̃2 , p̃3 , and the values of the invariants
Qm , Disc(Dm ), Invm for Mc2 (resp. Mc1 ). “Discrete” moduli are only needed when the field F is
not algebraically closed. In these tables we have written e1 2 for m(e1 , e1 ) etc and {ǫ1 , ǫ2 } for the
basis dual to {e1 , e2 }.
Proof. If [m] ∈ Mc2 , there is a basis of elements satisfying m(x, x) = kx and if [m] ∈ Mc1 there
is a unique one-dimensional nontrivial subalgebra. If [m] ∈ Mc2 it is relatively straightforward
to show that the normal forms in Table 1 are the only possible ones even if F is not algebraically
closed. If [m] ∈ Mc1 there is a unique idempotent e1 and those multiples of e1 which are
perfect squares define [λ] ∈ F∗ /F∗2 . It turns that this class completely characterises algebras in
Mc1 .
QED
Remark A.2 If F is algebraically closed Mc1 consists of a single point as was shown in [3].
Recall that
eCardano =
Γ

(

[a, b, c] ∈ P2 s.t. 27ca2 + 4b3 = 0
57

)

,

Normal form

Moduli

e21 = e1
e22 = e2
1
e1 + νe2
e1 e2 =
2

continuous:
ν ∈ F \ { 12 }.
marcus
discrete:
none.

e21
e22
e1 e2

= 0
= e2
1
e1 + e2
=
2

Automorphisms

Coordinates p̃2 , p̃3 ;
Invariants Qm , Disc(Dm ).

{1}

p̃2 = −9(2ν − 1)2 (ǫ1 ∧ ǫ2 )2
p̃3 = −18(2ν − 1)(ν + 1)(ǫ1 ∧ ǫ2 )2
1
Invm = − (2ν − 1)3 (ǫ1 ∧ ǫ2 )3
2
Qm = (1 − 2ν)x2 y(e1 ∧ e2 )
Disc(Dm ) = (1 − 2ν)(ǫ1 ∧ ǫ2 )2

{1}

p̃2 = −36(ǫ1 ∧ ǫ2 )2
p̃3 = −36(ǫ1 ∧ ǫ2 )2
Invm = −4(ǫ1 ∧ ǫ2 )3
Qm = −2x2 y(e1 ∧ e2 )
Disc(Dm ) = 0

continuous:
point.
marcus
discrete:
none.

Table 1: m ∈ Mc2

Normal form
e21
e22
e1 e2

= e1
= λe1
1
=
e2
2

Moduli
continuous:
point.
marcus
discrete:

Automorphisms

Z2

[λ] ∈ F∗ /F∗2 .
Table 2: m ∈ Mc1

58

Invariants
p̃2 = 0
p̃3 = −27λ(ǫ1 ∧ ǫ2 )2
Invm = 0
Qm = λy 3 (e1 ∧ e2 )
Disc(Dm ) = λ(ǫ1 ∧ ǫ2 )2

is the ‘projective’ extension of the ‘affine’ curve ΓCardano ⊂ F2 and that Φ : Mcst → P2 is the
projective extension of ν̂ : Mc3 → F2 .
It follows immediately from the Tables 1 and 2 that the properties given above for generic
algebras extend to stable algebras as follows:
e Cardano then Φ([m]) completely determines the equivalence class of m (even if
• if Φ([m]) 6∈ Γ
F is not algebraically closed);

eCardano \ {[0, 0, 1]} then Φ([m]) together with the equivalence class of a (at
• if Φ([m]) ∈ Γ
most) quadratic extension of F determines the equivalence class of m;
• if Φ(m) = [0, 0, 1] then Φ([m]) together with the equivalence class of the splitting field of
the fundamental cubic Qm completely determine the equivalence class of m;


c
c
e
• Φ([m]) ∈ ΓCardano \ {[0, 0, 1]} iff [m] ∈ Sing(Mst )\ Sing(Sing(Mst )) ;

• Φ([m]) = [0, 0, 1] iff [m] ∈ Sing(Sing(Mcst ));

eCardano iff m admits a non-trivial automorphism.
• Φ([m]) ∈ Γ

A.2

Non-stable commutative algebras

For completeness we give the classification of non-stable algebras which by definition are those
algebras on which all scalar-valued invariant polynomials vanish.
Theorem A.3 In Table 3 we give the normal forms, moduli, automorphisms, and the value of
the fundamental cubic Qm for non-stable commutative algebras. In these tables we have written
e1 2 for m(e1 , e1 ) etc and (ǫ1 , ǫ2 ) for the basis dual to (e1 , e2 ).
Proof. The proof of this theorem is very similar to the proof of the theorem above.

QED

Remark A.4 It is interesting to observe that non-stable commutative algebras are characterised
amongst all commutative algebras as being those whose automorphism group is isomorphic to F∗ .
Similarly stable commutative algebras are characterised as those whose automorphism group is
finite.
Remark A.5 Note that at the end of the article [3] a description of the moduli space of nonstable not necessarily commutative algebras is given.

59

Normal form
e21
e22
e1 e2

= e1
= 0
= νe2

Moduli

Automorphisms

Fundamental cubic

F∗

Qm = (1 − 2ν)yx2 (e1 ∧ e2 )2

F∗

Qm = −2yx2 (e1 ∧ e2 )2

F∗

Qm = y 3(e1 ∧ e2 )2

F∗

Qm = 0

continuous:
ν ∈ F \ { 12 }.
marcus
discrete:
none.

e21
e22
e1 e2

= 0
= 0
= e2

continuous:
Point.
marcus
discrete:
none.

e21
e22
e1 e2

= 0
= e1 + δe2
1
=
δe1
2

e21
e22
e1 e2

= e1
= 0
1
e2
=
2

continuous:
δ ∈ {0, 1}.
marcus
discrete:
none.
continuous:
Point.
marcus
discrete:
none.

Table 3: Non-stable algebras

60

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