The Arithmetic of Quaternion Algebra

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96 CHAPTER 4. APPLICATIONS TO ARITHMETIC GROUPS We denote i E = 2 + √ 3 , E′ i = 2 − √ + 2i , F + i, H = −1 √ , 3 5 � 0 u = √ 15 √ � − 15 , v = 0 1 � √ − √ 3 2 15 √ � − √ 15 . 3 The transformation u fixes F and exchange E and E ′ . The transformation v fixed H and exchange B and B ′ . Lemma 4.3.3. The hyperbolic quadrilateral BEE ′ B ′ is a fundamental domain of G. The transformations h, u/ √ 15, v/ √ 3 generates G. Its area is 8π/6. Here are tree figures: 1)fundamental domain of Γ . 2) fundamental domain of G 3)Unit disc. these pictures are designed by C.Leger. D Minimal geodesic curves. Definition 4.9. Let g be a hyperbolic matrix of Γ of norm N. Let P is a point of the geodesic of H joining the double points of ¯g. The image of ¯ Γ\H of the orient segment of the geodesic joining P to g(P ) is an orient closed curve, being independent of P , of length log N, called the minimal geodesic curve of ¯g. Definition 4.10. an element ¯g ∈ ¯ Γis primitive if it not the power of the other element of ¯ Γ with exponent strictly great than 1. Its conjugate class in ¯ Γ is said to be primitive too. If ¯g is primitive, hyperbolic, then it generates the cyclic group of elements of ¯Γ which has the same fixed points. Its minimal geodesic curve is passes through a single time. If g ′ = g m , m ∈ Z, m �= 0, the norm of g ′ is N m , and the minimal geodesic curve of g ′ is the curve obtain by passing through that of g m times, in the same direction if m > 0, and in the contrary direction otherwise. The minimal geodesic curve of the hyperbolic ¯g, ¯g ′ ∈ ¯ Γ are the same if and only if ¯g and ¯g ′ are conjugate in ¯ Γ. We have thus the following result. Lemma 4.3.4. The number of the minimal geodesic curves of length logN is equal to the number of conjugate classes of the elements of Γ with the characteristic polynomial X 2 − (N 1/2 + N −1/2 )X + 1. Denoted it by e(N). Notice that if g, g −1 are conjugate in Γ, ther exists then x ∈ Γ satisfying xgx −1 = g −1 ⇒ x 2 ∈ R(x) ∩ R(g) = R, hence x 2 = −1. If Γ does not contain such an element, e(N) is even. For the quaternion group, e(N) can be computed explicitly(III.5). EXAMPLE: Γ is the unit group of reduced norm 1 of a maximal order of a quaternion field over Q of its discriminant 26. We have the following results: 1. ) Γ is embedded in SL(2, R), since 26 = 2 · 13 is the product of two prime factors. 2. ) Γ does not contain any parabolic element according to 1.1.
4.3. EXAMPLES AND APPLICATIONS 97 3. ) Γ does not contain any elliptic element, since ( −1 13 to III,3.5. 4. ) The genus g of ¯ Γ\H is equal to 2, since by A, g = 1 + 1 (2 − 1)(13 − 1) = 2. 12 −3 ) = ( 13 ) = 1 according 5. ) The conjugate classes of hyperbolic ¯ Γ, have its norm ε 2m , m ≥ 1, where ε runs through the fundamental units of norm 1 of real quaternion field, In them not 2 nor 3 can be decomposed. 6. ) The number of the primitive conjugate classes of ¯ Γ with reduced norm ε2m is equal to (2)h(B) � (1 − ( B p )) p=2,13 where (2) = 1 or 2 according to Q(ε) containing a unit of −1 or not, where B runs the orders of Q(ε) of which the unit group of norm 1 is generated by ε 2m , and the number of classes of B is related to that of L = Q(ε) by the formula h(B) = hLf(B)[R × L : B× ] −1 prod p|f(B)(1 − ( L p )p−1 ) with RL = the integer ring of L, of class number hL, f(B) = conductor of B. EXAMPLE. ¯ Γ is the modular group P SL(2, Z). We have e2 = 1, e3 = 1, e∞ = 1 and the genus of the surface ¯ Γ\H ∗ 2 is 0, since g = 1 + 1/12 − e2/4 − e3/3 − e∞/2 = 0. The number of the hyperbolic primitive conjugate classes of a given norm is (2) � h(B) with the same notations as that in the precedent example. E The examples of Riemann surfaces which are isospectral but not isometric. There are some numerical invariants as follows. — vol( ¯ Γ\H) — eq = the number of elliptic points of order q in ¯ Γ\H — e∞ =the number of points of ¯ Γ\H — e(N) = the the number of the minimal geodesic of length log N of ¯ Γ\H which are not depend on the isometric class of the surface ¯ Γ\H. Using the properties of Selberg ( Cartier-Hejhal-Selberg) zeta function we can prove: — To give the spectral for the hyperbolic laplacian in L 2 ( ¯ Γ\H) is equivalent to give the invariants. – Two groups of the same invariants but for a finite number between them, have the same invariants. We may ask if two surfaces ¯ Γ\H and ¯ Γ ′ \H of the same numerical invariants are isometric. The answer is NO. We can restrict ourselves to the cocompact groups Γ without elliptic points. Our examples shall utilize the quaternion
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The Arithmetic of Quaternion Algebra

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