The Arithmetic of Quaternion Algebra

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88 CHAPTER 4. APPLICATIONS TO ARITHMETIC GROUPS We give am exact sequence of continuous mapping: 1 −−−−→ SO(2, R) i −−−−→ SL(2, R) ϕ −−−−→ H −−−−→ 1 where i is the natural inclusion, and ϕ(g) = ¯g(i), a Haar measure on SL(2, R) with the compatibility of the hyperbolic measure of H and a Haar measure dθ of SL(2, R). Denote it by y −2 dxdydθ. It is false in general that for a discrete subgroup of finite covolume Γ ⊂ SL(2, R) we could have for this measure: (1) vol( ¯ Γ\H)vol(SO(2, R) = vol(Γ\SL(3, R)), but it is true if Γ acts without fixed point in H. Corollary 4.2.6. The Tamagawa measure on SL(2, R) equals y −2 dxdydθ, where dθ is normalized by vol(SO(2, R) = π. Proof. In view of 1.6, the group SL(2, Z) possesses a subgroup Γ of finite index which does not contain the root of unit different from 1. A group with this property acts without fixed points and faithfully o H. according to 1.3, we have vol(Γ\SL(2, R)) = vol(SL(2, Z)\SL(2, R))[SL(2, Z) : Γ]. On the other hand, if F is a fundamental domain of P SL(2, Z) in H, then ∪γF , with γ ∈ ¯ Γ\P SL(2, Z) is a fundamental domain of ¯ Γ in H, thus vol( ¯ Γ\H) = vol(P SL(2, Z)\H)[P SL(2, Z) : ¯ Γ]. Since [SL(2, Z) : Γ] = 2[P SL(2, Z) : ¯ Γ], it follows from (1) the relation (2) vol(P SL(2, Z)\H)vol(SO(2, R)) = 2vol(SL(2, Z)\SL(2, R). We saw in the precedent example and 1) in §1 that vol(P SL(2, Z)\H) = π/3 for the hyperbolic measure, vol(SL(2, Z)\SL(2, R)) = π 2 /6 for Tamagawa measure. Corollary 2.6 then follows. In the proof we obtain also the following property. Corollary 4.2.7. Let Γ be an arithmetic group. The volume of ¯ Γ\H for the hyperbolic measure is equal to � 1 1, if −1 /∈ Γ vol(Γ\SL(2, R)) calculated for Tamagawa measure. π 2, if −1 ∈ Γ It allows to calculate by 1.7 the hyperbolic volume of ¯ � � Γ\H. We consider a a b nontrivial real homography associated with g = ∈ SL(2, R). It has two c d
4.2. RIEMANN SURFACES 89 double points in C ∩ ∞: (1) distinct, real if (a + d) 2 > 4, (2) distinct, complex conjugation, if (a + d) 2 < 4, (3) mingling if (a + d) 2 = 4. We obtain the above statement easily in virtue of the equalities: z = (az + b)(cz + d) −1 is equivalent to cz 2 + (d − a)z − b = 0. The discriminant of the quadratic equation is (d + a) 2 − 4. Definition 4.6. In the case (1), the homography is said to be hyperbolic. Its norm or its multiplicator is equal to N = λ 2 , where λ is the proper value of g which is great than 1 strictly. In case (2), it said to be elliptic. Ita angle or its multiplicatoris equal to N = λ 2 , where λ = e iθ is the proper value of g such that 0 ≤ θ ≤ π. In case (3), it is said to be parabolic. These definitions depend only on the conjugate class of g in GL(2, R), and hence can be extended to the conjugate classes. Proposition 4.2.8. Let ¯g be a homography of norm N. We have log N = d(z0, ¯g(z0) = inf d(z, ¯g(z)) z∈H for every element z0 belonging to the geodesic joining the double points of ¯g. Proof. Since GL(2, R) acts by isometry, it follows that ¯g(z) = Nz, and then use 2.2. compactificaton of H. We shall compactify H by embeding it in the space H ∪ R ∪ ∞ which is equipped a topology as follows: the system of basic neighborhoods at the infinity is the open neighborhoods Vy, y > 0 defined as below : two pictures here!!! for ∞ : Vy = {z ∈ H|Imz > 0}, for A ∈ R : Vy = {z ∈ H|d(B − z) < y}. Fundamental domains. We recall a certain number of classic results about the construction of fundamental domains. References: Poincare [1], Siegal [3]. Let Γ be a discrete subgroup of SL(2, R) of finite covolume, and ¯ Γ be the group of the homographies associate with Γ. 1. For every element z0∈H which is not the double point of any elliptic matrix of Γ ( the existence of such point is easy to prove), the set F = {z ∈ H|d(z, z0 ≤ d(¯g(z), z0) ∀¯g ∈ ¯ Γ} is a hyperbolic polygon and a fundamental set of Γ in H2. 2. The edges of F are even in number and congruent pairwise modulo ¯ Γ. We can also rearrange them in pairs (Ci, ¯gi(Ci)), 1 ≤ i ≤ n.
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The Arithmetic of Quaternion Algebra

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