The Arithmetic of Quaternion Algebra

More documents

Recommendations

Info

60 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD Theorem 3.3.9. (Maximal commutative subfield). For a quadratic extension L/K can be embedded in a quaternion field H if and only if that Lv is a field for v ∈ Ram(H). Two quaternion algebras have always some common maximal commutative subfields (up to isomorphism) and the group Quat(K) is defined. Proof. For a quadratic extension L/K to be contained in a quaternion field H/K it is obviously necessary that for every v of K the algebra Lv to be contained in Hv. therefore, Lv should be a field if Hv is a field. If v ∈ Ram(H), v then can not be decomposed in L. Conversely, if this condition is satisfied, we then choose an element θ of the set K × ∩ � ivn(L × a ) v∈Ram(H) which is non-empty since |Ram(H)| is even by 3.7. Because that θ ∈ n(L × u if u /∈ Ram(H), and θ /∈ n(L × v ) if v ∈ Ram(H), the quaternion algebra {L, θ} is isomorphic to H. If H and H ′ are two quaternion algebras over K, Lemma 3.6 allows to construct an extension L, such that Lv is a field if v ∈ Ram(H) ∪ Ram(H ′ ). The precedent results give then the inclusion in H and H ′ . The group Quat(K) is hence defined, see I, the end of §2. The structure of group Quat(K) is given by the following rules: if H < H ′ are two quaternion algebras over K, we define HH ′ up to isomorphism by It satisfies H ⊗ H ′ � M(2, K) ⊕ HH ′ . (HH ′ )v � HvH ′ v, ε(HH ′ )v = ε(Hv)ε(H ′ v). It follows that the ramification of HH ′ can be deduce from that of H and that of H ′ by Ram(HH ′ ) = {Ram(H) ∪ Ram(H ′ )} − {Ram(H) ∩ Ram(H ′ )}. The classification theorem results then from the property of existence: Property III. For two places v �= w of K there exists a quaternion algebra H/K such that Ram(H) = {v, w}. Proof. If L/K is a separable quadratic extension such that Lv, Lw are the fields (3.6), and θ ∈ iviwn(L × A ∩ k× (see their definition in the proof of 3.7), thus Ram({L, θ}) = {v, w}. Example: The quaternion algebra over Q . The quaternion algebras over Q denoted by {a, b} generated by i, j satisfying i 2 = a, j 2 = b, ij = −ji is ramified at the infinity if and only if a and b are both negative. Its reduced discriminant d is the product of a odd number of prime factors if a, b < 0 and of an even number otherwise. For example, {−1, −1}, d = 2; {−1, −3}, d = 3; {−2, −5}, d = 5; {−1, −7}, d = 7; {−1, −11}, d = 11; {−2, −13}, d = 13; {−3, −119}, d = 17; {−3, −10}, d = 30.
3.4. NORM THEOREM AND STRONG APPROXIMATION THEOREM 61 A rapid method for obtaining the examples is to use the likeness in order to avoid the study of {a, b}2, with remarking that if p is a prime number with p ≡ −1(mod4), then {−1, −p} has the discriminant p, finally for p ≡ 5mod(8), then {−2, −p} has discriminant p. A little attempt allows to find easily a quaternion algebra with a given discriminant, that is to say, two integer numbers , of which the local Hilbert symbols are given in advance. For example, {−1, 3}, d = 6; {3, 5}, d = 15; {−1, 7}, d = 14. If p is a prime, p ≡ −1(mod4), then {2, p} has the discriminant 2p; if p ≡ 5(mod8) , then {−2, p} has the discriminant 2p. 3.4 Norm theorem and strong approximation theorem . The norm theorem was proved in 1936-1937. Hasse and Schilling [1], Schilling [1],Maass [1], Eichler [3], [4] made contribution to its proof. Its application to the euclidean order, and to the functional equation of L function was made by Eichler [5]. The strong approximation theorem for for the unit group with reduced norm 1 of the central simple algebra over number fields is due to Kneser [1], [2], [3]. A recent article have proved this theorem in the case of function field(Prasad [1]). Theorem 3.4.1. (Norm theorem) Let KH be the set of the elements of K which are positive for the infinite real place of K and ramified in H. Then KH = n(H). Proof. The condition is natural since n(H) = R+. Conversely let x ∈ K × H ; we construct a separable quadratic extension L/K such that : – x ∈ n(L), – for every place v ∈ Ram(H), Lv/Kv is a quadratic extension. Therefore, L is isomorphic to a commutative subgroup of H by 3.8, and x ∈ n(H). This is an exercise of using the approximation theorem and the lemma on polynomials. Let S be a finite set of places of K. For the finite v we see that Hvcontains an element of reduced norm πv. Since H is dense in Hv, we see that H contains an element of reduced norm a uniform parameter of Kv, and by multiplying x with n(h) for a suitable element h ∈ H, we can assume that for a finite set S of places of K: x is a unit for p ∈ S ∩ P . We choose for every v ∈ S an extension Lv such that : – Lv − C if v is real, – Lv is the unramified quadratic extension of Kv if v ∈ P ∩ S. for every v ∈ S,it exists yv ∈ Lv of norm x. The minimal polynomial of yv over Kv can be written as pv(X) = X 2 − avX + x. We choose a ∈ K very near to av if v ∈ S (and the same if one needs an integer for every place of K excluding possibly one place v /∈ S), so that the polynomial p(X) = X 2 + aX + x is irreducible and defines an extension K � K(y) � K[X]/(p(X)) ⊂ Xs, such that K(y)v = Lv, if v ∈ S.
Page 1:
The Arithmetic of Quaternion Algebr
Page 4 and 5:
ii CONTENTS
Page 6 and 7:
2 CHAPTER 1. QUATERNION ALGEBRA OVE
Page 8 and 9:
Page 10 and 11:
Page 12 and 13:
Page 14 and 15: 10 CHAPTER 1. QUATERNION ALGEBRA OV
Page 84 and 85: 80 CHAPTER 4. APPLICATIONS TO ARITH
Page 104 and 105: 100 CHAPTER 4. APPLICATIONS TO ARIT
Page 110 and 111: 106CHAPTER 5. QUATERNION ARITHMETIC
Page 112 and 113: 108CHAPTER 5. QUATERNION ARITHMETIC
Page 114 and 115:
110CHAPTER 5. QUATERNION ARITHMETIC
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122:
show all

The Arithmetic of Quaternion Algebra

Create successful ePaper yourself

Delete template?

Save as template?