The Arithmetic of Quaternion Algebra

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20 CHAPTER 1. QUATERNION ALGEBRA OVER A FIELD . In particular if Card( ˜ G\C(h, B) is finite and is zero for almost every order B ⊂ L, then we have Card( ˜ G\C(h)) = � Card( ˜ G\C(h, B)). B The relation is useful for every explicit computation of the conjugate classes: the trace of Heck operators(Shimizu [2]), class number of ideal or of the type number of order(ch.V), the number of conjugate class of a quaternion group of reduced norm 1 and of a given reduced trace (ch.IV). The group of units of an order. The unit of an order is the invertible element which and its inverse both are contained in the order. They constitutes naturally a group denoted by O × . The units of reduced norm 1 form a group too denoted by O 1 . Lemma 1.4.12. An element of O is a unit if and only if its reduced norm is a unit of R. Proof. If x, x −1 belong to O, then n(x), n(x −1 = n(x) −1 are in R. Inversely, if x ∈ O, and n(x) −1 ∈ R, we then have x −1 = n(x) −1 x ∈ O, because x ∈ O Exercises 1. Prove, if the right order of an ideal is maximal, then its left order is maximal too. From this deduce that a maximal order is such an order which ties to one of the other orders. 2. Prove, if R is principal, the order M(2, R) is principal. Deduce from it that the maximal orders of M(2, K) are all conjugate each other, i.e. of the same type. 3. Let H be a quaternion algebra {−1, −1} over Q (cf. §1). Prove there exists in an integral ideal an element of minimal reduced norm. Prove, if h ∈ H, there exists x ∈ Z(1, i, j, ij) such that n(x − h) ≤ 1, and even in some case x(n − h) < 1. Deduce from it that Z(1, i, j, (1 + i + j + (1 + i + j + ij)/2) is principal. 4. Theorem of four squares(Lagrange). Every integer is a sum of 4 squares. Using 4.3 prove it. You can firstly verify the set of the sums of 4 squares in Z is stable under multiplication, then every prime number is the sum of 4 squares. 5. Abelian variety(Shimura [1]). Let H be a quaternion algebra over Q pos- � � a b sessing a R-representation f. If z ∈ C, and x = ∈ M(2, R), we � � c d z use e(z) to denote the column vector and x(z) = (az + b)(cz + d) 1 −1 . Let O be an order of H over Z. For every z ∈ C with its imaginary part being positive strictly , set D(z) = f(O)z = {f(x)z|x ∈ O}.
1.4. ORDERS AND IDEALS 21 Prove D(z) is a lattice of C 2 , i.e. a discrete sub group of C 2 of rank 4. If a ∈ H is an element with its square a 2 being a strictly negative rational number, we set ˆx = a −1 xa for x ∈ H. Prove that x ↦→ ˆx is an involution of H, and that t(xˆx)is strictly positive if x �= 0. Show it is possible to define a bilinear R-form < x, y > over C such that< f(x)z, f(y)z >= t(ax¯y) for all x, y ∈ H. Verify there exists an integer c ∈ N such that E(x, y) = c < x, y > is a riemannian form over the complex torus C 2 /D(z), i.e. – E(x, y) is an integer for every (x, y) ∈ D(z) × D(z), – E(x, y) = −E(y, x), – the R-bilinear form E(x, √ −1y) is bilinear and positively definite in (x, y). It is known that the existence of a riemannian form on a complex torus is equivalent to the existence of a structure of an abelian variety. 6. Normalization. Let H/K be a quaternion algebra, and h ∈ H. Demonstrate (1) Oh is an ideal if and only if h is invertible. (2) Oh is a two-sided ideal if and only if (1) is valid and Oh = hO. (3) the normalization of O is the group consisting of the elements h ∈ H such that Oh is a two-sided ideal. 7. The equation of polynomials over quaternion(Beck [1]). Let H/K be a quaternion field, and H[x] be the set of polynomials P (x) = � aix i , where the coefficients aibelong to H. H[x] is equipped with a ring structure such that the indeterminate x commute with the coefficients. (a) Prove that every polynomial P (x) can be factored in a unique way as the product of a monic polynomial with coefficients in K, a constant in H × , and a monic polynomial of H[x] which can not be divided by a non-unit polynomial in K[x]. (b) Prove that the equation P (x) = 0 has a solution of element x = a belonging to K if and only if x − a divides P (x). (c) Show the coefficients of the polynomial n(P ) = � aiājx i+j is in K. We call it thereduced norm of P . We want to find the solutions of the equation P (x) = 0 in H, then we study the associate solution in H of the equation n(P )(x) = 0. We can suppose P to be monic, and has no any solution in K, according what said above. If h is a Quaternion, we denote its minimal polynomial by Ph. (d) Prove that if Ph divides P , then every conjugation of h in H is the root of P . Particularly, the equation P (x) = 0 has an infinitely many solutions in H. (e) Prove that if Ph does not divide P , then the equation P (x) = 0 has at most a conjugation of h as a solution. This happens if and only if Ph divides n(P ).
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The Arithmetic of Quaternion Algebra

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