CONJUGACY CLASSES IN PSL 2K CHRISTOPHER-LLOYD SIMON Abstract. We first describe over a field Kof characteristic different from 2 the orbits for the

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CONJUGACY CLASSES IN PSL2(K)

CHRISTOPHER-LLOYD SIMON

Abstract. We ﬁrst describe, over a ﬁeld Kof characteristic diﬀerent from 2, the orbits for the

adjoint actions of the Lie groups PGL2(K)and PSL2(K)on their Lie algebra sl2(K). While the

former are well known, the latter lead to the resolution of generalised Pell-Fermat equations which

characterise the corresponding orbit. The synthetic approach enables to change the base ﬁeld,

and we illustrate this picture over the ﬁelds with three and ﬁve elements, in relation with the

geometry of the tetrahedral and icosahedral groups. While the results may appear familiar, they

do not seem to be covered in such generality or detail by the existing literature.

We apply this discussion to partition the set of PSL2(Z)-classes of integral binary quadratic

forms into groups of PSL2(K)-classes. When K=Cwe obtain the class groups of a given

discriminant. Then we provide a complete description of their partition into PSL2(Q)-classes in

terms of Hilbert symbols, and relate this to the partition into genera. The results are classical,

but our geometrical approach is of independent interest as it may yield new insights into the

geometry of Gauss composition, and unify the picture over function ﬁelds.

Finally we provide a geometric interpretation in the modular orbifold PSL2(Z)\Hfor when

two points or two closed geodesics correspond to PSL2(K)-equivalent quadratic forms, in terms

of hyperbolic distances and angles between those modular cycles. These geometric quantities are

related to linking numbers of modular knots. Their distribution properties could be studied using

the geometry of the quadratic lattice (sl2(Z),det) but such investigations are not pursued here.

Plan of the paper

Introduction 2

1. Geometric algebra of gl2(K)4

2. The Lie algebra sl2(K)6

3. Ptolemy’s theorem for quadrilaterals inscribed in P(X)8

4. The adjoint actions of PGL2(K)and PSL2(K)on P(sl2(K)) 10

5. Applications to binary quadratic forms 15

6. Arithmetic equivalence of singular moduli and modular geodesics 19

References 22

Acknowlegements. This article contains the main results obtained in the chapter 1 of my thesis.

I would thus like to thank my thesis advisors Etienne Ghys and Patrick Popescu-Pampu for their

guidance and encouragement; as well as Francis Bonahon, Louis Funar, Jean-Pierre Otal and Anne

Pichon who refereed and carefully read my work. I am also grateful to Nicolas Bergeron for sharing

stimulating discussions, and to the members of the arithmetics & dynamics teams at Penn State

for inviting me to present my works in their seminars. I owe Marie Dossin for helping me with the

ﬁgures in tikz. Finally, I thank the editors and referees of the MRR for their instructive comments.

Date: June 13, 2023.

arXiv:2210.02481v2 [math.GR] 11 Jun 2023

Introduction

Adjoint action of PSL2(K)on sl2(K).Let us work over a ﬁeld Kof characteristic diﬀerent from 2.

The automorphism group PGL2(K)of the projective line and its largest simple subgroup PSL2(K)

play a fundamental role in various areas of mathematics. They appear for instance in algebraic

geometry when K=Cand in hyperbolic geometry when K=R; in arithmetics when K=Qand

in Galois theory when K=Z/p.

The ﬁrst step to understand (the representation theory of) those linear algebraic groups is to

describe their conjugacy classes, and more precisely the adjoint actions on their Lie algebra sl2(K).

These actions preserve the Killing form, which is a multiple of the non-degenerate quadratic form

det: sl2(K)→K. It is well known that PGL2(K)acts transitively on every level set of det. After

introducing the cross-ratio bir(a,b)∈Kof two elements a,b∈sl2(K)with non-zero determinant,

we will precise this statement.

Proposition 0.1. Let a,b∈sl(V)have determinant −δ̸= 0 and bir(a,b)/∈ {1,∞}.

The matrices M∈PGL2(K)conjugating ato bhave a well deﬁned determinant in the quotient

K×/NrK(K[√δ]×), and its is equal to the class of bir(a,b).

In contrast, we will index the PSL2(K)-orbits inside {det = −δ}by the classes [χ]in the quotient

group K×/NrK(K[√δ]), and parametrize each orbit (δ, χ)by the solutions in K×Kto the generalised

Pell-Fermat equation x2−δy2=χ. In homological terms, the conjugacy problem in PSL2(K)has

obstructions measured by the group K×/NrK(K[√δ]×), and when they vanish the conjugacies form

a torsor under the group of units {γ∈K[√δ]|Nr(γ)=1}.

Theorem 0.2. Let a,b∈sl2(K)have determinant −δ̸= 0 and cross-ratio bir(a,b)=4χ /∈ {1,∞}.

The elements C∈SL2(K)such that CaC−1=bare parametrized by the Pell-Fermat conic:

(x, y)∈K×K:x2−δy2=χ

C(x, y) = x(1+ba−1) + y(a+b)

In particular, aand bare conjugate by an element C(x, y)∈SL2(K)if and only if bir(a,b)

belongs to the subgroup of norms NrKK[√δ]⊂K×of the quadratic extension, and by an element

C(x, 0) ∈SL2(K)∩K[{a,b}]if and only if bir(a,b)belongs to the subgroup of squares (K×)2⊂K×.

The proofs of these statements can be reduced to elementary linear algebra once we thoroughly

understand the geometry of the Lie algebra sl2(K)inside the quaternion algebra gl2(K). In short,

commutativity rhymes with colinearity whereas anti-commutativity rhymes with orthogonality.

We recall this background material in the ﬁrst two sections , and provide an amusing application

in the third to prove an analogue of Ptolemy’s identity for quadrilaterals inscribed in the isotropic

cone Xof (sl2(K),det), relying on a natural quadratic desingularization ψ:K2→X.

Classes of binary quadratic forms. By polarizing a binary quadratic form on K2with respect

to the canonical symplectic form det, we obtain an isomorphism:

Q=lx2+mxy +ry2∈ Q(K2)Q(v)=det(v,qv)

←−−−−−−−−→ q=1

2−m−2r

2l m ∈sl2(K)

between the Poisson algebra (Q(K2),disc) and the Lie algebra (sl2(K),−4 det), conjugating the ac-

tions of PSL2(K)by change of variables and by conjugacy. After interpreting the values of quadratic

forms as the scalar products with elements in the isotropic cone Q(v) = ⟨q, ψ(v)⟩, and computing

that the products Qa(u)Qb(v) = ⟨a, ψ(u)⟩⟨b, ψ(v)⟩for primitive vectors u, v ∈K2are equivalent to

the cross-ratio bir(Qa, Qb) = bir(a,b)in K×/Nr(K[√∆]×), we will deduce the following.

Proposition 0.3. The set ClK(∆) of PSL2(K)-orbits in Q(K2)with non-square discriminant ∆

embeds into the group K×/NrK(K[√∆]×)of exponent two, by sending the class of the norm x2−∆

4y2

of the K-extension K[√∆] to the identity, and using the multiplication of values for composition.

The initial motivation was to understand the space Q(Z2)of integral binary quadratic forms

Q(x, y) = lx2+mxy +ry2up to change of variables by PSL2(Z), and the class groups Cl(∆) of

primitive classes with non-square discriminant ∆ = m2−4lr ∈Zintroduced by Gauss in [Gau07].

We refer to [Cas78, Cox97] and [Wei84] for the relevant background and history.

For a ﬁeld Kof characteristic ̸= 2, the extension of scalars Z→Kinduces a map Q(Z2)→ Q(K2),

yielding a group morphism Cl(∆) →ClK(∆). We say that Qa, Qb∈ Q(Z2)are K-equivalent when

they are conjugate by C∈PSL2(K). When K⊃Qthis implies that they have the same discriminant

(not just modulo (K×)×), and Theorem 0.2 & Proposition 0.3 show that their K-equivalence is

measured by bir(Qa, Qb)≡Qa(1,0)Qb(1,0) = lalb∈K×/Nr(K[√∆]×).

Thus, when K=Cthis groups the PSL2(Z)-classes of Q(Z2)according to their discriminant ∆,

and as Kdecreases we obtain ﬁner partitions of the class groups Cl(∆) into K-classes. In section 5

we provide a computable characterisation of Q-equivalence in terms of the Hilbert symbols (δ, χ)p

at all primes p∈Z, which measures the obstruction to solving the equation x2−δy2=χin Qp.

We also deduce from Theorem 0.2 a relation between the partition of Cl(∆) into Q-classes and its

partition into genera, which are given by the cosets Cl(∆)/Cl(∆)2modulo the subgroup of squares.

Theorem 0.4. For all non-square discriminant ∆∈Z, genus equivalence implies Q-equivalence.

For all fundamental discriminants ∆, genus equivalence is also implied by Q-equivalence.

Arithmetic equivalence of singular moduli & modular geodesics. We conclude with a geo-

metric interpretation of K-equivalence, which one may compare with [Pen96]. The modular group

PSL2(Z)acts on the upper-half plane HP ={z∈C| ℑ(z)>0}by linear fractional transformations,

and the quotient is the modular orbifold M= PSL2(Z)\HP.

Consider primitive integral binary quadratic forms Qa, Qbwith non-square discriminant ∆, and

denote (α′, α),(β′, β)their roots (which one may order up to simultaneous inversion).

If ∆>0, then Qaand Qbare uniquely determined by α, β ∈HP, and their PSL2(Z)-classes

correspond to points [α],[β]∈M, called singular moduli.

Corollary 0.5. Two complex irrationals α, β ∈Q(√∆) are K-equivalent if and only if there exists

a hyperbolic geodesic arc in Mfrom [α]to [β]whose length λis of the form cosh λ

22=1

(2x)2−∆y2

for x, y ∈K, in which case all geodesic arcs from [α]to [β]have this property.

If ∆<0then Qaand Qbcorrespond to the oriented geodesics (α′, α),(β′, β)in HP and their

PSL2(Z)-classes correspond to primitive closed oriented geodesics in M, called modular geodesics.

Corollary 0.6. Two modular geodesics of the same length 2 sinh−1(√∆/2) are K-equivalent if and

only if one of the following equivalent conditions hold:

θThere exists one intersection point with angle θ∈]0, π[such that cos θ

22=1

(2x)2−∆y2for

x, y ∈K, in which case all intersections have this property.

λThere exists one co-oriented ortho-geodesic of length λsuch that cosh λ

22=1

(2x)2−∆y2for

x, y ∈K, in which case all such ortho-geodesics have this property.

In conclusion, the Q-equivalence is measured by the geometric quantities (cos θ

2)2or cosh λ

22

as elements in Q×mod NrQQ(√∆), and their multiplication implies a geometric interpretation for

the multiplication of genera.

1. Geometric algebra of gl2(K)

Consider a ﬁeld Kof characteristic diﬀerent from 2and a K-vector space Vof dimension 2.

Involutive algebra. The K-algebra gl(V)of linear endomorphisms of Vis isomorphic to V⊗V∗

with product deﬁned by (u⊗µ)·(v⊗ν) = u⊗µ(v)ν. It is endowed with the canonical linear

form Tr: gl(V)→Kdeﬁned by Tr(u⊗µ) = µ(u). Let us ﬁnd a canonical involution M7→ M#

on gl(V), that is an anti-commutative linear endomorphism of order two, to deduce a canonical

non-degenerate bilinear form (M, N)7→ Tr(M N #)isomorphic to the pairing gl(V)×gl(V)∗→K.

A non-degenerate bilinear form ω:V×V→Kis equivalent to an isomorphism ω∗:V→V∗. Its

associated adjoint involution #: gl(V)→gl(V)is the composition of (ω∗)⊗(ω∗)−1:V⊗V∗→V∗⊗V

with the canonical map switching factors, thus #: u⊗µ7→ (ω∗)−1(µ)⊗ω∗(u). Observe that #only

depends on ωup to scaling. The plane Vadmits a unique non-degenerate anti-symmetric bilinear

form ωup to scaling as Λ2V∗≃K, and this deﬁnes our canonical involution #.

Only after choosing a basis of Vdo we have the identiﬁcations V=K2and gl(V) = gl2(K). Then

ω(u, v) = det(u, v)and the associated adjoint involution on gl2(K)is the transpose-comatrix:

M=a b

c d 7−→ M#=d−b

−c a .

The ﬁxed subalgebra of M7→ M#is reduced to the center K1of gl(V). Composing (M, M#)with

addition or multiplication yields the central elements:

Tr(M)1:= M+M#and det(M)1:= M×M#

which recovers the linear trace map Tr: gl(V)→K, and deﬁnes the multiplicative determinant

map det: gl(V)→K. The involution #preserves the group GL(V)of invertible elements, which

consists in those A∈gl(V)such that det(A)∈K×, in which case A−1= det(A)−1A#.

For A∈GL(V)and M∈gl(V)we have (AMA−1)#=AM #A−1, so the left adjoint linear

action of GL(V)on gl(V)preserves the involution, whence all the structures which will follow.

The kernel SL(V)of the determinant morphism det: GL(V)→K×is called the subgroup of

units, thus A∈SL(V)⇐⇒ det(A)=1 ⇐⇒ A#=A−1. The kernel sl(V)of the trace form is

the anti-symmetric part for the involution, thus a∈sl(V)⇐⇒ Tr(a)=0 ⇐⇒ a#=−a.

Quadratic space. On the vector space gl(V)the determinant is a non degenerate quadratic form,

and as det(M+N)1= (M+N)(M+N)#=det(M) + Tr(MN#) + det(N)1, its polar symmetric

bilinear form is:

⟨M, N⟩= tr(MN#) where tr(P) := 1

2Tr(P)

The involution #has eigenvalues ±1and its eigenspaces provide a decomposition

gl(V) = K1⊕sl(V)

which is orthogonal with respect to the determinant form. Thus every element M∈gl(V)splits as

the sum of its symmetric and anti-symmetric parts with respect to the involution:

M= tr(M)1+ pr(M) where tr(M)1=M+M#

2and pr(M) := M−M#

In particular det(M)1= tr(M)21−pr(M)2which we may write det = tr2−pr2.

The 4-dimensional space gl(V), which contains the isotropic cone gl(V)\GL(V)deﬁned by

det(M) = ⟨M, M⟩= 0, decomposes as the direct sum of the anistropic line K1and its orthogonal

hyperplane sl(V)deﬁned by tr(M) = ⟨1, M⟩= 0. Denote by Xthe isotropic cone for the determinant

restricted to the kernel sl(V)of the trace, in formulae:

X={M∈gl(V)| ⟨1, M⟩=0=⟨M, M⟩} ={a∈sl(V)|det(a)=0}

Discriminant. The relation M2−(M+M#)M+(MM#) = 0 yields the Cayley-Hamilton identity

χM(M) = 0 for X2−Tr(M)X+ det(M)∈K[X]the characteristic polynomial of M. Hence a

non-central element M∈gl2(K)\K1generates a commutative subalgebra K[M] = Span(1, M)of

dimension 2, which is isomorphic to the quadratic extension K[X]/(χM)of Kwith Galois involution

given by the restriction of the involution #.

The discriminant of M∈gl(V)is deﬁned as that of its characteristic polynomial, equal to

disc(M) = Tr(M)2−4 det(M).

In particular disc(A) = Tr(A)2−4for A∈SL(V)and disc(a) = −4 det(a)for a∈sl(V).

We call M∈gl(V)semi-simple when disc(M)̸= 0, that is when χMhas simple roots in its

splitting ﬁeld. If these roots belong to Kthen K[M]is isomorphic to the direct product K×K,

otherwise K[M]is a simple K-algebra (no proper ideals). In both cases K[M]is a semi-simple

K-algebra (a product of simple algebras). When disc(M)=0we have χM(X)=(X−λ)2for λ∈K

so the algebra K[M]is not integral (it has zero divisors) as M−λ1is nilpotent.

The discriminant is preserved under the projection disc(M) = disc(pr M), so an element is

semi-simple if and only if its projection in sl(V)lies outside the cone X.

Projectivization. In the projective 3-space P(gl(V)), the point P(K1)and the plane P(sl(V)) are

mutually polar with respect the non-degenerate quadric P(gl(V)) \PGL(V). The point lies oﬀ

the quadric and its polar plane intersects the quadric transversely along the non-degenerate conic

P(X). Geometrically, the conic P(X)consists in the set of tangency points between the quadric

P({det = 0})and the pencil of lines through P(1).

The quadric P({det = 0})in P(GL(V)). The point P(1)lies oﬀ the quadric and its

polar plane P(sl(V)) intersects the quadric in the conic P(X).

Over K, the isomorphism types of the quadric P({det = 0})and of the conic P(X)are given, in

terms of the classes in K×/(K×)2of the diagonal elements appearing in the diagonalisation of the

quadratic form det, by {1,−1,1,−1}and {−1,1,−1}.

Lemma 1.1 (Equivariant ruling of the quadric).The map Ψ: M7→ (ker M, im M)deﬁnes a

bijective algebraic correspondence between the projective quadric P({det = 0})and P(V)×P(V)

sending the projective conic P(X)to the diagonal P(V). The map Ψconjugates the adjoint action

of PGL(V)restricted to P({det = 0})with its tautological diagonal action on P(V)×P(V).

Recall that the action of PGL(V)on P(V)is simply-transitive on triples of distinct lines. For a

symplectic form ωon V, we deﬁne the Maslov index of such a triple x1, x2, x3∈P(V)as the element

ω(⃗x1, ⃗x2)∈K×/(K×)2where the three vectors ⃗xi∈xi⊂Vsum up to zero. The level sets of the

Maslov index do not depend on the choice of ω. One may show [Sim22a, Proposition 1.39] that the

action of PSL(V)on P(V)is simply-transitive on triples of distinct lines with a given Maslov index.

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CONJUGACYCLASSESINPSL2(K)CHRISTOPHER-LLOYDSIMONAbstract.Wefirstdescribe,overafieldKofcharacteristicdifferentfrom2,theorbitsfortheadjointactionsoftheLiegroupsPGL2(K)andPSL2(K)ontheirLiealgebrasl2(K).Whiletheformerarewellknown,thelatterleadtotheresolutionofgeneralisedPell-Fermatequationswhichcharacter...

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CONJUGACY CLASSES IN PSL 2K CHRISTOPHER-LLOYD SIMON Abstract. We first describe over a field Kof characteristic different from 2 the orbits for the.pdf

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CONJUGACY CLASSES IN PSL 2K CHRISTOPHER-LLOYD SIMON Abstract. We first describe over a field Kof characteristic different from 2 the orbits for the

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