Embeddings of Integral Quadratic Forms

229 Pages·2009·1.073 MB·English

by  Rick Miranda, David R. Morrison

Embeddings of Integral Quadratic
Forms
Rick Miranda
Colorado State University
David R. Morrison
University of California, Santa Barbara
Copyright (cid:13)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 ρ 37
k
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 G 65
L
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 Z 85
p
4. The discriminant and the Gauss invariant 86
5. Milgram’s theorem: the signature 88
v
vi CONTENTS
Chapter IV. Quadratic Forms over Z 91
p
1. Indecomposable Forms of Ranks One and Two Over Z 91
p
2. Quadratic Forms over Z , p odd 94
p
3. Relations for Quadratic Forms over Z 100
2
4. Normal forms for 2-torsion quadratic forms 105
5. Normal forms for quadratic Z -modules 112
2
Chapter V. Rational Quadratic Forms 119
1. Forms over Q and Q 119
p
2. The Hilbert norm residue symbol and the Hasse invariant 120
3. Representations of numbers by forms over Q and Q 122
p
4. Isometries 124
5. Existence of forms over Q and Q 124
p
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 Q 129
p
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´e Theorem Recalled 141
2. The groups O(L) and O#(L) 142
3. The Generalized Eichler Isometry 146
4. Factorization for Forms Containing Wε 149
p,k
5. Factorization for Forms Containing U 154
k
6. Factorization for Forms Containing V 159
k
7. Cartan-Dieudonn´e-typeTheoremsforQuadraticZ -Modules163
2
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)andΣ#(L)inTermsoftheDiscriminant-Form186
0
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.
LetSymm2Lbe thequotientof L⊗ Lbythe submodulegenerated
R
by all tensors of the form x⊗y−y⊗x; Symm2L 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 Symm2L 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
formoverR. Theadjoint mapto(L,h, i)(ortoh, i),denotedbyAd,is
the R-linear map from L to Hom (L,F) defined by Ad(x)(y) = hx,yi.
R
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 KerL 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. Aninner product R-module,orinner product mod-
uleoverR,isanondegenerateR-valuedsymmetricbilinearform(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 , i : L×L → F defined by
Q
hx,yi = Q(x+y)−Q(x)−Q(y)
Q
is an F-valued symmetric bilinear form on L.
Again we often refer to Q as the form on L. The adjoint map Ad
Q
of Q is simply the adjoint map of h , i . We say Q is nondegenerate
Q
(respectively, unimodular) if h , i is. The bilinear form h , i is called
Q Q
the associated bilinear form to Q.
We denote the kernel of h , i by Ker(L,Q) (or just by Ker(L) or
Q
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
Rad (L,Q) = {x ∈ Ker(L,Q)|Q(x) = 0}.
q
(We denote the q-radical by Rad (L) or Rad (Q) when convenient.)
q q
Notice that 2Q(x) = hx,xi = 0 for x ∈ Ker(L,Q), so that if multi-
Q
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| . Also, since 2Q(x) = 0 for x ∈ Ker(Q), the
Ker(Q)
image of Q| is contained in the kernel of multiplication by 2 on
Ker(Q)
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/Rad (L), then Q descends to a quadratic form
q
¯¯
on L with trivial q-radical.