Arithmetic Fuchsian Groups Arising from Quaternion Algebras

Home , Algebra homomorphism, Quaternion algebra

HEIDELBERG UNIVERSITY

SEMINAR:ARITHMETICSOFTHE HYPERBOLIC PLANE LECTURE 8

MENELAOS ZIKIDIS

Arithmetic Fuchsian Groups arising from Quaternion Algebras

Contents

1 Motivation 2

2 Representation Theoretical construction of Fuchsian Groups 2

3 Quaternion Algebras 2 3.1 Quadratic forms and Orders ...... 6 3.2 Ramiﬁed Quaternions ...... 8

4 Groups from Orders 9 1 Motivation

Fuchsian groups are discrete subgroups of PSL(2, R), i.e. subgroups of the isometry 2 group of the hyperbolic plane H . One of the important features of those groups lies by the concept of uniformization of surfaces. Namely one can endow a compact Riemann surface of genus g ≥ 2 X with the structure of a quotient of the form:

2 X = H /Γ where Γ is Fuchsian group. This was the spark of an even greater and up to date still incomplete program, namely that of Thurston on geometrization of higher dimensional surfaces. We will construct here a class of Fuchsian groups, the arithmetic Fuchsian groups, which have the property of coding all of their structure in terms of number- theoretic data. We use the notion of orders of quaternion algebras from which arithmetic Fuchsian groups will arise.

2 Representation Theoretical construction of Fuchsian Groups

Before we go into the number-theoretic construction of Let us consider a ﬁnite dimensional linear representation of PSL(2, R):

T : PSL(2, R) −→ GL(n, R)

Then the subgroup T (PSL(2, R)) ∩ GL(n, Z) lifts to a discrete subgroup Γ < P SL(2, R) given by: −1 Γ := T (T (PSL(2, R)) ∩ GL(n, Z)) Definition: All subgroups of PSL(2, R) obtained in that manner and all their subgroups of finite index are called Arithmetic Fuchsian Groups. By the classification of all classical groups, due to A.Weil, we know that all arithmetic subgroups of SL(2, R), up to a relation called commensurability, are given by Fuchsian Groups derived from quaternion algebras over totally real number fields. In this lecture we will construct the Arithmetic Fuchsian groups in terms of quaternion algebras.

3 Quaternion Algebras

Deﬁnition: Let F be a ﬁeld of char(F ) 6= 2.A Quaternion Algebra A over F is a 4- dimensional F -Algebra with basis vectors {1, i, j, k}, where the algebra structure is given by 1 being the multiplicative identity, the relations:

i2 = a · 1, j2 = b · 1, k = i · j = −j · i, ∗ for some a, b ∈ F and by linear extension of this multiplication over F , so that A is an associative F -Algebra. For an object with that structure, we write: a, b A = . F

For a ﬁeld extension K/F , we can extend the scalars of the algebra in the obvious manner: a, b a, b A = ⊗ K =∼ . F F K Examples: 1. The ﬁrst example of a quaternion algebra over F , is the matrix algebra: 1, 1 M(2,F ) =∼ F

with generators being given by:

1 0 0 1 i = ; j = 0 −1 1 0

2. The ﬁrst non-trivial one are the Hamiltonian Quaternions: −1, −1 H = R

In general, a quaternion algebra A is isomorphic either a division algebra, i.e. A∗ = A − {0}, or it is isomorphic to the matrix algebra M(2,F ). In the following discussion we will try to ﬁnd out when each case shows up. To do so, we construct a representation of the quaternion algebra that will allow us to correspond to this abstract construction matrix algebras: √ Let A = a,b be a quaternion algebra and F ( a)/F a quadratic extension. Then F √ φ : A → M(2,F ( a)) is a representation given by: √ √ 1 0 a 0 0 1 0 a 1 7→ ; i 7→ √ ; j 7→ ; k 7→ √ , 0 1 0 − a b 0 −b a 0 √ which are linearly independent matrices which build a basis of a subalgebra of M(2,F ( a)) So for an element x ∈ A, x = x0 + ix1 + jx2 + kx3 we have: √ √ x0 + x1 a x2 + x3 a gx := φ(x) = √ √ b(x2 − x3 a) x0 − x1 a In particular one can perform a straight-forward check that φ deﬁnes a faithful algebra morphism, i.e.:

• gx + gy = gx+y

3 • gxy = gx · gy

• [x = y ⇐⇒ gx = gy] Remark: By the representation we just deﬁned, we see that if a is a square a ∈ (F ∗)2 then obviously the morphism becomes an isomorphism onto M(2,F ). Proposition: There exists an isomorphy:

a, b ax2, by2 A = =∼ ; x, y ∈ F. F F

Moreover quaternion algebras are simple central F -Algebras, i.e.:

1. Rad(A) = 0

2. Z((A)) = {x ∈ A|[x, y] = 1 ∀y ∈ A} = F · 1

Proof:

a,b 0 ax2,by2 0 0 0 • Let A = F and A = F with bases {1, i, j, k} and {1, i , j , k } respec- tively. Then the F -algebra homomorphism:

φ : A −→ A0 1 7−→ 1 i0 7−→ xi j0 7−→ yj k0 7−→ xyk

is indeed a F -algebra isomorphism: (xi)2 = ax2;(yj)2 = by2;(xi)(yj) = (xy)ij = −(xy)ji = −(yj)(xi) ¯ a,b a,b 2 • As F is algebraically closed, so ∀x ∈ F¯ ∃y ∈ F¯ such that x = y . So from the statement we just showed above follows:

a, b 1, 1 =∼ =∼ M(2, F¯). F¯ F¯

Further we know that Z(M(2, F¯)) = F¯ · 1. But we also know that one obtains a,b a,b ¯ ∼ a,b a,b 1 F¯ just by a scalar extension F ⊗ F = F¯ and therefore Z( F ) = F · . ¯ ¯ • Let 0 6= a C A, then we have for the extension of scalars 0 6= a⊗F F C M(2, F ).Where F¯ be the algebraic closure of F . But the matrix algebra over a ﬁeld is obviously simple as: ¯ ¯ 0 6= a = M(I); for I C F =⇒ I = F. So a is a 4-dimensional F -vector space, and already by the dimension follows a = A.

4 a,b Deﬁnition: Let x ∈ A = F

Standard involution : i : A −→ A

x 7−→ x¯ = x0 − x1i − x2j − x3k Reduced trace : tr : A −→ F

x 7−→ x +x ¯ = 2x0

and tr(x) = T r(gx) Reduced Norm : n : A −→ F 2 2 2 2 x 7−→ xx¯ = x0 − x1a − x2b − x3ab and n(x) = det(gx)

Theorem: A quaternion algebra A is a division algebra, if and only if the following equivalence holds: n(x) = 0 ⇐⇒ x = 0 Proof:

•⇐ : Assume [n(x) 6= 0 ⇐⇒ x 6= 0] ( ⇐⇒ [¬n(x) = 0 ⇐⇒ ¬x = 0]) ) x 6= 0 =⇒ n(x) 6= 0 x¯ =⇒ · x = 1 n(x) = x · x¯ n(x)

x¯ =⇒ = x−1 =⇒ ∀x ∈ A, x 6= 0, ∃x−1 ∈ A s.t. x · x−1 = 1 n(x) =⇒ A is a division algebra.

•⇒ : Let A be a division algebra and take 0 6= x ∈ A. So we have:

1 = n(1) = n(x · x−1) = n(x)n(x−1) =⇒ n(x) 6= 0

and n(x) 6= 0 =⇒ x 6= 0 is obvious.

a,b Theorem: If A = F M(2,F ), then A is a division algebra. Proof:

2 • First take a∈ / F ∗ , i.e. not a square.

• Let L = F (i) so we have A = L + Lj

5 Assumption: A is not a division algebra. Then by the last theorem we have that:

∃x ∈ A, x = x0 + x1i + x2j + x3k 6= 0 s.t. n(x) = 0

=⇒ nL(x0 + ix1) − b · nL(x2 − ix3) = 0

If x2 − ix3 = 0, then nL(x0 + ix1) = 0

=⇒ x0 + ix1 = 0 =⇒ x = 0; L integral dom. which leads to a contradiction. So x2 + ix3 6= 0 and we can write:

nL(x0 + ix1) 2 2 b = = nL(q0 − iq1) = q0 − aq1, nL(x2 + ix3) with q0, q1 ∈ F . From here we can construct a map ψ : A → M(2,F ) by sending the basis to: 1 0 0 1 q −q aq −q 1 7→ ; i 7→ ; j 7→ 0 1 ; k 7→ 1 0 0 1 a 0 aq1 −q0 aq0 aq1

So under this map one has: ψ(i)2 = a; ψ(j)2 = b and all matrices are linearly indepen- ∼ dent, which gives an isomorphism ψ : A −→ M(2,F ).

3.1 Quadratic forms and Orders a,b ∼ Theorem: As always let A = F .Then A is not a division algebra, i.e. A = M(2,F ) if and only if ∃(x, y) ∈ F ×F that solves the quadratic form/ conic equation: ax2 +by2 = 1. Proof: =⇒

• First we show that y 6= 0 exists in the subspace of A spanned by {i, j, k} = A0 such that the induced norm n0(y) = 0: Assume that there is no such y. [@] Since A is not a division algebra we know that an x = a0 + ia1 + ja2 + ka3 exists with n(x) = 0. If a0 = 0 then we are done, so lets assume that a0 6= 0, that means that one of a1, a2, a3 is non-zero. Without loss of generality take a1 6= 0. Moreover

2 2 2 2 n(x) = 0 =⇒ a0 − ba2 = a(a1 − ba3)

Under this relation we see that n0(y) = 0 for

2 2 y := b(a0a3 + a1a2)i + a(a1 − ba3)j + (a0a1 + ba2a3)ij

Because of [@] we have y = 0:

2 2 y = 0 =⇒ −a1 + aba3 = 0 =⇒ n(a1i + a3ij) = 0 =⇒ a1 = 0 [@]

which leads to a contradiction.

6 2 2 2 • Now we know that a1, a2, a3 exist with −aa1 − ba2 + aba3 = 0, from which at least two are non-zero. If a 6= 0 then the pair (x, y) = ( a2 , a1 ) satisﬁes the relation 3 aa3 ba3 ax2 + by2 = 1 If a = 0 then the pair (x, y) = ( 1+a , a2(1−a) ) satisﬁes the relation 3 2a 2aa1 ax2 + by2 = 1

That completes the first implication. ⇐: Now we want to find non-zero element in A with norm 0, i.e. non- invertible. 2 2 • We take first the existing solution√ of the quadratic equation : ax0 + by0 = 1, and show that a ∈ N √ (F ( b)). From x = 0 the result would follow trivially, as √ F ( b)/F 0 F ( b) = F . So assume x 6= 0, then we can write: 2 √ √ 1 y0 √ 1 y0 √ a = 2 + b 2 = NF ( b)/F + b =⇒ a ∈ NF ( b)/F (F ( b)) x0 x0 x0 x0 √ • Now if b is a square in F , i.e. ∃c ∈ F : b = c then c2 = b = j2 =⇒ (c + j)(c − j) = 0 √ and therefore A has zero divisors, so it can’t be a division√ algebra. So assume b ∈ √ F . From our last point we now that a ∈ NF ( b)/F (F ( b)), hence ∃x1, y1 ∈ F , of 2 2 which at least one is non-zero, with a = x1−by1 and follows that n(x1+i+y1j) = 0, so that A has a non-zero, non-invertible element.

The role of quaternion algebras in number theory, in particular in class field theory lies by the fact that they correspond to the constitutes of elements of order 2 in the Brauer group, a construction that aims to the classification of simple central algebras up to Morita-equivalence. In that context they are used for the construction of generalized reciprocity laws, which are the "fine structure" behind modular arithmetic. This is just a motivational appetizer, that may help to strengthen the nexus between algebraic number theory and differential geometry under the geometric constructions that appear from this abstract non-sense, a scent of which we describe here. But enough of this for now, let’s bring some "order ": Definition: Let F be a number-field, OF be the ring of integers of F and A a quaternion algebra over F .

• Let V be a F -vector space and L a ﬁnitely generated OF -module contained in V , then we call L an OF -Lattice in V . Moreover, we call L a complete OF -Lattice in ∼ V if L ⊗OF F = V

• An Ideal I of A is a complete OF -Lattice in A. • An Order O in A is an ideal, which is endowed with a structure of a ring with 1. • An order of A is maximal if it is maximal with respect to inclusion. • The group of units of reduced norm 1 is O1 := {x ∈ O|n(x) = 1}.

7 3.2 Ramiﬁed Quaternions

a,b Let A = F , F quadratic number field with [K : Q] = n and K/F a finite extension. For each Galois field embedding σ : F → K we denote:

σ(a), σ(b) Aσ = σ(F ) σ(a), σ(b) Aσ ⊗ K = F K

Deﬁnition

• An element α ∈ F is called totally positive if σ(α) ≥ 0 for every real Galois ﬁeld embedding σ : F → R • F is totally real, if every Galois embedding

σi : F → C; i ∈ {1, ..., n} =⇒ σi(F ) ⊂ R,

i.e. all Galois ﬁeld embeddings have real image. We set σ1 to be the one corre- sponding to the identity automorphism.

a,b • For σ : F → R of the number ﬁeld F , then we call F ramiﬁed at σ if:

σ(a), σ(b) ∃ρ : −→H∼ R and unramiﬁed at σ if: σ(a), σ(b) ∼ ∃ρ˜ : −→ M(2, R) R

Proposition: Let A a quaternion algebra over a totally real number ﬁeld F such that the only unramiﬁed "place" is at σ1. If F 6= Q then A is a division algebra. Proof: Suppose A is not a division algebra. Then the last theorem implies, that ∼ A = M(2, F). Therefore we have:

σi σi ∀i ≥ 2; A = M(2, σi(F )) =⇒ A ⊗ R = M(2, R) which contradicts the assumption for ramiﬁcation. In what follows we are interested in the situation of the last proposition to construct the arithmetic groups.

8 4 Groups from Orders

∼ For an order O in A and the isomorphism ρ1 : σ(a),σ(b) −→ M(2, ), it is ρ1(O1) R R 1 1 subgroup of SL(2,R) and Γ = ρ (O )/{±1} a subgroup of PSL(2, R). ∼ Theorem: For an order O in A and the isomorphism ρ1 : σ(a),σ(b) −→ M(2, ), it is R R 1 1 1 1 ρ (O ) subgroup of SL(2,R) and Γ = ρ (O )/{±1} a subgroup of PSL(2, R). Then Γ, is a Fuchsian group. Proof: Although the statement is true in general we will prove it here for simplicity without going into number theoretic details for: a, b A = and O = {x ∈ A|x0, x1, x2, x3 ∈ Z}. Q What is to be shown, is that Γ is a discrete subgroup of SL(2, R) under the constructed embedding φ of the algebra A into SL(2, R), from paragraph 1. We do this by showing 1 1 that there is a neighborhood U of the identity in SL(2, R) disjoint to ρ (O ). For this purpose choose: g11 g12 1 1 1 1 U := {g = ∈ SL(2, R)| |g11 − 1| ≤ ; |g12| ≤ ; |g21| ≤ ; |g22 − 1| ≤ } ; g21 g22 2 2 2 2

1 1 take gx ∈ ρ (O ) ∩ U and show that gx = id: √ √ √ √ g11 = x0+ ax1; g12 = x2+ ax3; g21 = b(x2− ax3); g22 = x0− ax1; x0, x1, x2, x3 ∈ Z The condition that g lives in U implies:

|g11 + g22 − 2| < 1 =⇒ |2x0 − 2| < 1 =⇒ x0 = 1.

For b > 1: √ 1 1 |x − ax | < < =⇒ |2x | < 1 =⇒ x = 0 2 3 2b 2 2 2 Further it is: √ 1 √ 1 |x a| < ; |x a| < =⇒ x = x = 0 1 2 3 2 1 3 1 1 That gives us: gx = id, and therefore the quotient Γ = ρ (O )/{±1} is a Fuchsian group, as it is a discrete subgroup of PSL(2, R). Definition: A subgroup Γ˜ < Γ is said to be a Fuchsian group derived from a quaternion algebra. Definition: Two subgroups Γ1, Γ2 < Γ of any group Γ are said to be commensurable if: [Γj :Γ1 ∩ Γ2] < ∞; j = 1, 2. Definition: A group that is commensurable to a group of the form of Γ constructed as above is said to be an Arithmetic Fuchsian group. I general, for a totally real number field, we obtain an arithmetic group in that manner if and only if there is exactly one real Galois embedding such that the quaternion algebra is not a division algebra.