The Arithmetic of Quaternion Algebra

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114CHAPTER 5. QUATERNION ARITHMETIC IN THE CASE WHERE THE EICHLER CONDITI C Classical isomorphism. We are going to explain How certain isomorphisms of finite groups can be proved by taking advantage of quaternion. Let q = p n , n ≥ 0 be a power of a prime number p, we have then Card(GL(2, Fq)) = (q 2 − 1)(q 2 − q) and Card(SL(2, Fq)) = (q − 1)q(q + 1). In particular, Car(SL(2, F3)) = 24, Card(GL(2, F3)) = 48, Card(SL(2, F4)) + 60, Card(SL(2, F5)) = 120. Proposition 5.3.4. The tetrahedral binary group E24 of order 24 is isomorphic to SL(2, F3). The Alternate group A5 of order 60 is isomorphic to SL(2, F4) and the icosahedral binary group E120 of order 120 is isomorphic to SL(2, F5). Proof. E24 is isomorphic to the unit group of a maximal order(uniquely determined up to isomorphism) O of a quaternion field {−1, −1} over Q of reduced discriminant 2, and the natural homomorphism O → O/3O = M(2, F3) induces an isomorphism of E24 onto SL(2, F3. E120 is isomorphic to the unit group of reduced norm 1 of a maximal order (unique up to isomorphism) O of the quaternion field {−1, −1} over Q( √ 5) which is unramified at the finite places. The natural homomorphism O → O/2O = M(2, F4 induces a homomorphism of E120 onto SL(2, F4) with kernel {∓1}, hence A5 = E120/{∓1} is isomorphic to SL(2, F4). The natural homomorphism O → / √ 5O induces an isomorphism of E120 onto SL(2, F5). D The construction of Leech lattice. Recently Jacques Tits gave a nice construction of Leech lattice in virtue of quaternions which we shall give as an example of the application of the arithmetic theory of quaternion. We shall point out that J.Tits in this manner obtained an elegant geometric description of the twelve among the twenty-four sporadic groups defined in practice (these 12 groups appear as the subgroups os automorphisms of Leech lattice). Definition 5.3. A Z-lattice of dimension n is a subgroup of R n which is isomorphic to Z n . We denote by x · y the usual scalar product in R n . We say that the lattice L is even if all the scalar product x · y are integers for x, y ∈ L, and if all the scalar products x · x are even for x ∈ L. We say that L is Unimodular if it is equal to its dual lattice L ′ = {x ∈ R n |x · L ⊂ Z} with respect to the scalar product. We say that two lattices are equivalent if it exists an isomorphism from one group to another such that it conserves the scalar product invariant. We can show easily that a unimodular even lattice is of dimension divided by 8, and even classify these lattices in the dimension 8, 16, 24 where they have 1, 224 classes respectively. In the higher dimension, the Minkowski-Siegel formula, the mass formula analogue to what we have proved for the quaternion algebra, and that it amounts to as a formula for a Tamagawa number, show that the class number is gigantic: it increases with the number of variables, and it in dimension 32 is already great than 80 millions! Leech discovered that one of these lattices in dimension 24 has a remarkable property which characterizes the following Proposition 5.3.5. The Leech lattice is the only lattice which is even,unimadular, of dimension 24, and not containing any vector x with x · x = 2. The method for constructing the even unimodular lattice. Choose a commutative field K which is totally real and of even degree 2n such
5.3. EXAMPLES 115 that the difference of K is totally principal in the restrict sense and denote by H the unique quaternion field( up to isomorphism )which is totally defined over K and unramified at the finite places. Let R, Rd be the integer ring of K and its difference respectively. Proposition 5.3.6. The maximal R-orders of H equipped with the scalar product: x · y = T K/Q(d −1 t(x¯y) are the unimodular, even lattice of dimension 8n. Proof. Recall that the inverse of the difference is the dual of the integer ring R of K with respect to the bilinear form T K/Q(x¯y) defined by the trace T K/Q of K over Q. Let O be a maximal R-order of H. It is clear that O is isomorphic to a Z-lattice of dimension 8n. It should prove that the bilinear form defined in the proposition is equivalent to the usual scalar product, or in another words, the quadratic form defined by q(x) = 2T K/Q(d −1 n(x)) is positively definite. In fact, x ∈ H × implies That d −1 n(x) is totally positive and the trace is strictly positive. We verify that a) x · y ∈ Z and x · x ∈ 2Z, because the inverse of the difference Rd −1 were sent to each other by the trace in Z. b) O is equal to its dual O ′ = {x ∈ H|T K/Q(d −1 t(xO)) ⊂ Z = {x ∈ H|t(xO) ∈ R} because H is not ramified at the finite places. The construction of Leech lattice. For the reasons prior to the curiosity of a non-specialist in the theory od finite groups which is justified by the presence of the binary icasahedral group in the automorphisms of Leech’s lattice, The construction of Tits for the lattice utilizes the quternion field H which is totally defined and unramified over K = Q( √ 5). We have seen that a maximal R-order O equipped with the scalar product in the precedent proposition is a unimodular even latticeof order 8, and recall that, if τ = (1 + √ 5)/2, Then R = Z[1, τ] and x · y = T K/Q(2x¯y/(5 + √ 5)). We shall observe later that the only integers x which are totally positive in R of trace T K/Q(x) ≤ 4 are (1) 0, 1, 2, τ 2 = (3 + √ 5)/2, τ −2 = (3 − √ 5)/2. Although it is not used here, it bears in mind that we have given an explicit R-basis of an order O in exercise. The unit group of reduced norm 1 of O denoted by O 1 is isomorphic to the icosahedral binary group of order 120, and contains the cubic roots of unit. Let x be one of them, put e = x + τ. We can prove immediately that n(e) = 2, and e 2 = emod(2). We denote by h the standard hermitian form of the H-vector space H 3 : h(x, y) = � xi¯yi, if x = (xi) and y = (yi) belong to H 3 , from it we deduce on that on R 24 there is a scalar product induced by the Q-bilinear form of Q-vector space H 3 of dimension 24: x · y = T K/Q(2h(x, y)/(5 + √ 5))
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The Arithmetic of Quaternion Algebra

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