Chapter 6 Arithmetic of quaternion algebras Here we’ll investigate some arithmetic properties and consequences of quaternion algebras. Namely, we’ll try to get a more concrete understanding of orders and ideals in quaternion al- gebras, including a class number formula and questions of factorization. We’ll also give some applications to ternary and quaternary quadratic forms, with more impressive applications to follow in Chapter 7. The primary references are Vigneras [Vig80] and some papers of Pizer [Piz76], [Piz77], [Piz80]. Note that some of what we do will be in greater generality than these references (e.g., Pizer’s paper are just over Q and Vigneras restricts to Eichler orders for some things), and some things we cover are not treated by Vigneras or Pizer. Our focus is on explaining (some of) what is known and how to do concrete computations rather than giving complete proofs. (In other words, I don’t have time to prove everything. Proofs of some results, e.g. mass and class number formulas, involve a fair amount of auxiliary material.) Throughout this chapter, F is a local or global field of characteristic 0 unless stated otherwise. The letter B will denote a quaternion algebra over F .IfF is a p-adic or number field, o denotes the ring of integers of F ,and will be an order in B. (Unless stated F O otherwise all orders are oF -orders, including the discussion of orders in other fields K which contain F .) If F is a p-adic field, $ = $F is a uniformizer for F and p = $oF is the unique maximal ideal in o .IfB is a p-adic division algebra, then $ , then denotes the unique F B OB maximal order and P = $ denotes the unique maximal 2-sided ideal in . BOB OB 6.1 Quaternionic orders We have defined orders, ideals and (reduced) norms of ideals, ideal classes and class numbers for central simple algebras. For the arithmetic of quaternion algebras, we will want a couple more notions: discriminants and levels. We will define level below, first locally, then globally. Now we give a uniform definition of discriminant. Let F be a number or p-adic field, B/F be a quaternion algebra, and an order of B. O The dual lattice to is O ? = ↵ B :tr(↵ ) o . O { 2 O ⇢ F } This is the algebraic notion of a dual space with respect to a bilinear form where the bilinear 147 QUAINT Chapter 6: Arithmetic of quaternion algebras Kimball Martin form is the trace form defined in Exercise 3.2.3. Lemma 6.1.1. The dual is a 2-sided -ideal and its inverse ( ) 1 is a 2-sided integral O? O O? − ideal, called the different of . O Exercise 6.1.1. Prove this lemma. 1 The discriminant disc of is the (reduced) norm N(( ?)− ) of the different of , O O O 1 O i.e., the (integral) ideal of oF generated by N(x) for x ( ?)− . In the cases F = Q or 2 O F = Qp, n N and disc = nZ or disc = nZp, we sometimes simply say the discriminant 2 O O is n. These definitions of different and discriminant are analogous to the case of number and p-adic field extensions K/F : the dual lattice (ideal) a of an ideal a o with respect to ? ⇢ K the trace form is the inverse different and the norm of the different from K/F is shown to to be the discriminant of a. Recall that an order is not necessarily a free o -module. In general, it may not be if O F F is a number field with class number > 1. However, is free if o is a PID, i.e., if F is O F p-adic or hF =1. In these cases, it is easier to compute discriminants: Proposition 6.1.2. Suppose o is a PID and an order in B with o -basis ↵ ,...,↵ as F O F 1 4 afreemodule.Thendisc =(det(↵ ↵ ) )2. O i j i,j Proof. See [Vig80, Lem I.4.7]. Proposition 6.1.3. Suppose and are orders of B.Then implies disc O O0 O⇢O0 O⇢ disc ,i.e.,disc disc ,withdisc =disc if and only if = . O0 O0| O O O0 O O0 Proof. See [Vig80, Cor I.4.8]. This proposition is useful in determining if an order is a maximal order, just like the discriminant is a useful tool to determine if an order is the full ring of integers in a number field. E.g., we will use this in Theorem 6.1.15 below. Of course it may be that and have the same discriminant if one is not contained O O0 in the other—this happens for instance when and conjugate orders. We’ll compute O O0 discriminants for orders “with level” below, and see that the discriminant is the level. This includes the case of maximal orders, and the discriminant will be independent of the choice of a maximal order, and we will define the discriminant of these quaternion algebras to be the discriminant of a maximal order. 1, 1 Exercise 6.1.2. Let F = and B = = − − ,andconsidertheorders = Q HQ Q 1+i+j+k O Z[i, j, k] (the Lipschitz integers) and 0 = Z[i, j, ] (the Hurwitz integers). Using O 2 Proposition 6.1.2,showdisc =4and disc 0 =2. O O Note this is compatible with Proposition 6.1.3. We will use this discriminant calculation in Example 6.1.5 below to conclude the Hurwitz integers are a maximal order in HQ. 148 QUAINT Chapter 6: Arithmetic of quaternion algebras Kimball Martin 6.1.1 Local orders In this section, let F be a p-adic field and B be a quaternion algebra over F . Then, up to isomorphism, either B = D (nonsplit) or B = M2(F ) (split), where D denotes the unique quaternion division algebra over F . Here we summarize the theory of (a large class of) oF -orders in B. First we will describe the maximal orders. Recall from Lemma 5.1.2, the unique quaternion division algebra D/F contains the unique quadratic unramified field extension K = F (pu). Since $ is not a norm from K,by Proposition 3.3.7, we can express u, $ ↵$β D = = : ↵,β K M (K), (6.1.1) F β ↵ 2 ⇢ 2 ✓ ◆ ⇢✓ ◆ where ↵ ↵ denotes Galois conjugation in K/F . 7! Lemma 6.1.4. With D as in (6.1.1),theorderinD given by ↵$β = : ↵,β o , OD β ↵ 2 K ⇢✓ ◆ consists of all oF -integral elements of D. Proof. It is easy to see that is an order, so we will just check the integrality assertion. OD Write ↵ = x + ypu and β = z + wpu, where u o⇥ is a nonsquare. Note the element 2 F ↵$β γ = D is integral if and only if tr γ =tr↵ =2x o and det γ = N↵ $Nβ β ↵ 2 2 F − 2 ✓ ◆ oF . Since K/F is unramified, the image of the norm map is precisely the set of elements of even valuation, so det γ o if and only if the “parts of even and odd valuation” N↵ and 2 F $Nβ lie in o , i.e., if and only if N↵,Nβ o , i.e., if and only if v(N↵)=2v (↵) 0 F 2 F K ≥ and v(Nβ)=2v (β) 0, i.e., if and only if ↵,β o . K ≥ 2 K For global applications, it will also useful to describe maximal orders in terms of real- izations of D in M2(K) where K/F is a ramified quadratic extension. Exercise 6.1.3. Let p be an odd rational prime, and u anonsquareinZ⇥.Let$ p, up p 2{ } and K = Qp(p$) the associated ramified quadratic extension. First check that the ↵uβ quaternion division algebra D/ can be realized as the set of matrices , where Qp β ↵ ✓ ◆ ↵,β K and bar denotes Galois conjugation in K. Then show the maximal order of 2 OD D is given by the set of such matrices with ↵,β o .(RecallExercise 1.2.11.) 2 K Theorem 6.1.5. When B = D,thereisauniquemaximalorder ,consistingofall OD -integral elements. When B = M (F ),anymaximalorder is GL (F )-conjugate to OF 2 OB 2 M ( ). 2 OF Proof. The case of B = D follows from the above lemma together with Proposition 4.2.2. It is also a special case of Theorem 4.3.2. When B = M2(F ), this is a special case of Theorem 4.2.7. See also [Vig80, Sec 2.1, 2.2] or [MR03, Sec 6.4, 6.5] for complete proofs in just the quaternionic setting. 149 QUAINT Chapter 6: Arithmetic of quaternion algebras Kimball Martin n $ 1 Example 6.1.1. Let gn = for some n Z. Then n = gnM2( F )g− = 1 2 O O n n ✓ ◆ F p O n is a maximal order in M2(F ).(WeessentiallysawthisinExample 4.2.5.) p− ✓ OF ◆ Thus the maximal orders of B are easy to describe. What about non-maximal orders? In the split case, we can construct other orders by intersecting two or more maximal orders. Orders in B which are the intersection of two maximal orders play a special role in number theory, and are called Eichler orders (cf. Section 4.2). The two maximal orders need not be distinct, so Eichler orders include maximal orders. Example 6.1.2. Let B = M (F ) and recall from Example 6.1.1. Then (n)= 2 On OB F F n 0 n = On O is an Eichler order for n 0.Wesaythisorderisoflevel p .