Hasse-Minkowski Theorem

HASSE-MINKOWSKI THEOREM 1. Introduction In rough terms, a local-global principle is a statement that asserts that a certain property is true globally if and only if it is true everywhere locally. We will give a proof of Hasse-Minkowski theorem over Q, which is the best known example for local-global principles Theorem 1.1. (Hasse-Minkowski) Let K be a number field and let q be a quadratic form in n variables with coefficients in K. Then q represents 0 in K if and only if it represents 0 in every completion of K. 2. Basic results on quadratic forms Let V be a K-vector space of finite dimension n. Recall that a quadratic form on V is a map q from V to K such that q(ax) = a2 q(x) for all x ∈ V and a ∈ K, and b(x, y) = (q(x+y)−q(x)−q(y))/2 isPa bilinear form so that q(x) = b(x, x). Thus, for a given basis {ei } for V , and x = i xi ei , we have q(x) = X t QX, where X is the column vector of the xi , and Q is a symmetric matrix having b(ei , ej ) as ij-th entry. If {e0i } is another bases of V and if P is the matrix expressing e0i in terms of the ei , then with X = P X 0 , we have q(x) = X t QX = X 0t P t QP X 0 ; hence the matrix of q in the new basis is equal to P t QP . In particular, det(P t QP ) = det(Q)det(P )2 , so the class of det(Q) modulo nonzero squares of K is independent of the chosen basis and called the discriminant of q, denoted by d(q). If V = K n and (ei ) is the canonical basis of V , we can identify a quadratic form on V with a homogeneous polynomial of degree 2 in n variables over K by the formula X q(x1 , · · · , xn ) = qi,j xi xj , 1≤i,j≤n where the qi,j are the entries of the symmetric matrix Q. We call Q the coefficient matrix of q. Definition 2.1. Two quadratic forms q and q 0 with coefficient matrices Q, and Q0 respectively, are equivalent if there exists P ∈ GLn (K) such that Q0 = P t QP . Denoted q ∼ q 0 . Using Gauss’s reduction of quadratic forms into sums of squares, gives Theorem 2.1. Let q be a quadratic form in n variables. There exists an equivalent form thatPis a diagonal quadratic form; in other words, there exist ai ∈ K such that q ∼ 1≤i≤n ai x2i . We say that a form q represents a ∈ K if there exists x ∈ K n such that q(x) = a, with added condition that x = 6 0 when a = 0. 1 2 Theorem 2.2. Let q be a nondegenerate quadratic form in n variables and let c ∈ K ∗ . The following conditions are equivalent: (1) The form q represents c. (2) There exists a quadratic form q1 in n − 1 variables such that q ∼ cx20 ⊕ q1 . (3) The quadratic form q cx20 represents 0 in K. 3. quadratic forms over finite and local fields We begin with the simplest possible fields, the finite fields. Let q = pk be a prime power, with p 6= 2. Theorem 3.1. A quadratic form over Fq of rank n ≥ 2 represents all elements of F∗q , and a quadratic form of rank n ≥ 3 represents all elements of Fq . P 2 Proof. By Theorem2.1, we assume that q = 1≤i≤m ai xi with a1 a2 6= 0. Let a ∈ Fq . We choose xi = 0 for all i ≥ 3. Since q is odd the map x 7→ x2 is a group homomorphism of F∗q into itself, and its kernel has two elements. It follows that its image has (q − 1)/2 elements, so adding 0, there are (q + 1)/2 squares in Fq . Since a1 a2 6= 0 it follows that the subsets {a1 x21 } and {a − a2 x22 } of Fq also have (q + 1)/2 elements. Hence they have nonempty intersection.  Corollary 3.1. Let c ∈ F∗q that is not a square. A nondegenerate quadratic form over Fq is equivalent to x21 + · · · + x2n−1 + ax2n with a = 1 if its discriminant is a square, and with a = c otherwise. Proof. Use induction on n and previous theorem.  Corollary 3.2. Two nondegenerate quadratic forms over Fq are equivalent if and only if they have the same rank and the same discriminant in F∗q /F∗2 q . 4. definition of the local hilbert symbol We introduce the Hilbert symbol, which will be sufficient for the local study of quadratic forms. We shall omit the proof here. Definition 4.1. If a and b are in K∗ , we set (a, b) = 1 if the equation ax2 +by 2 = z 2 has a nontrivial solution, and (a, b) = −1 otherwise. The number (a, b) is called the Hilbert symbol of a and b. Proposition 4.1. Let a and b be in K∗ . We have (a, b) = 1 if and only if a ∈ √ ∗ N (K( b) ). Proposition 4.2. We have the following formulas, where all the elements that occur are assumed to be nonzero: (1) (a, b) = (b, a) and (a, c2 ) = 1. (2) (a, −a) = (a, 1 − a) = 1. (3) (a, b) = 1 implies (aa0 , b) = (a0 , b). (4) (a, b) = (a, −ab) = (a, (1 − a)b). We state the explicit computation of the Hilbert symbol when K = Qp . We denote as usual by Up the group of p-adic units. HASSE-MINKOWSKI THEOREM 3 Theorem 4.1. (1) For K = R, we have (a, b) = 01 if a < 0 and b < 0, and (a, b) = 1 if a or b is positive. (2) For K = Qp with p 6= 2, write a = pα a1 , b = pβ b1 with a1 and b1 in Up . Then a1
β b1
α (a, b) = (−1)αβ(p−1)/2 . p p (3) For K = Q2 , with the same notation we have a1
β b1
α (a, b) = (−1)(a1 −1)(b1 −1)/4 . 2 2 From this theorem, we know that the Hilbert symbol is bilinear on F2 -vector space K∗ /K∗2 . Proposition 4.3. Let q(x, y, z) = ax2 + by 2 + cz 2 be a nondegenerate quadratic form in three variables with coefficients in Qp (including p = ∞). Set  = (q) = (a, b)(b, c)(a, c), and let d = d(q) = abc be the discriminant of q. Then q represents 0 in Qp if and only if (−1, −d) = . Proof. The form q represents 0 if and only if the form −cq does, hence if and only if −acx2 − bcy 2 = z 2 has a nontrivial solution, in other words by definition (−ac, −bc) = 1. By bilinearity this condition is 1 = (−ac, −bc) = (−1, −1)(−1, a)(−1, b)(a, b)(a, c)(b, c)(c, c), and since (c, c) = (−1, c), this can be written (−1, −abc) = (a, b)(b, c)(a, c), proving the proposition.  Corollary 4.1. Let c ∈ Q∗p , and let q(x, y) = ax2 + by 2 be a nondegenerate quadratic form in two variables. Then q represents c in Qp if and only if (c, −ab) = (a, b). Proof. Use above proposition.  5. quadratic forms over Qp We define a second invariant. Up to equivalence, we can assume that q is in P diagonal form as q(x) = 1≤i≤n ai x2i , and we set Y ((a1 , · · · , an )) = (ai , aj ), 1≤i,j≤n where (ai , aj ) is the Hilbert symbol. We have the following theorem. Theorem 5.1. The value of ((a1 , · · · , an )) is independent of the linear change of variables that transforms q into diagonal form, hence is an invariant of the quadratic form itself, which we will denote by (q). It follows from this theorem that just as for the discriminant d(q), (q) is an invariant of the equivalence class of q. Theorem 5.2. Let q be a nondegenerate quadratic form in n variables, and set d = d(q), and  = (q). Then q represents 0 nontrivially in Qp if and only if one of the following holds: (1) n = 2 and d = −1. (2) n = 3 and (−1, −d) = . (3) n = 4 and either d 6= 1, or d = 1 and (−1, −d) = . (4) n ≥ 5. 4 Corollary 5.1. Let c ∈ Q∗p /Q∗2 p . A nondegenerate form q in n variables with invariants d and  represents c if and only if one of the following holds: (1) n = 1 and c = d. (2) n = 2 and (c, −d) = . (3) n = 3 and either c 6= −d or c = −d and (−1, −d) = . (4) n ≥ 4. Corollary 5.2. Two quadratic forms over Qp are equivalent if and only if they have the same rank, discriminant, and invariant (q). 6. quadratic forms over Q Theorem 6.1. If a and b are in Q∗ then (a, b)v = 1 for almost all v ∈ P , and we have the product formula Y (a, b)v = 1. v∈P Proof. Use Theorem 4.1.  Theorem 6.2. Let (ai )i∈I be a finite set of elements of Q∗ and let (i,v )i∈I,v∈P be a set of numbers equal to ±1. There exists x ∈ Q∗ such that (ai , x) = i,v for all i ∈ I and all v ∈ P if and only if the following three conditions are satisfied: (1) Almost all of the i,v Q are equal to 1. (2) For all i ∈ I we have v∈P i,v = 1. (3) For all v ∈ P there exists xv ∈ Q∗v such that (ai , xv )v = i,v for all i ∈ I. Theorem 6.3. (Hasse-Minkowski Theorem for n ≤ 2) Lemma 6.1. Over any field K of characteristic diffferent from 2 the form ax2 + bxy + cy 2 represents 0 nontrivially if and only if b2 − 4ac is a square in K. Proof. Use the identity (2ax + by)2 − y 2 (b2 − 4ac) = 4a(ax2 + bxy + cy 2 ).  Since q represents 0 nontrivially in R, we must have d ≥ 0. If d = 0 thenQ q is a square of a linear form hence represents 0 nontrivially. If d > 0 then let d = i pvi i be the prime power decomposition of d. Since q represents 0 nontrivially in every Qp , by above lemma d is a square in Qp . This implies that vpi (d) = vi is even for all i, hence that d is a square. Theorem 6.4. (Hasse-Minkowski Theorem for n = 3) Proof. We may assume that our quadratic form is a diagonal form q(x, y, z) = ax2 + by 2 + cz 2 . If one of the coefficients is 0 then q has a nontrivial zero in Q by the case n = 2. Thus we may assume abc 6= 0. Furthermore, we may assume that q(x, y, z) = x2 − ay 2 − bz 2 with a, b square-free integers, where we assume |a| ≤ |b|. We prove the theorem by induction on m = |a| + |b|. If m = 2 then q(x, y, z) = x2 ± y 2 ± z 2 , and since the case x2 + y 2 + z 2 is excluded since q represents 0 in R, in other cases the form represents 0. Q Thus assume now that m > 2, in other words |b| ≥ 2, and let b = ± 1≤i≤k pi be the prime factorization of the square-free number b. Let p = pi for some i. We claim that a is a square modulo p. This is trivial if a ≡ 0 (mod p). Otherwise a is a p-adic unit, and by assumption there exists a nontrivial p-adic solution to ay 2 + bz 2 = x2 , where x, y, z ∈ Zp , and at least one in Up . Thus x2 ≡ ay 2 (mod pZp ). Now y is a p-adic unit. It follows that a ≡ (x/y)2 (mod pZp ), so a is a square modulo p. Since HASSE-MINKOWSKI THEOREM 5 this is true for all p|b, by the Chinese remainder theorem this implies that a is a square modulo b, in other words that there exist b0 and k such that k 2 = a + bb0 , 0 2 0 where k may be chosen √ such that |k| ≤ |b|/2. Since bb = k − a, bb is a norm in the extension K( a)/K, where K = Q or any Qv . Thus, q represents 0 in K if and only if the same is true for q 0 (x, y, z) = x2 − ay 2 − b0 z 2 . In particular, by assumption q 0 represents 0 in all the Qv . But since |b| ≥ 2 and |a| ≤ |b|, we have |b0 | = |b| k2 − a ≤ + 1 < |b|. b 4 Thus we may apply our induction hypothesis to the form q 0 (more precisely to the form q 00 , where b0 is replaced by its square-free part);hence q 0 represents 0 in Q, and so the same is true for the form q.  Theorem 6.5. (Hasse-Minkowski Theorem for n = 4) Before the proof in this case, we slightly strengthen the case n = 3. Proposition 6.1. Let q(x, y, z) be a quadratic form in three variables, and assume that q(x, y, z) = 0 has a nontrivial solution in every completion of Q except perhaps in one. Then it has a nontrivial solution in Q, hence in all places. Proof. Assume q(x, y, z) = ax2 + by 2 + cz 2 . By Proposition 4.3, q represents 0 in Qv if and only if (−1, −abc)v = (a, b)v (b, c)v (a, c)v . By assumption this is true for all v except perhaps one. Since both sides satisfy the product formula(Theorem 6.1), it follows that this equality is true for all v; by 4.3 again, q represents 0 in Qv for all v. Hence by the proof of case n = 3, q represents 0 in Q.  Proof. We may assume that q = a1 x21 + a2 x22 − a3 x23 − a4 x24 . Let v be a place of Q. Since q represents 0 in Qv , an application of Theorem 2.2 shows that there exists cv ∈ Q∗v that is represented both by a1 x21 + a2 x22 and by a3 x23 + a4 x24 , and Corollary 4.1 implies that for all v we have (cv , −a1 a2 )v = (a1 , a2 )v and (cv , −a3 a4 )v = (a3 , a4 )v . By the product formula for the Hilbert symbol, we deduce from Theorem 6.2 that there exists c ∈ Q∗ such that for all places v, (c, −a1 a2 )v = (a1 , a2 )v and (c, −a3 a4 )v = (a3 , a4 )v . a1 x21 + a2 x22 − cx20 The form thus represents 0 in each Qv , hence by the proof of the case n = 3 also in Q, so c is represented by a1 x21 + a2 x22 . Similarly c is represented  by a3 x23 + a4 x24 , so q represents 0. Theorem 6.6. (Hasse-Minkowski Theorem for n ≥ 5) Proof. Write Q = Q1 − Q2 where Q1 (x1 , x2 ) = a1 x21 + a2 x22 and Q2 (x3 , · · · , xn ) = −a3 x23 − · · · − an x2n . Let S be the set consisting of v = 2, v = ∞ and v such that not every ai ∈ Z∗v for i ≥ 3. For all v ∈ S, Q1 and Q2 represent some common nonzero αv over Qv since Q represents 0 over Qv . The set of nonzero squares Q∗2 v is open so the coset 6 of Q∗2 v containing αv is an open set. The quadratic form Q1 is continuous so the inverse image of the coset containing αv is an open set Av in Qv × Qv . By the approximation theorem there are x1 , x2 ∈ Q such that (x1 , x2 ) ∈ Av for all v ∈ S. Thus a := Q1 (x1 , x2 ) is in Q and a/αv ∈ Q∗2 v for all v ∈ S. Consider the quadratic form Q0 = at2 −Q2 . There is a nontrivial solution to Q0 = 0 over every Qv for v ∈ S 0 since a/αv ∈ Q∗2 v for all v ∈ S. Furthermore, the equation Q = 0 has a nontrivial solution over every Qv where v is not in S since char(Qv ) 6= 2 and n − 2 ≥ 3. So there is a nontrivial solution to Q0 = 0 over Q by the induction hypothesis since Q0 is an (n − 1)-dimensional quadratic form. This means the equation Q2 = a has a solution over Q. We now have solutions over Q to Q1 = a and Q2 = a so Q = Q1 − Q2 = 0 has a nontrivial solution over Q.