18.782 Arithmetic Geometry, Lecture Note 11

18.782 Introduction to Arithmetic Geometry Lecture #11 11.1 Quadratic forms over Qp The Hasse-Minkowski theorem reduces the problem of determining whether a quadratic form f over Q represents 0 to the problem of determining whether f represents zero over Qp for all p ≤ ∞. At first glance this might not seem like progress, since there are infinitely many p to check, but in fact we only need to check p = 2, p = ∞ and a finite set of odd primes. Theorem 11.1. Let p be an odd prime and let f be a diagonal quadratic form of dimension n > 2 with coefficients a1 , . . . , an ∈ Z× p . Then f represents 0 over Qp . Proof. The equation f (x1 , . . . , xn ) ≡ 0 mod p is a homogeneous equation of degree 2 in n > 2 variables over Fp . It follows from the Chevalley-Warning theorem that it has a non-trivial solution (y1 , . . . , yn ) over Fp ' Z/pZ. Assume without loss of generality that y1 6= 0 and let g(z) be the univariate polynomial g(y) = f (y, y2 , . . . , yn ) over Zp . Then g(y1 ) ≡ 0 mod p and g 0 (y1 ) = 2a1 y1 6≡ 0 mod p, so by Hensel’s lemma there is a root z1 of g(y) over Zp . We then have f (z1 , y2 , . . . , yn ) = g(z1 ) = 0, so f represents 0 over Qp . Corollary 11.2. Every quadratic form of dimension n > 2 over Q represents 0 over Qp for all but finitely many primes p. Proof. In diagonal form the coefficients a1 , . . . , an lie in Z× p for all odd p - a1 · · · an . For quadratic forms of dimension n ≤ 2, we note that a nondegenerate unary form never represents 0, and the nondegenerate form ax2 + by 2 represents 0 if and only if −ab is square (this holds over any field). But when −ab is not square it may still be the case that ax2 +by 2 represents a given nonzero element t, and having a criterion for identifying such t will be useful in our proof of the Hasse-Minkowski theorem. Lemma 11.3. The nondegenerate quadratic form ax2 + by 2 over Qp represents t ∈ Q∗p if and only if (a, b)p = (t, −ab)p . Proof. Since t 6= 0, the equation ax2 + by 2 = t has a non-trivial solution in Qp if and only if (a/t)x2 + (b/t)y 2 = 1 has a solution, which is equivalent to (a/t, b/t)p = 1. We have (a/t, b/t)p = (at, bt)p = (a, bt)p (t, bt)p = (a, b)p (a, t)p (t, bt) = (a, b)p (t, abt)p = (a, b)p (t, abt)p (t, −t)p = (a, b)p (t, −ab)p , where we have used that the Hilbert symbol is symmetric, bilinear, invariant on square classes, and satisfies (x, −x)p = 1. Thus (a/t, b/t)p = 1 if and only if (a, b)p (t, −ab)p = 1, which is equivalent to (a, b)p = (t, −ab)p since both are ±1. Corollary 11.4. The nondegenerate form ax2 + by 2 + cz 2 over Qp represents 0 if and only if (a, b)p = (−c, −ab)p Proof. By the lemma, if suffices to show that ax2 + by 2 + cz 2 represents 0 if and only if the binary form ax2 + by 2 represents −c. The reverse implication is clear (set z = 1). For the forward implication, if ax20 + by02 + cz02 = 0 then either z0 6= 0, in which case a(x0 /z02 ) + b(y0 /z0 )2 = −c or z0 = 0 in which case ax2 + by 2 represents 0 and therefore every element of Qp , including −c. 1 Corollary 11.5. A ternary quadratic form over Q that represents 0 over all but at most one completion of Q represents 0 over every completion of Q. Proof. The corollary is trivially true if the form is degenerate and otherwise it follows from the product formula for Hilbert symbols and the corollary above. 11.2 Approximation We now prove two approximation theorems that we will need to prove the Hasse-Minkowski theorem for Q. These are quite general theorems that have many applications, but we will state them in a particularly simple form that suffices for our purposes here. Before proving them we first note/recall that Q is dense in Qp and Z is dense in Zp . Theorem 11.6. Let p ≤ ∞ be any prime of Q. Under the metric d(x, y) = |x − y|p , the set Q is dense in Qp and the set Z is dense in Zp . Proof. We know that Q∞ = R is the completion of Q and we proved that Qp is (isomorphic to) the completion of Q for p < ∞, and any field is dense in its completion (this follows immediately from the definition). We note that the completion Z∞ = Z (any Cauchy sequence of integers must be eventually constant), and for p < ∞ the we can apply the fact that Zp = {x ∈ Qp : |x|p ≤ 1} and Z = {x ∈ Q : |x|p ≤ 1}. Theorem 11.7 (Weak approximation). Let S be a finite set of primes p ≤ ∞, and for each p ∈ S let xp ∈ Qp be given. Then for every  > 0 there exists x ∈ Q such that |x − xp |p <  Q for all p ∈ S. Equivalently, the image of Q in p∈S Qp dense under the product topology. Proof. If S has cardinality 1 we can apply Theorem 11.6, so we assume S contains at least 2 primes. For any particular prime p ∈ S, we claim that there is a yp ∈ Q such that |yp |p > 1 and |yp |q < 1 for q ∈ S − {p}. Indeed, let P be the product of the finite primes in S, and for each p < ∞ choose r ∈ Z>0 so that p−r P < 1. Then define ( P if p = ∞, yp = −r p P otherwise. We now note that for any q ∈ S, ( ∞ lim |yp |q = n→∞ 0 if q = p, if q = 6 p. It follows that for each q ∈ S ypn lim = n→∞ 1 + ypn ( 1 with respect to | |q for q = p, 0 with respect to | |q for q = 6 p, since limn→∞ |1 − ypn /(1 + ypn )|p = limn→∞ |1/(1 + ypn )|p = 0 and limn→∞ |ypn /(1 + ypn )|q = 0 for q = 6 p. For each n ∈ Z>0 define X xp ypn zn = . 1 + ypn p∈S Then limn→∞ zn = xp with respect to | |p for each p ∈ S. So for any  > 0 there is an n for which x = zn satisfies |x − xp |p <  for all p ∈ S. 2 Theorem 11.8 (Strong approximation). Let S be a finite set of primes p < ∞, and for each p ∈ S let xp ∈ Zp be given. Then for every  > 0 there exists x ∈ Z such that |x − xp |p <  Q for all p ∈ S. Equivalently, the image of Z in p∈S Zp is dense under the product topology. Proof. Fix  > 0. By Theorem 11.6, for each xp we can pick yp ∈ Z≥0 so that |yp − xp |p < . Let n be a positive integer such that pn > yp for all p ∈ S. By the Chinese remainder theorem, there exists x ∈ Z such that x ≡ yp mod pn for all p ∈ S, and for this x we have |x − xp |p <  for all p ∈ S. Remark 11.9. In more general settings it is natural to consider the infinite product of all the rings of p-adic integers Y Ẑ = Zp . p<∞ Recall that for infinite products, the product topology is defined using a basis of open sets that consists of sequences (Up ), where each Up is an open subset of Zp , and for all but ˆ is finitely many p we have Up = Zp . It follows from Theorem 11.8 that the image of Z in Z dense. There is another way to define Ẑ, which is to consider the inverse system of rings (Z/nZ), where n ranges over all positive integers n and we have reduction maps from Z/mZ to Z/nZ whenever n|m (note that we now have an infinite acyclic graph of maps, not just a linear chain). The inverse limit Ẑ = lim Z/nZ ←− ˆ are is called the profinite completion of Z. One can show that these two definitions of Z canonically isomorphic. So a more pithy statement of Theorem 11.8 is that Z is dense in its profinite completion (this statement applies to profinite completions in general). Remark 11.10. Note the difference between weak and strong approximation. With weak approximation we obtain a rational number x that is p-adically close to xp for each p in a finite set S, but we have no control on |x|p for p 6∈ S. With strong approximation we obtain a rational number (in fact an integer) x that is p-adically close to xp for each p ∈ S and also satisfies |x|p ≤ 1 for all p 6∈ S, except the prime p = ∞; in order to apply the CRT we may need to make |x|∞ very large. More generally, we could allow ∞ ∈ S if we grant ourselves the freedom to make |x|p0 large for one prime p0 6∈ S; in this case x would be a rational number, not an integer, but its denominator would be divisible by no primes other than p0 , so that x ∈ Zp for all p = 6 p0 . This is characteristic of strong approximation theorems, we obtain an element whose absolute value is bounded at all but one prime. The following lemma follows from the strong approximation theorem and Dirichlet’s theorem on primes in arithmetic progressions: for any relative prime integers a and b there are infinitely many primes congruent to a mod b. Lemma 11.11. Let S be a finite set of primes p ≤ ∞, and for each p ∈ S let xp ∈ Q× p be given. Then there exists an x ∈ Q such that (i) x ∈ xp Q×2 p for each p ∈ S. (ii) |x|p = 1 for all but at most one finite prime p0 6∈ S. 3 Proof. Let S0 = S − {∞}, and define the rational number Y y=± pvp (xp ) , p∈S0 where the sign of y is negative if ∞ ∈ S and x∞ < 0, and positive otherwise. Then |y|p = |xp |p for all p ∈ S0 , and it follows that for each p ∈ S0 we have y = up xp for some ep up ∈ Z× p . By the strong approximation theorem there exists an integer z ≡ up mod p , for ×2 all p ∈ S0 , where ep = 1 for odd p and ep = 3 for p = 2. It follows that z ∈ up Qp for all p ∈ S0 , since the square class Q of up depends only on its reduction mod pep . The integers z and m = p∈S0 pep are relatively prime, so it follows from Dirichlet’s theorem that there are infinitely many primes congruent to z mod m. Let p0 be the least such prime. Then p0 ∈ zQ×2 p for all p ∈ S0 , and x = p0 y satisfies both (i) and (ii). 11.3 Proof of the Hasse-Minkowski theorem Before proving the Hasse-Minkowski theorem for Q we make one final remark. The definition of the Hilbert symbol we gave in the last lecture makes sense over any field, in particular Q, and the proofs of Lemma 10.2 and Corollary 10.3 still apply. In the proof below we use (a, b) to denote the Hilbert symbol of a, b ∈ Q× . Theorem 11.12 (Hasse-Minkowski). A quadratic form over Q represents 0 if and only if it represents 0 over every completion of Q. Proof. The forward implication is clear, we only need to prove the reverse implication. So let f be a quadratic form over Q that represents 0 over every completion of Q. We may assume without loss of generality that f is a diagonal form a1 x21 + · · · + an x2n , which we may denote ha1 , . . . , an i. We write ha1 , . . . , an ip to denote the same form over Qp . If any 6 j), so we assume ai = 0, then f clearly represents 0 over Q (set xi = 1 and xj = 0 for i = f is nondegenerate and proceed by induction on its dimension n. Case n = 1: The theorem holds trivially (f cannot represent 0 over any Qp ). Case n = 2: The form ha, bip represents 0 if and only if −ab is square in Qp . Thus vp (−ab) ≡ 0 mod 2 for all p < ∞ and −ab > 0. It follows that −ab is square in Q, and therefore ha, bi represents 0. Case n = 3: Let f (x, y, z) = z 2 − ax2 − by 2 , where a and b are nonzero square-free integers with |a| ≤ |b|. We know (a, b)p = 1 for all p ≤ ∞ and wish to show (a, b) = 1. We proceed by induction on m = |a| + |b|. The base case m = 2 has a = ±1 and b = ±1, in which case (a, b)∞ = 1 implies that either a or b is 1 and therefore (a, b) = 1. We now suppose m ≥ 3, and that the result has been proven for all smaller m. For each prime p|b there is a primitive solution (x0 , y0 , z0 ) ∈ Z3p to z 2 − ax2 − by 2 = 0. We must have p|(z02 − ax20 ), since p|b, but we cannot have p|x0 since then we would have p|z0 , contradicting 2 primitivity. So x0 ∈ Z× p and a = (z0 /x0 ) is a square modulo p. This holds for every prime p|b, and b is square-free, so a is a square modulo b. It follows that a + bb0 = t2 for some t, b0 ∈ Z with t ≤ |b/2|. This implies (a, bb0 ) = 1, √ √ since bb0 = t2 − a is the norm of t + a in Q( a). Therefore (a, b) = (a, b)(a, bb0 ) = (a, b2 b0 ) = (a, b0 ). We also have (a, bb0 )p = 1, and therefore (a, b0 )p = (a, b)p = 1, for all p ≤ ∞. But |b0 | = t2 − a t2 a |b| ≤ + ≤ + 1 < |b|, b b b 4 4 so |a| + |b0 | < m and the inductive hypothesis implies (a, b0 ) = 1. Thus (a, b) = 1, as desired. Case n = 4: Let f = ha1 , a2 , a3 , a4 i and let S consist of the primes p|2a1 a2 a3 a4 and ∞. × Then ai ∈ Z× p for all p 6∈ S. For each p ∈ S there exists tp ∈ Qp such that ha1 , a2 ip represents tp and ha3 , a4 ip represents −tp (we can assume tp = 6 0: if 0 is represented, by both forms, so is every element of Qp ). By Lemma 11.11, there is a rational number t and a prime p0 6∈ S such that t ∈ tp Q×2 p for all p ∈ S and |t|p = 1 for all p 6∈ S ∪ {p0 }. The forms ha1 , a2 , −tip and ha3 , a4 , tip represent 0 for all p 6∈ S ∪ {p0 } because all such p are odd, and ai , ±t ∈ Z× p , so (a1 , a2 )p = 1 = (t, −a1 a2 )p and (a3 , a4 )p = 1 = (−t, −a3 a4 )p , 2 and we may apply Corollary 11.4. Since t ∈ tp Q× p for all p ∈ S, the forms ha1 , a2 , −tip and ha3 , a4 , tip also represent 0 for all p ∈ S. Thus ha1 , a2 , −tip and ha3 , a4 , tip represent 0 for all p 6= p0 , and by Corollary 11.5, also for p = p0 . By the inductive hypothesis ha1 , a2 , −ti and ha3 , a4 , ti both represent 0, therefore ha1 , a2 , a3 , a4 i represents 0. Case n ≥ 5: Let f = ha1 , . . . , an i. Let S be the set of primes for which ha3 , . . . , an ip does not represent 0. The set S is finite, by Corollary 11.2. If S is empty then ha3 , . . . , an i, and therefore f , represents 0, by the inductive hypothesis, so we assume S is not empty. 2 2 For each p ∈ S pick tp ∈ Q× p represented by ha1 , a2 i, say a1 xp + a2 yp = tp , such that ha3 , . . . , an ip represents −tp (such a tp exists since f represents 0 over Qp and, as above, we can always pick tp 6= 0). By the weak approximation theorem there exists x, y ∈ Q that are simultaneously close enough to all the xp , yp ∈ Qp so that t = a1 x2 + a2 y 2 is close enough to all the tp to guarantee that t ∈ tp Q×2 p for all p ∈ S (for p < ∞ the square class only depends on at most the first three nonzero p-adic digits, and over R = Q∞ we can ensure that x and y have the same signs as x∞ and y∞ ).1 It follows that ht, a3 , . . . , an ip represents 0 for all p ∈ S, and since ha3 , . . . , an ip represents 0 for all p 6∈ S, so does ht, a3 , . . . , an ip . Thus ht, a3 , . . . , an ip represents 0 for all p, and by the inductive hypothesis, ht, a3 , . . . , an i represents 0. Therefore ha3 , . . . , an i represents −t = −a1 x2 − a2 y 2 , hence ha1 , . . . , an i represents 0. 1 ×2 Equivalently, the set of squares Q×2 is an open subset of Q× p p , hence so is every square class tp Qp . 5