Extensions of representations of integral quadratic forms

EXTENSIONS OF REPRESENTATIONS OF INTEGRALQUADRATIC FORMSWAI KIU CHAN, BYEONG MOON KIM, MYUNG-HWAN KIM,AND BYEONG-KWEON OHAbstract. Let N and M be quadratic Z-lattices, and K be a sublatticeof N. A representation σ : K→M is said to be extensible to N if thereexists a representation ρ : N→M such that ρ| K = σ. We prove in thispaper a local-global principle for extensibility of representation, whichis a generalization of the main theorems on representations by positivedefinite Z-lattices by Hsia-Kitaoka-Kneser [4] and Jöchner- Kitaoka [7].Applications to almost n-universal lattices and systems of quadraticequations with linear conditions are discussed.1. IntroductionLet f(x) = ∑ mi,j=1 a ijx i x j be a quadratic form in m variables x 1 , . . . , x m ,where (a ij ) is a m×m symmetric matrix over Q. We say that f represents aquadratic form g(y) = ∑ ni,j=1 b ijy i y j in n variables y 1 , . . . , y n over Z if thereexist b 1 , . . . , b n ∈ Z m such thatf(y 1 b 1 + · · · + y n b n ) = g(y).The representation problem asks for a complete determination of the set offorms that are represented by f. This is equivalent to deciding when thematrix equation (which can be viewed as a system of Diophantine equations)t T (a ij )T = (b ij )has a solution T ∈ M m×n (Z). For the recent development of this problem,we refer the readers to the recent surveys [3] and [12] and the referencescited there.Using the geometric language of quadratic spaces and lattices (see [8] and[11]), the representation problem of quadratic forms can be rephrased asfollows. The equivalence class of the quadratic form f(x) corresponds to theisometry class of a Z-lattice M with a basis {v 1 , . . . , v m } and a symmetricbilinear form B such that B(v i , v j ) = a ij . A Z-lattice N of rank n with2000 Mathematics Subject Classification. Primary 11E12, 11E20.Key words and phrases. Extension of representations, integral quadratic forms.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).1

2 W. K. Chan, B. M. Kim, M.-H. Kim and B.-K. Ohbilinear form B ′ is said to be represented by M, written N → M, if thereexists a Z-linear map σ : N → M such that B(σ(x), σ(y)) = B ′ (x, y) forall x, y ∈ N. This map σ is called a representation of N by M. If g(y) isthe quadratic form corresponding to N, then N → M if and only if g(y)is represented by f(x). Therefore, the representation problem of quadraticforms is equivalent to deciding when a Z-lattice is represented by anotherZ-lattice.Throughout this paper, we shall conduct our discussion in the abovegeometric language. Any unexplained terminologies and notations can befound in [8] and [11]. The term lattice always refers to a Z-lattice on anon-degenerate quadratic space with a bilinear form B and its associatedquadratic map Q. Note that under this condition every representation of alattice by another lattice must be injective. Let p be a prime. For a latticeM, its p-adic completion M p is the Z p -lattice Z p ⊗M, where Z p is the ring ofp-adic integers. The genus of M, denoted gen(M), is the set of all lattices Lon the space QM such that L p∼ = Mp for all primes p. A necessary conditionfor N → M is that N p → M p for all p and R ⊗ N → R ⊗ M, and we say thatN is represented by gen(M). If this is the case, then Hasse-Minkowski’slocal-global principle asserts that there will be a representation of the spaceQN by QM. Therefore, when discussing the representation problem, thereis 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 restrictionof an isometry of QM.Let K be a sublattice of N and σ : K → M be a representation. Wesay that a representation ρ : N → M is an extension of σ if ρ| K = σ.This definition has its analog for Z p -lattices. In the next section, we shallprove a result concerning extension of representation which can be viewedas a generalization of the main representation theorems for positive definitelattices proved by Hsia, Kitaoka and Kneser [4] and by Jöchner and Kitaoka[7]. In Sections 3 and 4 we shall apply our results to study almost 2-universallattices and quadratic equations with linear conditions.We close this section with some additional notations and terminologieswhich will be used throughout this paper. If M is a positive definite latticeof rank m, for any i = 1, . . . , m, the i-th successive minimum of M is denotedby µ i (M). By reduction theory [2, Chapter 12], there exists a constant α,depending only on m, such that d(M) > αµ 1 (M) · · · µ m (M).Let R be Z or Z p . For a basis {v 1 , . . . , v n } of a R-lattice M, we writeM ∼ = (B(v i , v j )).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

Extensions of representations of integral quadratic forms 3{v 1 , . . . , v n }, we call M diagonal and simply writeM ∼ = 〈Q(v 1 ), . . . , Q(v n )〉.The determinant of the matrix (B(x i , x j )) is called the discriminant of M,denoted d(M). Unless stated otherwise, every R-lattice M is assumed tobe integral in the sense that B(M, M) ⊆ R. A representation σ : N → Mof R-lattices is said to be primitive if σ(N) is a direct summand of M asR-modules. If φ and ψ are representations from N to M and a ∈ R, wewrite φ ≡ ψ mod aM if φ(x) − ψ(x) ∈ aM for all x ∈ N.2. Extensibility of representationsThroughout 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 QKand W be the orthogonal complement of U in V . Let π be the orthogonalprojection 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 ofprimes which contains all primes that divide 2d(M)d(K). If the orthogonalcomplement of σ(K) in M is positive definite, then there exists a constantC 1 = C 1 (K, M, n, T, s) with the following property:Suppose that for each prime p, there exists a representationρ (p) : N p →M p which extends σ. If µ 1 (π(N)) > C 1 , thenthere exists a representation ρ : N→M which extends σ andρ ≡ ρ (p) mod p s M p for all p ∈ T .Proof. Suppose that the theorem holds for the special case when K ⊆ N ∩Mand 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) : N p →σ −1 (M) p to K p is theinclusion map. Let C 1 be the constant for K, σ −1 (M), n, T and s. Ifµ 1 (π(N)) > C 1 , then there exists a representation γ : N→σ −1 (M), whichis an extension of the inclusion K ↩→ σ −1 (M), such that γ ≡ σ −1 ρ (p) modp s σ −1 (M p ) for all p ∈ T . Let ρ be the isometry σγ. Then ρ| K = σ andhence ρ extends σ. Moreover, ρ ≡ ρ (p) mod p s M p for all p ∈ T .So, from now on we assume that σ : K→M is the inclusion map. We mayenlarge T to contain all the primes dividing d(π(M)). Then select a positiveinteger a such that p a π(M p ) ⊆ p s M p for all p ∈ T . Pick a prime q outsideT and a positive integer e. Let C 1 be the constant C 1 (m, n, π(M), T, a, q, e)obtained from the main theorem in [7, page 96]. The prime q and the integere will not play any role in the subsequent discussion; they are chosen so thatthe constant C 1 can be obtained. Note that C 1 does not depend on N.

4 W. K. Chan, B. M. Kim, M.-H. Kim and B.-K. OhFor each p ∈ T , write ρ (p) = 1 Up ⊥ τ (p) for some τ (p) ∈ O(W p ). Nowsuppose that µ 1 (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(M p ) for all p ∈ T . Let ρ bethe isometry 1 U ⊥ γ ∈ O(V ).Let x be any vector in N. If p ∉ T , then Q p K p ∩ N p = K p and π(M p ) ⊆M p . Therefore, ρ(x) = (x − π(x)) + γ(π(x)) ∈ M p . If p ∈ T , thenρ(x) − ρ (p) (x) = γ(π(x)) − τ (p) (π(x)) ∈ p a π(M p ) ⊆ p s M p .As a result, ρ(N) ⊆ M and ρ ≡ ρ (p) mod p s M p for all p ∈ T . It is clear thatρ extends σ.□Remark 2.2. Theorem IV’ of [1] gives a version of the above theorem withadditional primitivity conditions imposed on both the ρ (p) and ρ.When M itself is positive definite, the following version of Theorem 2.1will be found more useful in later discussion.Theorem 2.3. Let σ : K→M be a representation. Suppose that M ispositive definite and that m ≥ k+2(n−k)+3. Let s be a positive integer andT be a finite set of primes which contains all primes that divide 2d(M)d(K).There exists a constant C 2 = C 2 (K, M, n, T, s) with the following property:Suppose that for each prime p, there exists a representationρ (p) : N p −→ M p which extends σ. If µ k+1 (N) > C 2 , thenthere exists a representation ρ : N→M which extends σ andρ ≡ ρ (p) mod p s M p 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. Thenthere exists a constant C ′ = C ′ (K) such that µ 1 (π(N)) > C ′ µ k+1 (N).Proof. The lemma is clear when k = 0. Suppose that k ≥ 1, and fix abasis {e 1 , . . . , e k } of K. Let x ∈ N such that π(x) is a minimal vectorof π(N). Let Λ be the lattice spanned by {e 1 , . . . , e k , x}. Then d(Λ) =d(K)Q(π(x)) = d(K)µ 1 (π(N)). However, since {e 1 , . . . , e k , x} is a set ofk+1 linearly independent vectors in N, there exists a constant α (dependingonly on k) such thatd(Λ) ≥ αµ 1 (N) · · · µ k (N)µ k+1 (N).Therefore,µ 1 (π(N)) ≥ αµ 1(N) · · · µ k (N)µ k+1 (N).d(K)

Extensions of representations of integral quadratic forms 5Take C ′ to be the quantity αµ 1(N)···µ k (N)d(K). It is independent of N because Ncontains K whose rank is k and so µ 1 (N), . . . , µ k (N) are bounded above bya constant independent of N.□Remark 2.5. Let F be a totally real number field and O be its ring ofintegers. We define the minimum of a totally positive O-lattice L by µ 1 (L) =min{tr(Q(v)) : 0 ≠ v ∈ L}, where tr is the trace from F to Q. Since themain theorem in [7] holds in this general setting (see [7, Remark (ii), page100]), Theorem 2.1 holds for O-lattices as well.3. New almost 2-universal lattices of rank 6In this section, all lattices are assumed to be positive definite. A latticeM is called n-universal if M represents all lattices of rank n, and is calledalmost n-universal if it represents all but finitely many lattices of rank n.It is not hard to see that an almost n-universal lattice must have rank atleast n + 3. It was shown in [10] that there are only finitely many isometryclasses 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 [9].Therefore there are infinitely many isometry classes of almost 2-universallattices 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 rank5. This leads to the following definition.Definition 3.1. An almost n-universal lattice M is called new if it does notcontain an almost n-universal sublattice of smaller rank.As an application of Theorem 2.3, we shall prove in this section that thereare 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 suchthat 5 ∤ ab, and let M(a, b) be the rank 6 lattice 〈1, 1, 2, 5, a, b〉. For anypositive 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 pthere is a representation of N p by M(a, b) p which extends Z p [z].Proof. Suppose that M(a, b) has the matrix presentation 〈1, 1, 2, 5, a, b〉 withrespect to some orthogonal basis {x 1 , . . . , x 6 }. Since a and b are fixedthroughout this proof, we shall denote M(a, b) simply by M. It is easyto verify that the lemma is true when α = 1; so we assume in the followingthat α > 1. We claim that there exists ɛ ∈ Z with the following properties:(a) α − ɛ 2 is represented by Z[x 2 , x 3 , x 4 ] ∼ = 〈1, 2, 5〉;(b) ɛ ∈ Z × p whenever p | α.

6 W. K. Chan, B. M. Kim, M.-H. Kim and B.-K. OhA direct computation shows that the above claim holds for all α ≤ 25.For the convenience of the readers, we list the choices of ɛ for all α ≤ 25 inthe following table:ɛα1 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 22, 242 11, 13, 153 14, 16, 20, 224 17, 19, 21, 23, 25For α ≥ 26, here are the choices for ɛ:ɛ2 or 4 if 5 ∤ α and 2 ∤ α5 if 5 ∤ α but 2 | α2 if 5 | α but 2 ∤ α1 if 5 | α and 2 | αSince 〈1, 2, 5〉 has class number 1 (see, for example, [5]), it suffices to checkthat α − ɛ 2 is represented by the genus of 〈1, 2, 5〉, which can be done in aroutine manner.So, there exists a vector z ∈ Z[x 1 , x 2 , x 3 , x 4 ] such thatQ(z) = α and ɛ = B(z, x 1 ).Let N be a binary lattice which contains z primitively. If α ∈ Z × p , thenz extends to an orthogonal basis {z, u} for N p . Moreover, Z p [z] ⊥ in M prepresents all elements in Z p . Therefore, M p contains a vector e such thatB(z, e) = 0 and Q(e) = Q(u), and hence there is a representation of N p byM p which sends z to itself and u to e.From now on, α is assumed to be divisible by p. Let J = Z[z, x 1 ] andK be the orthogonal complement of J in M. Note that J p is unimodularbecause d(J p ) = α − ɛ 2 ∈ Z × p . Suppose that N has a matrix presentation( ) α β. It suffices to show that for each p, there exists w ∈ J p satisfyingβ γ( ) ( )B(z, z) B(z, w) α β= and 〈γ − δ〉→K p .B(z, w) B(w, w) β δSince z is a primitive vector in the unimodular lattice J p , there existsw ∈ J p such that B(z, w) = β; see [11, 82:17]. It remains to show thatγ − Q(w) is represented by K p . It suffices to show that K p is universal, thatis, K p represents all elements in Z p . Note that K p is split by a unimodularsublattice of rank at least 3. Therefore, K p is universal when p > 2. Supposenow that p = 2. Then K 2∼ = 〈a, b, c1 , 2c 2 〉 for some c 1 , c 2 ∈ Z × 2 . So, K 2 ∼ =

Extensions of representations of integral quadratic forms 7H ⊥ 〈d〉 ⊥ 〈2d ′ 〉, where H is the hyperbolic plane and d, d ′ ∈ Z × 2 . This showsthat K 2 is also universal.□Theorem 3.3. There are infinitely many isometry classes of new almost2-universal lattices of rank 6.Proof. Let a, b be a pair of integers satisfying the hypothesis( )of Lemmaα β3.2. Let N be a binary lattice with matrix presentation in reducedβ γform, 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) byTheorem 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-universallattices of rank 5 by [10], there exists an a such that 〈1, 1, 2, 5, a〉 does notcontain 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 conditionsLet M m,n (Z) be the set of m × n integral matrices and S + n (Z) be the setof positive definite symmetric matrices of rank n. The following is a matrixversion of Theorem 2.3.Theorem 4.1. Let M ∈ S + m(Z), H ∈ S + n−k (Z), A ∈ M k,m(Z) and B ∈M k,n−k (Z) with k < n such that m ≥ k + 2(n − k) + 3. Then there existsa constant C = C(M, A, B) > 0 satisfying the following property: If thesystem of Diophantine equations(4.1) X t MX = H and AX = Bhas a solution X (p) ∈ M m,n−k (Z p ) for every prime p and µ 1 (H) > C, thenit has a solution X ∈ M m,n−k (Z).Proof. Let T := t (A · adj(M)) and K := t T MT . Then, any ˜T ∈ M m,n−k (Z)which satisfies the matrix equation()t (T, ˜T )M(T, ˜T K det(M)B) =t (det(M)B) Halso satisfies (4.1). Therefore, this theorem follows from Theorem 2.3.□As an example to illustrate the above theorem, let us consider the simplestcase in which n = 2 and k = 1.Corollary 4.2. Let a 1 , . . . , a m be pairwise relatively prime positive odd integersand l 1 , . . . , l m be any integers such that gcd(l 1 , . . . , l m ) = 1. Suppose

8 W. K. Chan, B. M. Kim, M.-H. Kim and B.-K. Ohthat m ≥ 6 and b is a fixed integer. There exists a constant C, dependingonly on b, the a i s and the l j s, such that the system of Diophantine equations{a1 x 2 1 + a 2x 2 2 + · · · + a mx 2 m = hl 1 x 1 + l 2 x 2 + · · · + l m x m = bhas an integral solution provided (i) h > C and (ii) (h − b) ∏ mi=1 l i is even.Proof. Let M be the lattice 〈a 1 , . . . , a m 〉 and {x 1 , . . . , x m } be an orthogonalbasis of M such that Q(x i ) = a i for all i. Let A = a 1 · · · a m , and for anyi = 1, . . . , m, let A i := A/a i and M(i) be the orthogonal complement ofZ[x i ] in M. Letx := l 1 A 1 x 1 + · · · + l m A m x m .and Q(x) = α. Then, by virtue of Theorem 2.3, it suffices to show that foreach prime p, there exists a vector y ∈ M p such that( )(4.2) Z p [x, y] ∼ α bA=.bA hWe first treat the case when p is an odd prime. Without loss of generality,we may assume that p ∤ l 1 . There exists a primitive vector z ∈ M(1) p suchthat x−l 1 A 1 x 1 = p t z for a nonnegative integer t. Suppose that p ∤ A 1 . ThenM(1) p is unimodular of rank ≥ 5, and hence it contains a pair of primitivevectors e, f such that Q(e) = Q(z), Q(f) = h − l −21 b2 a 1 and B(e, f) = 0. By[8, Corollary 5.4.1], there exists ρ ∈ O(M(1) p ) such that ρ(e) = z. We maythen take y to be the vector l −11 bx 1 + ρ(f).If p | A 1 , we may assume that p | a m and p ∤ A m . In this case, M(m) p isunimodular of rank ≥ 5. Let z ′ be a primitive vector in M(m) p such thatp k z ′ = x − l m A m x m . By a similar reasoning as above, there exists a vectory ∈ M(m) p such thatB(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 M 2 is unimodular. Supposethat α ∈ Z × 2 . One can readily check that the orthogonal complementof Z 2 [x] in M(1) 2 is even if and only if ∏ mi=1 l i is odd. Since (h − b) ∏ mi=1 l iis even, there exists a vector w ∈ Z 2 [x] ⊥ such that Q(w) = h − b 2 A 2 α −1 .We can then take y to be vector w + bAα −1 x.Finally, assume that α ∈ 2Z 2 . Without loss of generality, we may assumethat l 1 is odd. Let v be the vector x − l 1 A 1 x 1 ∈ M(1) 2 . Since Q(v) ∈ Z × 2 ,Z 2 [v] ⊥ in M(1) 2 is a unimodular Z 2 -lattice. Moreover, it is even if andonly if ∏ mi=2 l i is odd. As is explained in the last paragraph, there exists avector w in the orthogonal complement of Z 2 [v] in M(1) 2 such that Q(w) =h − l −21 b2 a 1 . Then we may take y to be l −11 bx 1 + w. □

Extensions of representations of integral quadratic forms 9References[1] S. Böcherer and S. Raghavan, On Fourier coefficients of Siegel modular forms, J.Reine Angew. Math. 384 (1988), 80-101.[2] J. W. S. Cassels, Rational quadratic forms, Academic Press, London, 1978.[3] J. S. Hsia, Arithmetic of indefinite quadratic forms, Contemporary Math. 249(1999), Amer. Math. Soc., Providence, RI, 1-15.[4] J. Hsia, Y. Kitaoka, and M. Kneser, Representations of positive definite quadraticforms, J. Reine Angew. Math. 301 (1978), 132–141.[5] W. C. Jagy, I. Kaplansky, and A. Schiemann, There are 913 regular ternary forms,Mathematika 44 (1997), 332-341.[6] M. Jöchner, On the representation theory of positive definite quadratic forms, ContemporaryMath. 249 (1999), 73-86.[7] M. Jöchner and Y. Kitaoka, Representation of positive definite quadratic formswith congruence and primitive conditions, J. Number Theory 48 (1994), 88-101.[8] Y. Kitaoka, Arithmetic of quadratic forms, Cambridge Tracts in Mathematics 106,Cambridge University Press, 1993.[9] B. M. Kim, M.-H. Kim, and B.-K. Oh, 2-universal positive definite integral quinaryquadratic forms, Contemporary Math. 249 (1999), 51-62.[10] B.-K. Oh, The representation of quadratic forms by almost universal forms ofhigher rank, Math. Z. 244 (2003), 399-413.[11] O.T. O’Meara, Introduction to quadratic forms, Grundlehren der mathematischenWissenschaften 117, Springer-Verlag, Berlin, 1963.[12] R. Schulze-Pillot, Representation by integral quadratic forms - a survey, ContemporaryMath. 344 (2004), Amer. Math. Soc., Providence, RI, 303-321.Department of Mathematics and Computer Science, Wesleyan University,Middletown CT, 06459 USAE-mail address: wkchan@wesleyan.eduDepartment of Mathematics, Kangnung National University, Kangwondo210-702, KoreaE-mail address: kbm@knusun.kangnung.ac.krDepartment of Mathematical Science, Seoul National University, Seoul151-747, KoreaE-mail address: mhkim@math.snu.ac.krDepartment of Applied Mathematics, Sejong University, Seoul 143-747, KoreaE-mail address: bkoh@sejong.ac.kr