The Arithmetic of Quaternion Algebra M-F Vigneras August 2, 2006 2 Contents 1 Quaternion Algebra over a ¯eld 1 1.1 Quaternion Algebra . 1 1.2 The theorem of automorphisms and the neutralizing ¯elds . 5 1.3 Geometry . 8 1.4 Orders and ideals . 13 2 Quaternion algebra over a local ¯eld 23 2.1 Classi¯cation . 23 2.2 Study of M(2;K)........................... 28 2.3 Orders embedded maximally . 32 2.4 Zeta function . 35 3 Quaternion algebra over a global ¯eld 43 3.1 Adeles . 43 3.2 Zeta function, Tamagawa number . 48 3.3 Cassi¯cation . 57 3.4 Norm theorem and strong approximation theorem . 61 3.5 Orders and ideals . 63 4 Applications to Arithmetic Groups 79 4.1 Quaternion groups . 79 4.2 Riemann surfaces . 85 4.3 Examples and Applications . 92 5 Quaternion arithmetic in the case where the Eichler condition is not satis¯ed any more 105 5.1 Units . 106 5.2 Class number . 108 5.3 Examples . 111 i ii CONTENTS Chapter 1 Quaternion Algebra over a ¯eld In this chapter K always denotes a commutative ¯eld of arbitrary characteristics if no particular mention, and Ks is the separable closure of K. 1.1 Quaternion Algebra De¯nition 1.1. A quaternion algebra H with center K is a central algebra over K of dimension 4, such that there exists a separable algebra of dimension 2 over K, and an invertible element θ in K with H = L + Lu, where u 2 H satis¯es u2 = θ; um = mu (1.1) for all m 2 L, and where m 7! m is the nontrivial K-automorphism of L. We denote H sometimes by fL; θg, but H does not determine this pair fL; θg uniquely. For example, it is clear that one could replace θ by θmm, if m is an element of L such that mm 6= 0. the element u is not determined by (1). If m 2 L is an element with mm = 1, we could replace u by mu. the de¯nition can be used in the case of arbitrary character. .we can verify easily that HK is a central simple algebra, i.e. an algebra with center K and without any nontrivial two-sided ideal.conversely, we can prove that every cental simple algebra of dimension 4 over K is a quaternion algebra.The rule of multiplication is deduced from (1). If m 2 L, we have (m1 + m2u)(m3 + m4u) = (m1m3 + m2m4θ) + (m1m4 + m2m3)u: (1.2) De¯nition 1.2. The conjugation is the K-endomorphism of H: h ! h, which is the extension of the nontrivial K-automorphism of L de¯ned by u = ¡u. It is easy to verify it is an involutive anti-automorphism of H. It can be expressed by the following relations: if h; k 2 H and a; b 2 K, we have ah + bk = ah + bk; h = h; hk = kh: De¯nition 1.3. Assume h 2 H. The reduced trace of h is t(h) = h + h. The reduced norm of h is n(h) = hh. 1 2 CHAPTER 1. QUATERNION ALGEBRA OVER A FIELD If h2 = K, its' minimal polynomial on K is (x ¡ h)(x ¡ h) = x2 ¡ t(h)x + n(h) The algebra K(h), generated by h over K, is a quadratic extension over K. The reduced trace and the reduced norm of h are simply the images of h by the operations trace and norm on K(h)=K. The conjugation and the identity are the K-automorphism of K(h). Under the usual de¯nition of the trace and the norm of a K-algebra [cf.Bourbaki[1]], the trace of H=K is T = 2t, The norm of H=K is N = 2n. We denote the group of the units in a ring X by X£. Lemma 1.1.1. The invertible elements in H are that with their reduced norm nontrivial. The reduced norm de¯nes a multiplicative homomorphism from H£ into K£. The reduced trace is K-linear, and the mapping (h; k) 7! t(hk) is a bilinear form non-degenerated on H. Proof. We leave the veri¯cation of the following very simple properties to reader as an exercise: n(hk) = n(h)n(k) n(h) 6= 0 is equivalent to The invertibility of h , and in this case h¡1 = hn(h)¡1, t(ah + bk) = at(h) + bt(k); t(hk) = t(kh), if a; b 2 K; andh; k 2 H. The fact that the mapping (h; k) ! t(hk) being non-degenerate comes from the assumption that L=K has been separable. In fact, if t(hk) = 0 whatever k 2 H , we have t(m1m = 0 for every m 2 L, if h = m1 + m2u,then m1 = 0. Similarly t(m2m = 0 for every m 2 L, whence m2 = 0 and h = 0. we note that, one of the advantage of the reduced trace is that, in the case of characteristic 2 the trace T = 2t is zero, but the reduced trace is non-degenerate. In case characteristic not 2, we recover the classical de¯nition of the quaternion algebra. the couple fL; θg is equivalent with a couple fa; bg formed by two nontrivial elements a; b in K and the relations (1) are determine H as the the K-algebra with basis 1; i; j; ij, where the elements i; j 2 H satisfy i2 = a; j2 = b; ij = ¡ji: (1.3) The transition between(1)and(3) carries out for example by setting L = K(i); θ = b; u = j. Setting k = ij one can write the table of multiplication of i; j; k , which shows these three elements acting symmetrically . The entities in the table are the products hh0: hn h' t j k i a k -j j -k b i k j -i -ab The conjugation,the reduced trace, and the reduced norm have their expression as follows: if h = x + yi + zj + tk, then h = x ¡ yi ¡ zj ¡ tk; t(h) = 2x; and n(h) = x2 ¡ ay2 ¡ bz2 + abt2; the coe±cient of k in h should not be confused with the reduced trace. We notice the another important property: the reduced norm de¯nes a quadratic form on the K-vector space V of the subjacent H. 1.1. QUATERNION ALGEBRA 3 We shall see that, the quaternion algebra H is de¯ned by the relations(1) or (3) to the forms fL; θg or fa; bg when the case is permitted. we also shall consider these notation u; i; j; t(h); h as the standard notation. The fundamental example of a quaternion algebra over K is given by the algebra M(2; k), the matrices of 2 with entities in K. The reduced trace and the reduced norm are the trace and the determinant as the usual sense in M(2;K). It can be identify K with its image in M(2;K) of the K-homomorphismµ ¶ which convert a b the unit of K to the identity matrix. Explicitly if h = 2 M(2;K), µ ¶ c d d ¡b h = , t(h) = a + d, n(h) = ad ¡ bc . We are going to show that ¡c a M(2; k) satis¯es the de¯nition of a quaternion algebra as follows. We choose a matrix with a distinguish value, and set L = K(m). Since m has the same distinguish value as m , it is similar to m.: there exists then an u 2 GL(2; k) such that umu¡1 = m. We verify that t(u) = 0, since t(um) = t(u)m 2 K for each m 2 L,from this we deduce u2 = θ 2 K0. In the following remark we are likely going to explain why is M(2;K) the fundamental example: Over a separably closed ¯eld, M(2;K) is the unique quaternion algebra up to an isomorphism. In fact, every separable algebra of dimension 2 over K can not be a ¯eld sent by the norm on K£ surjectively, and being included in M(2; k)(an inclusion is an injective K-homomorphism). From this we derive that, it is isomorphic to fK + K; 1g ' M(2;K) thanks to the realization of M(2;K) as a quaternion algebra done above. tensor product. Let F be a commutative ¯eld containing K. We verify directly by the de¯nition the tensor product of a quaternion algebra with F over K is a quaternion algebra over F , and that F ­ fL; θg = fF ­ L; θg . We write the obtained quaternion algebra by HF too. The algebra H is included in HF in a natural way. Taking the separable closure Ks of K as F we see that H is included in M(2;Ks). De¯nition 1.4. The ¯elds F=K such that HF to be isomorphic to M(2;F ) is called the neutralized ¯elds of H in M(2; k). The inclusions of H in M(2;F ) is called the F -representations. Examples. (1) The quaternion algebra over K has no any K-representation if it is not isomorphic to M(2;K). (2)We de¯neµ the¶ followingµ matrices:¶ µ ¶ 1 0 0 1 0 1 I = , J = , IJ = : These matrices satisfy the 0 ¡1 1 0 ¡1 0 relations (3) with a = b = 1. We derived from them in the case of characteristic unequal to 2, a quaternion algebra fa; bg is isomorphic to µ p p p ¶ x + ay b(z + at f p p p jx; y; z; t 2 Kg; b(z ¡ at) x ¡ ay p p where a and b are two roots of a and b in Ks .