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Embeddings of Integral Quadratic Forms Rick Miranda Colorado State Embeddings of Integral Quadratic Forms Rick Miranda Colorado State University David R. Morrison University of California, Santa Barbara Copyright c 2009, Rick Miranda and David R. Morrison Preface The authors ran a seminar on Integral Quadratic Forms at the Institute for Advanced Study in the Spring of 1982, and worked on a book-length manuscript reporting on the topic throughout the 1980’s and early 1990’s. Some new results which are proved in the manuscript were announced in two brief papers in the Proceedings of the Japan Academy of Sciences in 1985 and 1986. We are making this preliminary version of the manuscript available at this time in the hope that it will be useful. Still to do before the manuscript is in final form: final editing of some portions, completion of the bibliography, and the addition of a chapter on the application to K3 surfaces. Rick Miranda David R. Morrison Fort Collins and Santa Barbara November, 2009 iii Contents Preface iii Chapter I. Quadratic Forms and Orthogonal Groups 1 1. Symmetric Bilinear Forms 1 2. Quadratic Forms 2 3. Quadratic Modules 4 4. Torsion Forms over Integral Domains 7 5. Orthogonality and Splitting 9 6. Homomorphisms 11 7. Examples 13 8. Change of Rings 22 9. Isometries 25 10. The Spinor Norm 29 11. Sign Structures and Orientations 31 Chapter II. Quadratic Forms over Integral Domains 35 1. Torsion Modules over a Principal Ideal Domain 35 2. The Functors ρk 37 3. The Discriminant of a Torsion Bilinear Form 40 4. The Discriminant of a Torsion Quadratic Form 45 5. The Functor τ 49 6. The Discriminant of a Good Special Torsion Quadratic Form 54 7. The Discriminant-Form Construction 56 8. The Functoriality of GL 65 9. The Discriminant-Form and Stable Isomorphism 68 10. Discriminant-forms and overlattices 71 11. Quadratic forms over a discrete valuation ring 71 Chapter III. Gauss Sums and the Signature 77 1. Gauss sum invariants for finite quadratic forms 77 2. Gauss Sums 80 3. Signature invariants for torsion quadratic forms over Zp 85 4. The discriminant and the Gauss invariant 86 5. Milgram’s theorem: the signature 88 v vi CONTENTS Chapter IV. Quadratic Forms over Zp 91 1. Indecomposable Forms of Ranks One and Two Over Zp 91 2. Quadratic Forms over Zp, p odd 94 3. Relations for Quadratic Forms over Z2 100 4. Normal forms for 2-torsion quadratic forms 105 5. Normal forms for quadratic Z2-modules 112 Chapter V. Rational Quadratic Forms 119 1. Forms over Q and Qp 119 2. The Hilbert norm residue symbol and the Hasse invariant 120 3. Representations of numbers by forms over Q and Qp 122 4. Isometries 124 5. Existence of forms over Q and Qp 124 6. Orthogonal Groups and the surjectivity of (det,spin) 125 7. The strong approximation theorem for the spin group 126 Chapter VI. The Existence of Integral Quadratic Forms 129 1. The monoids Q and Qp 129 2. The surjectivity of Q → T 130 3. Hasse invariants for Integral p-adic Quadratic Forms 133 4. Localization of Z-modules 134 5. Nikulin’s Existence Theorem 135 6. The Genus 140 Chapter VII. Local Orthogonal Groups 141 1. The Cartan-Dieudonn´eTheorem Recalled 141 2. The groups O(L) and O#(L) 142 3. The Generalized Eichler Isometry 146 ε 4. Factorization for Forms Containing Wp,k 149 5. Factorization for Forms Containing Uk 154 6. Factorization for Forms Containing Vk 159 7. Cartan-Dieudonn´e-type Theorems for Quadratic Z2-Modules163 8. Scaling and Spinor Norms 168 9. Computation of Σ#(L) and Σ+(L) for p 6= 2 169 10. Computation of Σ#(L) for p = 2 172 11. Computation of Σ+(L) for p = 2 179 # # 12. Σ(L), Σ (L) and Σ0 (L) in Terms of the Discriminant-Form186 Bibliographical note for Chapter VII 195 Chapter VIII. Uniqueness of Integral Quadratic Forms 197 1. Discriminant Forms and Rational Quadratic Forms 197 2. A consequence of the strong approximation theorem 198 3. Uniqueness of even Z-lattices 201 CONTENTS vii 4. Milnor’s theorem and stable classes 203 5. Surjectivity of the map between orthogonal groups 205 6. Computations in terms of the discriminant-form 206 7. A criterion for uniqueness and surjectivity 209 8. Bibliographical Note for Chapter VIII 215 List of Notation 217 Bibliography 221 CHAPTER I Quadratic Forms and Orthogonal Groups 1. Symmetric Bilinear Forms Let R be a commutative ring with identity, and let F be an R- module. Definition 1.1. An F -valued symmetric bilinear form over R is a pair (L, h−, −i), where L is an R-module, and h−, −i : L × L → F is a symmetric function which is R-linear in each variable. We will often abuse language and refer to h , i as the bilinear form, and say that h , i is a symmetric bilinear form over R on L. A symmetric bilinear form is also sometimes referred to as an inner product. 2 Let Symm L be the quotient of L⊗R L by the submodule generated by all tensors of the form x ⊗ y − y ⊗ x; Symm2 L is the 2nd symmetric power of L. An F -valued symmetric bilinear form over R on L can also be defined as an R-linear map from Symm2 L to F . The first example of such a form is the R-valued symmetric bilinear form on R, which is the multiplication map. Definition 1.2. Let (L, h , i) be an F -valued symmetric bilinear form over R. The adjoint map to (L, h , i) (or to h , i), denoted by Ad, is the R-linear map from L to HomR(L, F ) defined by Ad(x)(y) = hx, yi. We will say that h , i is nondegenerate if Ad is injective, and h , i is unimodular if Ad is an isomorphism. The kernel of h , i, denoted by Kerh , i (or Ker L if no confusion is possible) is the kernel of Ad. Note that h , i is nondegenerate if and only if Kerh , i = (0). If L¯ = L/ Kerh , i, then h , i descends to a nondegenerate form on L¯. In this book we will deal primarily with nondegenerate forms, and we will largely ignore questions involving nondegeneracy. By modding out the kernel, one can usually reduce a problem to the nondegenerate case, so we feel that this is not a serious limitation. 1 2 I. QUADRATIC FORMS AND ORTHOGONAL GROUPS In this book we will often deal with the special case of an R-valued bilinear form on a free R-module. We give this type of form a special name. Definition 1.3. An inner product R-module, or inner product mod- ule over R, is a nondegenerate R-valued symmetric bilinear form (L, h−, −i) over R such that L is a finitely generated free R-module. If R is a field, this is usually referred to as a inner product vector space over R. We sometimes abuse notation and refer to L as the inner product module; the bilinear form h−, −i is assumed to be given. 2. Quadratic Forms Let R be a commutative ring, and let F be an R-module. Definition 2.1. An F -valued quadratic form over R is a pair (L, Q), where L is an R-module and Q : L → F is a function sat- isfying (i) Q(r`) = r2Q(`) for all r ∈ R and ` ∈ L (ii) the function h , iQ : L × L → F defined by hx, yiQ = Q(x + y) − Q(x) − Q(y) is an F -valued symmetric bilinear form on L. Again we often refer to Q as the form on L. The adjoint map AdQ of Q is simply the adjoint map of h , iQ. We say Q is nondegenerate (respectively, unimodular) if h , iQ is. The bilinear form h , iQ is called the associated bilinear form to Q. We denote the kernel of h , iQ by Ker(L, Q) (or just by Ker(L) or Ker(Q) when that is convenient). The kernel of a quadratic form has a refinement called the q-radical of (L, Q). This is defined to be Radq(L, Q) = {x ∈ Ker(L, Q)|Q(x) = 0}. (We denote the q-radical by Radq(L) or Radq(Q) when convenient.) Notice that 2Q(x) = hx, xiQ = 0 for x ∈ Ker(L, Q), so that if multi- plication by 2 is injective in F , then the q-radical coincides with the kernel. When restricted to the kernel of Q, Q is Z-linear, and its q-radical is just the kernel of Q|Ker(Q). Also, since 2Q(x) = 0 for x ∈ Ker(Q), the image of Q|Ker(Q) is contained in the kernel of multiplication by 2 on F . If this kernel is finite of order N, then the “index” of the q-radical of Q in the kernel of Q is a divisor of N. ¯ Note that if L = L/ Radq(L), then Q descends to a quadratic form on L¯ with trivial q-radical. 2. QUADRATIC FORMS 3 The reader will note the difference between our definition of h , iQ 1 and the more usual 2 [Q(x + y) − Q(x) − Q(y)]; this definition requires multiplication by 2 to be an isomorphism on F , and we do not want to restrict ourselves to this case. The price we pay is that not every symmetric bilinear form can occur as the associated bilinear form to some quadratic form. Definition 2.2. An F -valued symmetric bilinear form (L, h , i) over R is even if there exists an F -valued quadratic form Q over R on L such that h , i = h , iQ.
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