Ramanujan J (2008) 17: 145–153 DOI 10.1007/s11139-007-9023-y

Extensions of representations of integral quadratic forms Wai Kiu Chan · Byeong Moon Kim · Myung-Hwan Kim · Byeong-Kweon Oh

Received: 5 April 2006 / Accepted: 14 March 2007 / Published online: 18 September 2007 © Springer Science+Business Media, LLC 2007

Abstract Let N and M be quadratic Z-lattices, and K be a sublattice of N. A representation σ : K → M is said to be extensible to N if there exists a representation ρ : N → M such that ρ|K = σ . We prove in this paper a local–global principle for extensibility of representation, which is a generalization of the main theorems on representations by positive definite Z-lattices by Hsia, Kitaoka and Kneser (J. Reine Angew. Math. 301:132–141, 1978) and Jöchner and Kitaoka (J. Number Theory 48:88–101, 1994). Applications to almost n-universal lattices and systems of quadratic equations with linear conditions are discussed. Keywords Extension of representations · Integral quadratic forms Mathematics Subject Classification (2000) 11E12 · 11E20

Research of the first author was partially supported by the National Science Foundation. The third author was partially supported by KRF Research Fund (2003-070-C00001). W.K. Chan Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT 06459, USA e-mail: [email protected] B.M. Kim Department of Mathematics, Kangnung National University, Kangwondo 210-702, Korea e-mail: [email protected] M.-H. Kim Department of Mathematical Science, Seoul National University, Seoul 151-747, Korea e-mail: [email protected] B.-K. Oh () Department of Applied Mathematics, Sejong University, Seoul 143-747, Korea e-mail: [email protected]

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1 Introduction  Let f (x) = m i,j =1 aij xi xj be a quadratic form in m variables x1 , . . . , xm , where (aij ) is an m × m symmetric matrix over Q. We say that f represents a quadratic form g(y) = ni,j =1 bij yi yj in n variables y1 , . . . , yn over Z if there exist b1 , . . . , bn ∈ Zm such that f (y1 b1 + · · · + yn bn ) = g(y). The representation problem asks for a complete determination of the set of forms that are represented by f . This is equivalent to deciding when the matrix equation (which can be viewed as a system of Diophantine equations) t

T (aij )T = (bij )

has a solution T ∈ Mm×n (Z). For the recent development of this problem, we refer the readers to the recent surveys [3] and [12] and the references cited therein. Using the geometric language of quadratic spaces and lattices (see [9, 11]), the representation problem of quadratic forms can be rephrased as follows. The equivalence class of the quadratic form f (x) corresponds to the isometry class of a Z-lattice M with a basis {v1 , . . . , vm } and a symmetric bilinear form B such that B(vi , vj ) = aij . A Z-lattice N of rank n with bilinear form B  is said to be represented by M, written N → M, if there exists a Z-linear map σ : N → M such that B(σ (x), σ (y)) = B  (x, y) for all x, y ∈ N . This map σ is called a representation of N by M. If g(y) is the quadratic form corresponding to N , then N → M if and only if g(y) is represented by f (x). Therefore, the representation problem of quadratic forms is equivalent to deciding when a Z-lattice is represented by another Z-lattice. Throughout this paper, we shall conduct our discussion in the above geometric language. Any unexplained terminologies and notations can be found in [9] and [11]. The term lattice always refers to a Z-lattice on a non-degenerate quadratic space with a bilinear form B and its associated quadratic map Q. Note that under this condition every representation of a lattice by another lattice must be injective. Let p be a prime. For a lattice M, its p-adic completion Mp is the Zp -lattice Zp ⊗ M, where Zp is the ring of p-adic integers. The genus of M, denoted gen(M), is the set of all lattices L on the space QM such that Lp ∼ = Mp for all primes p. A necessary condition for N → M is that Np → Mp for all p and R ⊗ N → R ⊗ M, and we say that N is represented by gen(M). If this is the case, then Hasse-Minkowski’s local– global principle asserts that there will be a representation of the space QN by QM. Therefore, when discussing the representation problem, there is no harm to assume that N is a lattice in the space QM. Furthermore, Witt’s theorem implies that any representation of N by M is the restriction of an isometry of QM. Let K be a sublattice of N and σ : K → M be a representation. We say that a representation ρ : N → M is an extension of σ if ρ|K = σ . This definition has its analog for Zp -lattices. In the next section, we shall prove a result concerning extension of representation which can be viewed as a generalization of the main representation theorems for positive definite lattices proved by Hsia, Kitaoka and Kneser [4] and by Jöchner and Kitaoka [7]. In Sects. 3 and 4 we shall apply our results to study almost 2-universal lattices and quadratic equations with linear conditions.

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We close this section with some additional notations and terminologies which will be used throughout this paper. If M is a positive definite lattice of rank m, for any i = 1, . . . , m, the i-th successive minimum of M is denoted by μi (M). By reduction theory ([2], Chap. 12) there exists a constant α, depending only on m, such that d(M) > αμ1 (M) · · · μm (M). Let R be Z or Zp . For a basis {v1 , . . . , vn } of a R-lattice M, we write M∼ = (B(vi , vj )). The matrix on the right hand side is called a matrix presentation of M, which we often identify with M itself. If M admits an orthogonal basis {v1 , . . . , vn }, we call M diagonal and simply write M∼ = Q(v1 ), . . . , Q(vn ). The determinant of the matrix (B(xi , xj )) is called the discriminant of M, denoted d(M). Unless stated otherwise, every R-lattice M is assumed to be integral in the sense that B(M, M) ⊆ R. A representation σ : N → M of R-lattices is said to be primitive if σ (N) is a direct summand of M as R-modules. If φ and ψ are representations from N to M and a ∈ R, we write φ ≡ ψ mod aM if φ(x) − ψ(x) ∈ aM for all x ∈ N .

2 Extensibility of representations Throughout this section, K, N and M are lattices of rank k, n and m, respectively, in the quadratic space V = QM. Let U be the space QK and W be the orthogonal complement of U in V . Let π be the orthogonal projection from V onto W . Theorem 2.1 Let σ : K→M be a representation. Suppose that m ≥ k + 2(n − k) + 3. Let s be a positive integer and T be a finite set of primes which contains all primes that divide 2d(M)d(K). If the orthogonal complement of σ (K) in M is positive definite, then there exists a constant C1 = C1 (K, M, n, T , s) with the following property: Suppose that for each prime p, there exists a representation ρ(p) : Np →Mp which extends σ . If μ1 (π(N )) > C1 , then there exists a representation ρ : N →M which extends σ and ρ ≡ ρ(p) mod p s Mp for all p ∈ T . Proof Suppose that the theorem holds for the special case when K ⊆ N ∩ M and σ is the inclusion map. Now, K ⊆ N ∩ σ −1 (M) and for each prime p, the restriction of the representation σ −1 ρ(p) : Np →σ −1 (M)p to Kp is the inclusion map. Let C1 be the constant for K, σ −1 (M), n, T and s. If μ1 (π(N )) > C1 , then there exists a representation γ : N →σ −1 (M), which is an extension of the inclusion K → σ −1 (M), such that γ ≡ σ −1 ρ(p) mod p s σ −1 (Mp ) for all p ∈ T . Let ρ be the isometry σ γ . Then ρ|K = σ and hence ρ extends σ . Moreover, ρ ≡ ρ(p) mod p s Mp for all p ∈ T . So, from now on we assume that σ : K→M is the inclusion map. We may enlarge T to contain all the primes dividing d(π(M)). Then select a positive integer a such that p a π(Mp ) ⊆ p s Mp for all p ∈ T . Pick a prime q outside T and a positive

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integer e. Let C1 be the constant C1 (m, n, π(M), T , a, q, e) obtained from the main theorem in [7], p. 96. The prime q and the integer e will not play any role in the subsequent discussion; they are chosen so that the constant C1 can be obtained. Note that C1 does not depend on N . For each p ∈ T , write ρ(p) = 1Up ⊥ τ(p) for some τ(p) ∈ O(Wp ). Now suppose that μ1 (π(N )) > C. Since rank(π(M)) = m − k ≥ 2(n − k) + 3 = 2 rank(π(N )) + 3, there exists an isometry γ on W such that γ : π(N)→π(M), and that γ ≡ τ(p) mod p a π(Mp ) for all p ∈ T . Let ρ be the isometry 1U ⊥ γ ∈ O(V ). Let x be any vector in N . If p ∈ T , then Qp Kp ∩ Np = Kp and π(Mp ) ⊆ Mp . Therefore, ρ(x) = (x − π(x)) + γ (π(x)) ∈ Mp . If p ∈ T , then ρ(x) − ρ(p) (x) = γ (π(x)) − τ(p) (π(x)) ∈ p a π(Mp ) ⊆ p s Mp . As a result, ρ(N ) ⊆ M and ρ ≡ ρ(p) mod p s Mp for all p ∈ T . It is clear that ρ extends σ .  Remark 2.2 Theorem IV of [1] gives a version of the above theorem with additional primitivity conditions imposed on both the ρ(p) and ρ. When M itself is positive definite, the following version of Theorem 2.1 will be found more useful in later discussion. Theorem 2.3 Let σ : K→M be a representation. Suppose that M is positive definite and that m ≥ k + 2(n − k) + 3. Let s be a positive integer and T be a finite set of primes which contains all primes that divide 2d(M)d(K). There exists a constant C2 = C2 (K, M, n, T , s) with the following property: Suppose that for each prime p, there exists a representation ρ(p) : Np → Mp which extends σ . If μk+1 (N ) > C2 , then there exists a representation ρ : N→M which extends σ and ρ ≡ ρ(p) mod p s Mp for all p ∈ T . Proof This is a direct consequence of the next lemma.



Lemma 2.4 Suppose that M is positive definite and K ⊆ N ∩ M. Then there exists a constant C  = C  (K) such that μ1 (π(N )) > C  μk+1 (N ). Proof The lemma is clear when k = 0. Suppose that k ≥ 1, and fix a basis {e1 , . . . , ek } of K. Let x ∈ N such that π(x) is a minimal vector of π(N ). Let  be the lattice spanned by {e1 , . . . , ek , x}. Then d() = d(K)Q(π(x)) = d(K)μ1 (π(N )). However, since {e1 , . . . , ek , x} is a set of k + 1 linearly independent vectors in N , there exists a constant α (depending only on k) such that d() ≥ αμ1 (N ) · · · μk (N )μk+1 (N ). Therefore, μ1 (π(N )) ≥

αμ1 (N ) · · · μk (N ) μk+1 (N ). d(K)

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)···μk (N ) Take C  to be the quantity αμ1 (Nd(K) . It is independent of N because N contains K whose rank is k and so μ1 (N ), . . . , μk (N ) are bounded above by a constant independent of N . 

Remark 2.5 Let F be a totally real number field and O be its ring of integers. We define the minimum of a totally positive O-lattice L by μ1 (L) = min{tr(Q(v)) : 0 = v ∈ L}, where tr is the trace from F to Q. Since the main theorem in [7] holds in this general setting (see [7], Remark (ii), p. 100), Theorem 2.1 holds for O-lattices as well.

3 New almost 2-universal lattices of rank 6 In this section, all lattices are assumed to be positive definite. A lattice M is called n-universal if M represents all lattices of rank n, and is called almost n-universal if it represents all but finitely many lattices of rank n. It is not hard to see that an almost nuniversal lattice must have rank at least n + 3. It was shown in [10] that there are only finitely many isometry classes of almost n-universal lattices of rank n + 3 for n ≥ 2. When n = 2, there are exactly 11 isometry classes of 2-universal lattices of rank 5; see [8]. Therefore there are infinitely many isometry classes of almost 2-universal lattices of rank 6 or higher. A natural question is whether every almost 2-universal lattice of rank 6 contains some almost 2-universal lattices of rank 5. This leads to the following definition. Definition 3.1 An almost n-universal lattice M is called new if it does not contain an almost n-universal sublattice of smaller rank. As an application of Theorem 2.3, we shall prove in this section that there are infinitely many new almost 2-universal lattices of rank 6. Lemma 3.2 Let a, b be a pair of relatively prime positive odd integers such that 5  ab, and let M(a, b) be the rank 6 lattice 1, 1, 2, 5, a, b. For any positive integer α, there exists z ∈ M(a, b) such that (1) Q(z) = α; (2) if N is a binary lattice which contains z primitively, then for each p there is a representation of Np by M(a, b)p which extends Zp [z]. Proof Suppose that M(a, b) has the matrix presentation 1, 1, 2, 5, a, b with respect to some orthogonal basis {x1 , . . . , x6 }. Since a and b are fixed throughout this proof, we shall denote M(a, b) simply by M. It is easy to verify that the lemma is true when α = 1; so we assume in the following that α > 1. We claim that there exists ∈ Z with the following properties: (a) α − 2 is represented by Z[x2 , x3 , x4 ] ∼ = 1, 2, 5; whenever p | α. (b) ∈ Z× p

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A direct computation shows that the above claim holds for all α ≤ 25. For the convenience of the readers, we list the choices of for all α ≤ 25 in the following table:

α

1 2 3 4

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 22, 24 11, 13, 15 14, 16, 20, 22 17, 19, 21, 23, 25

For α ≥ 26, here are the choices for : 2 or 4

if 5  α and 2  α

5

if 5  α but 2 | α

2

if 5 | α but 2  α

1

if 5 | α and 2 | α

Since 1, 2, 5 has class number 1 (see, for example, [5]), it suffices to check that α − 2 is represented by the genus of 1, 2, 5, which can be done in a routine manner. So, there exists a vector z ∈ Z[x1 , x2 , x3 , x4 ] such that Q(z) = α

and = B(z, x1 ).

Let N be a binary lattice which contains z primitively. If α ∈ Z× p , then z extends to ⊥ an orthogonal basis {z, u} for Np . Moreover, Zp [z] in Mp represents all elements in Zp . Therefore, Mp contains a vector e such that B(z, e) = 0 and Q(e) = Q(u), and hence there is a representation of Np by Mp which sends z to itself and u to e. From now on, α is assumed to be divisible by p. Let J = Z[z, x1 ] and K be the orthogonal complement of J in M. Note that Jp is unimodular   because d(Jp ) =

α − 2 ∈ Z× p . Suppose that N has a matrix presentation β γ . It suffices to show that for each p, there exists w ∈ Jp satisfying     B(z, z) B(z, w) α β = and γ − δ→Kp . B(z, w) B(w, w) β δ α β

Since z is a primitive vector in the unimodular lattice Jp , there exists w ∈ Jp such that B(z, w) = β; see [11], 82:17. It remains to show that γ − Q(w) is represented by Kp . It suffices to show that Kp is universal, that is, Kp represents all elements in Zp . Note that Kp is split by a unimodular sublattice of rank at least 3. Therefore, Kp is universal when p > 2. Suppose now that p = 2. Then K2 ∼ = a, b, c1 , 2c2  for  , where H is the hyperbolic plane and ∼ . So, K some c1 , c2 ∈ Z× H ⊥ d ⊥ 2d = 2 2  d, d  ∈ Z× 2 . This shows that K2 is also universal.

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Theorem 3.3 There are infinitely many isometry classes of new almost 2-universal lattices of rank 6. Proof Let a, b be a pair of integers satisfying  the hypothesis of Lemma 3.2. Let N be α β a binary lattice with matrix presentation β γ in reduced form, that is, 2|β| ≤ α ≤ γ . If α is sufficiently large, then N→M(a, b) by [6]. Furthermore, for any fixed α, if γ is sufficiently large, then N→M(a, b) by Theorem 2.1 and Lemma 3.2. Consequently, M(a, b) is almost 2-universal. Since there are only finitely many isometry classes of almost 2-universal lattices of rank 5 by [10], there exists an a such that 1, 1, 2, 5, a does not contain any almost 2-universal lattice. Therefore, for all sufficiently large b, M(a, b) does not contain any almost 2-universal lattice of rank 5.  4 Quadratic equations with linear conditions Let Mm,n (Z) be the set of m × n integral matrices and Sn+ (Z) be the set of positive definite symmetric matrices of rank n. The following is a matrix version of Theorem 2.3. + (Z), H ∈ S + (Z), A ∈ M Theorem 4.1 Let M ∈ Sm k,m (Z) and B ∈ Mk,n−k (Z) n−k with k < n such that m ≥ k + 2(n − k) + 3. Then there exists a constant C = C(M, A, B) > 0 satisfying the following property: If the system of Diophantine equations

X t MX = H

and AX = B

(4.1)

has a solution X(p) ∈ Mm,n−k (Zp ) for every prime p and μ1 (H ) > C, then it has a solution X ∈ Mm,n−k (Z). Proof Let T := t (A · adj(M)) and K := t T MT . Then, any T˜ ∈ Mm,n−k (Z) which satisfies the matrix equation   K det(M)B t ˜ ˜ (T , T )M(T , T ) = t (det(M)B) H also satisfies (4.1). Therefore, this theorem follows from Theorem 2.3.



As an example to illustrate the above theorem, let us consider the simplest case in which n = 2 and k = 1. Corollary 4.2 Let a1 , . . . , am be pairwise relatively prime positive odd integers and 1 , . . . , m be any integers such that gcd(1 , . . . , m ) = 1. Suppose that m ≥ 6 and b is a fixed integer. There exists a constant C, depending only on b, the ai s and the j s, such that the system of Diophantine equations  2 = h, a1 x12 + a2 x22 + · · · + am xm 1 x1 + 2 x2 + · · · + m xm = b

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has an integral solution provided (i) h > C and (ii) (h − b)

m

i=1 i

is even.

Proof Let M be the lattice a1 , . . . , am  and {x1 , . . . , xm } be an orthogonal basis of M such that Q(xi ) = ai for all i. Let A = a1 · · · am , and for any i = 1, . . . , m, let Ai := A/ai and M(i) be the orthogonal complement of Z[xi ] in M. Let x := 1 A1 x1 + · · · + m Am xm and Q(x) = α. Then, by virtue of Theorem 2.3, it suffices to show that for each prime p, there exists a vector y ∈ Mp such that   α bA . (4.2) Zp [x, y] ∼ = bA h We first treat the case when p is an odd prime. Without loss of generality, we may assume that p  1 . There exists a primitive vector z ∈ M(1)p such that x − 1 A1 x1 = p t z for a nonnegative integer t. Suppose that p  A1 . Then M(1)p is unimodular of rank ≥ 5, and hence it contains a pair of primitive vectors e, f such that Q(e) = 2 Q(z), Q(f) = h − −2 1 b a1 and B(e, f) = 0. By [9], Corollary 5.4.1, there exists ρ ∈ O(M(1)p ) such that ρ(e) = z. We may then take y to be the vector −1 1 bx1 + ρ(f). If p | A1 , we may assume that p | am and p  Am . In this case, M(m)p is unimodular of rank ≥ 5. Let z be a primitive vector in M(m)p such that p k z = x − m Am xm . By a similar reasoning as above, there exists a vector y ∈ M(m)p such that B(z , y) = p −k bA and Q(y) = h. One can check easily that this vector y satisfies (4.2). Now, we consider the case when p = 2. Note that M2 is unimodular. Suppose that of Z2 [x] in M(1)2 α ∈ Z× 2 . One can readily  check that the orthogonal complement m is even if and only if m i=1 i is odd. Since (h − b) i=1 i is even, there exists a vector w ∈ Z2 [x]⊥ such that Q(w) = h − b2 A2 α −1 . We can then take y to be vector w + bAα −1 x. Finally, assume that α ∈ 2Z2 . Without loss of generality, we may assume that 1 × ⊥ is odd. Let v be the vector x − 1 A1 x1 ∈ M(1)2 . Since Q(v) ∈ Z 2m, Z2 [v] in M(1)2 is a unimodular Z2 -lattice. Moreover, it is even if and only if i=2 i is odd. As is explained in the last paragraph, there exists a vector w in the orthogonal comple2 ment of Z2 [v] in M(1)2 such that Q(w) = h − −2 1 b a1 . Then we may take y to be −1 1 bx1 + w.  References 1. Böcherer, S., Raghavan, S.: On Fourier coefficients of Siegel modular forms. J. Reine Angew. Math. 384, 80–101 (1988) 2. Cassels, J.W.S.: Rational Quadratic Forms. Academic Press, London (1978) 3. Hsia, J.S.: Arithmetic of indefinite quadratic forms. In: Contemporary Mathematics, vol. 249, pp. 1–15. Am. Math. Soc., Providence (1999) 4. Hsia, J., Kitaoka, Y., Kneser, M.: Representations of positive definite quadratic forms. J. Reine Angew. Math. 301, 132–141 (1978)

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5. Jagy, W.C., Kaplansky, I., Schiemann, A.: There are 913 regular ternary forms. Mathematika 44, 332–341 (1997) 6. Jöchner, M.: On the representation theory of positive definite quadratic forms. In: Contemporary Mathematics, vol. 249, pp. 73–86. Am. Math. Soc., Providence (1999) 7. Jöchner, M., Kitaoka, Y.: Representation of positive definite quadratic forms with congruence and primitive conditions. J. Number Theory 48, 88–101 (1994) 8. Kim, B.M., Kim, M.-H., Oh, B.-K.: 2-universal positive definite integral quinary quadratic forms. In: Contemporary Mathematics, vol. 249, pp. 51–62. Am. Math. Soc., Providence (1999) 9. Kitaoka, Y.: Arithmetic of Quadratic Forms. Cambridge Tracts in Mathematics, vol. 106. Cambridge University Press, Cambridge (1993) 10. Oh, B.-K.: The representation of quadratic forms by almost universal forms of higher rank. Math. Z. 244, 399–413 (2003) 11. O’Meara, O.T.: Introduction to Quadratic Forms. Grundlehren der mathematischen Wissenschaften, vol. 117. Springer, Berlin (1963) 12. Schulze-Pillot, R.: Representation by integral quadratic forms—a survey. In: Contemporary Mathematics, vol. 344, pp. 303–321. Am. Math. Soc., Providence (2004)