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Quaternion Algebras and Q 1.1. What Are Quaternion Algebras?

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Quaternion Algebras and Q 1.1. What Are Quaternion Algebras? INTRODUCTION TO THE LOCAL-GLOBAL PRINCIPLE LIANG XIAO Abstract. This is the notes of a series lectures on local-global principle and quaternion algebras, given at Connecticut Summer School in Number Theory. 1. Day I: Quaternion Algebras and Qp 1.1. What are Quaternion Algebras? 1.1.1. Hamiltonian H. Recall that we setup mathematics in such a way starting with positive integers N and integers Z to build Q as its quotient field, and then defining R using several equivalent axioms, e.g. Dedekind cut, or as certain completions. After that, we introduced the field of complex numbers as C = R ⊕ Ri, satisfying i2 = −1. One of the most important theorem for complex numbers is the Fundamental Theorem of Algebra: all complex coefficients non-constant polynomials f(x) 2 C[x] has a zero. In other words, C is an algebraically closed field; so there is no bigger field than C that is finite dimensional as an R-vector space. (With the development of physics), Hamilton discovered that there is an \associative-but- non-commutative field” (called a skew field or a division algebra) H which is 4-dimensional over R: H = R ⊕ Ri ⊕ Rj ⊕ Rk = ai + bj + cj + dk a; b; c; d 2 R ; where the multiplication is R-linear and subject to the following rules: i2 = j2 = k2 = −1; ij = −ji = k; jk = −kj = i; and ki = −ik = j: This particular H is called the Hamiltonian quaternion. One simple presentation of H is: 2 2 H := Chi; ji i + 1; j + 1; ij + ji : Question 1.1.2. Are there other quaternions that look like the Hamiltonian quaternion? Can we classify them? 1.1.3. Quaternion algebra. Over R, such quaternion algebra is unique, but over other fields, there are plenty. For example, we can consider 2 2 (1.1.1) D = Qhi; ji i + 1; j + 1; ij + ji ; or more generally, for any field k (of characteristic zero or just Q, at least today), and any two elements a; b 2 k× we put 2 2 (1.1.2) Dk;a;b := khi; ji i − a; j − b; ij + ji : 1 For example, the quaternion in (1.1.1) is DQ;−1;−1. These Dk;a;b are called quaternion alge- a;b bras over k. In other literature, the quaternion algebras are often denoted by k . It is conventional to put k = ij so that ij = −ji = k; jk = −kj = i; and ki = −ik = j: Like in the case of complex numbers, every element α = x + yi + zj + wk has a conjugate: α¯ := x − yi − zj − wk. We note: Tr(α) := α +α ¯ = 2x 2 k; Nm(α) := αα¯ = x2 − ay2 − bz2 + abw2 2 k: (It looks like we are solving quadratic equations in a non-commutative ring.) In particular, if Nm(α) 6= 0, then α has an inverse: 1 α−1 = α: Nm(α) Quaternion algebras are called division algebras if every nonzero element of Dk;a;b has a multiplicative inverse (both left and right), this is equivalent to the condition that x2 −ay2 − bz2 + abw2 is always nonzero unless all x; y; z; w are zero. Example 1.1.4. For a; b 2 Q<0, DQ;a;b is a division algebra, because for rational numbers x; y; z; w, x2 − ay2 − bz2 + abw2 = 0 would force x = y = z = w = 0 by positivity. The matrix algebra M2(Q) is also a quaternion algebra (but not a division algebra), isomor- 0 −1 1 0 phic to DQ;−1;1, in the sense that taking i = 1 0 and j = 0 −1 defines an isomorphism ∼ DQ;1;−1 = M2(Q). Question 1.1.5. Are these Dk;a;b genuinely different? In Exercise 1.4.6, you will verify some × obvious equalities, like Dk;a;b = Dk;a;−ab = Dk;ac2;b for a; b; c 2 k . When are these Dk;a;b division algebras? Example 1.1.6. When k = R, there are (up to isomorphism) two quaternion algebras: ∼ • H = D ;a;b whenever a; b are both negative; R∼ • M2(R) = DR;a;b whenever at least one of a and b is positive. 1.1.7. Local-global approach. To answer to Question 1.1.5 (for general field k) directly is very difficult; but the question is easier for some particular fields, like R. In this lecture series, we address Question 1.1.5 in the case when k = Q, and our approach takes the following form: for a quaternion algebra DQ;a;b with a; b 2 Q, we consider ∼ DQ;a;b ⊗Q R = DR;a;b: 0 0 ∼ So if for two pairs (a; b) and (a ; b ), if D ;a;b 6= D ;a0;b0 (which can be tested easily using ∼ R R 1.1.6), then DQ;a;b 6= DQ;a0;b0 . In other words, what we are doing is: • testing whether DQ;a;b is isomorphic to DQ;a0;b0 by tensoring over a bigger field k=Q (which in our case will be \local fields”), where the corresponding question is easier (namely, a local question); and • there is a local-global principal we will prove at the end, which shows that, if DQ;a;b and DQ;a0;b0 are isomorphic after tensoring with \all" local fields, then they are iso- morphic. Such type of argument is called a \local-global argument." 2 1.2. Introduction to Qp. 1.2.1. Local fields. The way we define the real numbers R (from Q) can be viewed as taking the completion of Q with respect to the usual absolute value j · j1, namely to adjoin limits of Cauchy sequences in Q. Q2 |·|2 |·|1 |·|3 R Q Q3 |·|p Qp: In fact, besides the usual absolute value, there are other \absolute values" or norms on Q, essentially one for each prime number p. Completion with respect to the norm associated to the prime p gives the main player today: Qp. Definition 1.2.2. Fix a prime p from now on. We define the p-adic valuation of a non-zero integer a, denoted by vp(a), to be the maximal (non-negative integer) exponent n such that n vp(a) p ja (or sometimes we write p jja). We put vp(0) = +1. Clearly we have the following properties: (1.2.1) vp(ab) = vp(a) + vp(b) and vp(a + b) ≥ minfvp(a); vp(b)g: This p-adic valuation extends to a p-adic valuation on Q, given by vp : Q / Z [ f1g a a x = b / vp b := vp(a) − vp(b): We can check that the expression above does not depend on the choice of the writing of x a as a rational number b . The above properties (1.2.1) are still preserved. Let us put it another way, define: −vp(a) jajp := p for a 2 Q: 14 14 1 For example, p = 5, v5( 75 ) = −2, and 75 5 = 25. For another, v5(100) = 2 and 100 5 = 25 . Let us formalize everything. Definition 1.2.3. A valuation on a field k is a map v : k −! Z [ f+1g; such that (a) v(x) = +1 if and only if x = 0, (b) v(xy) = v(x) + v(y) for all x; y 2 k, and (c) v(x + y) ≥ minfv(x); v(y)g for all x; y 2 k. We point out that if v(x) 6= v(y), the inequality in (c) is an equality, namely v(x + y) = minfv(x); v(y)g (see Exercise 1.4.8). A norm on a field k is a map j · j : k ! R≥0 such that (1) jxj = 0 if and only if x = 0, 3 (2) jxyj = jxj · jyj for all x; y 2 k, and (3) (Triangle inequality) jx + yj ≤ jxj + jyj for all x; y 2 k. It is called non-archimedean, if the norm satisfies (3') (Strong triangle inequality) jx + yj ≤ maxfjxj; jyjg. In this sense, we have just defined a valuation vp(−) on Q for each prime number p, and a norm j · jp on Q for each prime number p. Theorem 1.2.4. (Ostrowski) The following lits all the norms of Q: (1) the trivial norm j · jtriv, namely jxjtriv = 1 if x 6= 0, and 0 if x = 0; s (2) (Essentially the real norm) j · j1 for s 2 (0; 1]; and s −svp(·) (3) (Essentially the p-adic norm) j · jp = p for s 2 R>0. Notation 1.2.5. In view of Ostrowski's theorem, we say a place of Q to mean either a prime number, or the real norm (often denoted by 1). We may complete Q with respect to the norm at each place, and get either Q1 := R or Qp for some prime p. Construction 1.2.6. We define the ring of p-adic integers Zp to be the completion of Z with respect to the p-adic valuation. We can understand this using one of the following equivalent ways. Understanding I:A p-adic integer is a sequence of integers: a1; a2;::: n n such that an ≡ an+1 (mod p ). (Note that an ≡ an+1 (mod p ) implies that jan+1 − anjp ≤ p−n which converges to zero.) We can extend the p-adic valuation vp to Zp by setting vp((an)n2 ) := lim vp(an); N n!1 this limit stabilizes as n 0 (or the limit is 0 2 Zp). Understanding II:A p-adic integer is a sequence of congruences 2 3 a1 mod p; a2 mod p ; a3 mod p ;:::; n such that an ≡ an+1 (mod p ).
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