FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 10 definitions, 8 theorems, 4 propositions, 5 lemmas, 6 corollaries, 8 examples, 4 constructions, 3 remarks, 1 convention and 1 criterion.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction / notation
In this chapter \(L\) , \(L ^ { \prime }\) , \(L ^ { \prime \prime }\) , \(M\) are free \(\mathbb { Z }\) -modules of full rank within a quadratic innerproduct space \(( V , q )\) over \(\nsubseteq\) . In particular, \(V = L _ { \mathbb { Q } } = L _ { \mathbb { Q } } ^ { \prime } = L _ { \mathbb { Q } } ^ { \prime \prime } = M _ { \mathbb { Q } }\) . We write \(b , b ^ { \prime } , \ldots\) , for the polar forms of \(q , q ^ { \prime } , \ldots\) . Moreover the forms \(b , b ^ { \prime } , \ldots\) , induced on \(L , L ^ { \prime } , \ldots\) are assumed to be integer-valued. The localizations of \(V\) are \(( V _ { p } , q _ { p } )\) containing the local lattices \(L _ { p } , L , L _ { p } ^ { \prime } , \ldots\) .
Definition / construction / notation
As explained in Section 14.1, the appropriate way to compare integral lattices in a given genus is to consider these as a full rank lattice of a fixed (quadratic) inner product space over the rational numbers, say \(V\) . Observe that this fixes the signature so that the place at infinity does not play a role. Isometric lattices \(L , L ^ { \prime }\) have isometric localizations \(L _ { v } \simeq L _ { v } ^ { \prime }\) and so by definition belong to the same genus and belong to the localization \(V _ { v }\) of \(V\) .
Proposition 14.1.1
Proposition 14.1.1. Let \(L , M \subset V\) be free ℤ-submodules of \(V\) of maximal rank. Then 1. \(M = \gamma ( L )\) for some vector space isomorphism \(\gamma : V \to V\) . 2. \(L _ { p } = M _ { p }\) for almost all primes \(p\) . 3. \(L _ { p } = M _ { p }\) for all primes \(p\) if and only if \(L = M\) .
Theorem 14.1.2
Theorem 14.1.2 (Glueing local lattices, [36, Ch.11, Thm. 1.1]). Let \(V\) be a finite dimensional vector space over \(\mathbb { Q }\) and let \(L\) be \(a\) free \(\mathbb { Z }\) -submodule of \(V\) of maximal rank. For every prime \(p\) , let \(M ^ { ( p ) }\) be a \(p\) -adic free \(\mathbb { Z } _ { p }\) -module of maximal rank in the localization \(V _ { p }\) of \(V\) . Then there exists a free \(\mathbb { Z }\) -submodule \(M\) of \(V\) of maximal rank whose localizations are \(M ^ { ( p ) }\) if and only if \[ M ^ { ( p ) } = L _ { p } \quad f o r \ a l m o s t \ a l l \ p r i m e s \ p . \] If \(M\) exists, it is uniquely determined by the condition \(M ^ { ( p ) } = M _ { p }\) for all primes \(p\) .
Definition / construction / notation
When lattices in one genus are identified inside a fixed rational quadratic space \(V\), two sublattices are called equivalent if a rational isometry of \(V\) sends one to the other; they are genus-equivalent when this holds locally at every prime.
Definition / construction / notation
If we follow the preceding convention, first of all, two sublattices \(L ^ { \prime } , L ^ { \prime \prime }\) of \(V\) are isometric precisely if there is a vector space isometry sending \(L ^ { \prime }\) to \(L ^ { \prime \prime }\) in which case we call \(L ^ { \prime }\) and \(L ^ { \prime \prime }\) equivalent lattices. Secondly, \(L ^ { \prime }\) and \(L ^ { \prime \prime }\) belong to the same genus precisely if for all primes \(p\) local isometries \(g _ { p }\) of \(V _ { p }\) exist with \(g _ { p } L _ { p } ^ { \prime } = L _ { p } ^ { \prime \prime }\) .
Definition / construction / notation
It is natural to consider a more restricted form of equivalence between sublattices \(L ^ { \prime } , L ^ { \prime \prime }\) of \(V\) , namely equivalence under the rotation group \({ \mathsf { S O } } ( V , q )\) in which case we call the sublattices properly equivalent. The isometry group of the vector space \(V\) always contains isometries other than rotations, e.g., if \(\{ e _ { 1 } , \ldots , e _ { n } \}\) is an orthogonal basis, the linear map \(\tau\) sending \(e _ { 1 }\) to \(- e _ { 1 }\) and fixing all other basis elements is such an isometry. Then, by definition, the lattices \(L\) and \(L ^ { \prime } = \tau ( L )\) are equivalent, but they need not be properly equivalent as shown in Example 3 below.
Example / comparison
The above argument shows that any form equivalent to a diagonal form is also properly equivalent to it.
Example / comparison
If \(\dim V\) is odd, the isometry −id has determinant \(^ { - 1 }\) and all lattices are preserved by it. So in this case there is no difference between proper equivalence classes and equivalence classes.
Example / comparison
In Example 6.5.5.4 we showed that \(\pm\) id are the only isometries of the integral quadratic form \(q ( x , y ) = a x ^ { 2 } + 2 x y + c y ^ { 2 }\) on \(\mathbb { Z } ^ { 2 }\) with \(a , c \in \mathbb { Z }\) and \(a \ge 2\) and \(c > a\) . Hence \({ \mathsf { S O } } ( q ) = { \mathsf { O } } ( q )\) . Now \(V = \mathbb { Q } e _ { 1 } \oplus \mathbb { Q } ( e _ { 1 } - a e _ { 2 } )\) and the reflection \(\tau\) in \(V\) with matrix \(\tau = { \binom { - 1 } { 0 } } ^ { - 2 a ^ { - 1 } } 1\) with respect to \(e _ { 1 } , e _ { 2 }\) is in \(\operatorname { O } ( V , q )\) , so maps \(\mathbb { Z } ^ { 2 }\) to the equivalent lattice \(\tau ( \mathbb { Z } ^ { 2 } )\) with basis \(\{ e _ { 1 } , - ( 2 / a ) e _ { 1 } + e _ { 2 } \}\) , and form (in corresponding coordinates) \(q ^ { \prime } = a u ^ { 2 } - 2 u v + c v ^ { 2 }\) . It is, however, not properly equivalent to \(q\) . This last example leads to the following observation.
Lemma 14.1.4
Lemma 14.1.4. Let \(L\) be a sublattice of \(V\) of maximal rank. If every automorphism of \(L\) is a rotation, the equivalence class of \(L\) splits into two proper equivalence classes. If on the contrary \(L\) admits an isometry with determinant \(^ { - 1 }\) , then every lattice equivalent to \(L\) is also properly equivalent to it. In particular, if the genus of \(L\) consists of one equivalence class, then SO(??) is of index 2 in \(\mathrm { O } ( L )\) .
Construction / comparison
For \(p\)-adic lattices, equivalence and proper equivalence coincide. Consequently two global lattices lie in the same genus exactly when at every place one can carry one localization to the other by an element of the local special orthogonal group.
Corollary 14.1.5
Corollary 14.1.5. Let \(L ^ { \prime }\) be a lattice in \(V\) of maximal rank. 1. For every prime \(p\) let \(g _ { p } ^ { \prime } \in \mathsf { S O } ( V _ { p } , q _ { p } )\) . If \(g _ { p } ^ { \prime } \in \mathsf { S O } ( L _ { p } ^ { \prime } )\) for almost all \(p\) , then there is a unique lattice \(L ^ { \prime \prime }\) in \(V\) for which \[ L _ { p } ^ { \prime \prime } = g _ { p } ^ { \prime } L _ { p } ^ { \prime } \quad f o r \ a l l \ p . \] In particular \(L ^ { \prime }\) and \(L ^ { \prime \prime }\) belong to the same genus. 2. Conversely, if \(L ^ { \prime }\) and \(L ^ { \prime \prime }\) belong to the same genus, then for every prime \(p\) there exists \(g _ { p } ^ { \prime } \in \mathsf { S O } ( V _ { p } , q _ { p } )\) with \(L _ { p } ^ { \prime \prime } = g _ { p } ^ { \prime } L _ { p } ^ { \prime }\) , and \(g _ { p } ^ { \prime } \in \mathsf { S O } ( L _ { p } ^ { \prime } )\) for almost all \(p\) .
Definition 14.2.1
Definition 14.2.1. We say that two sublattices \(L ^ { \prime } , L ^ { \prime \prime }\) of \(( V , q )\) (maximal rank) are spinor-equivalent or belong to the same spinor genus if there exist \(\gamma ^ { \prime } \in\) \({ \mathsf { S O } } ( V , q )\) and \(g _ { p } \in \mathsf { S O } ^ { + } ( V _ { p } , q _ { p } )\) for all primes \(p\) , such that \[ L _ { p } ^ { \prime \prime } = \gamma ^ { \prime } { \circ } g _ { p } ( L _ { p } ^ { \prime } ) \mathrm { ~ f o r ~ a l l ~ p r i m e s ~ } p . \]
Lemma 14.2.2
Lemma 14.2.2. 1. Properly equivalent lattices are spinor equivalent; 2. Spinor equivalent lattices are in the same genus.
Convention / notation
For a lattice and its localizations, the book extends the notation for the image of the spinor norm to the corresponding global and local orthogonal groups.
Lemma 14.2.3
Lemma 14.2.3. Suppose that \(n = \dim V \geq 3\) . Then \(\mathsf { S } _ { V }\) contains \(\mathbb { Q } _ { > 0 } \cdot ( \mathbb { Q } ^ { \times } ) ^ { 2 }\) . If \(q\) is indefinite, the spinor norm map is surjective: \(\mathsf { S } _ { V } = \mathsf { D } ( \mathbb { Q } )\) .
Example / comparison
- If \(p\) is odd this holds for any \(p\) -adic lattice of rank \(\geq 2\) since such lattices are diagonalizable by Proposition 10.2.2. Moreover, in this case, by the Cartan–Dieudonn´e theorem in the shape of Corollary 7.2.5, this is the full image, that is, \(\mathsf { S } _ { L } = \mathbb { Z } _ { p } ^ { \times } \cdot ( \mathbb { Q } _ { p } ^ { \times } ) ^ { 2 }\) .
Example / comparison
Consider the integral lattice \(L = A _ { 2 }\) with quadratic form \(x ^ { 2 } + x y + y ^ { 2 }\) . This lattice is positive definite and has discriminant 3. For primes \(p \neq 3\) the lattice \(L _ { p }\) is unimodular and then, by the argument of example 1, \(\mathsf { S } _ { L _ { p } } = \mathbb { Z } _ { p } ^ { \times } \cdot ( \mathbb { Q } _ { p } ^ { \times } ) ^ { 2 }\) . For the prime 3 the situation is different since the lattice \(L _ { 3 }\) is not unimodular. The basis \(e _ { 1 } = ( 1 , 0 )\) , \(e _ { 2 } = ( - 1 , 2 )\) exhibits \(L _ { 3 } \simeq [ 1 ] \odot\) [3] and the corresponding reflections \(\sigma _ { e _ { 1 } }\) and \(\sigma _ { e _ { 2 } }\) have spinor norm 1, respectively 3. Since \(q ( x e _ { 1 } + y e _ { 2 } ) =\) \(x ^ { 2 } + 3 y ^ { 2 }\) cannot be of the form \(- z ^ { 2 }\) in \(\mathbb { Q } _ { 3 }\) (a square is 1 modulo 3), the group \(\mathsf { S } _ { L _ { 3 } }\) is the cyclic group of order 2 generated by the class of \({ 1 \cdot 3 }\) and so again is strictly smaller than \(\mathsf { D } ( \mathbb { Q } _ { 3 } )\) . The next result demonstrates how to use information about the images of the local spinor norms to deduce that the genus of \(L\) contains only one spinor genus:
Theorem 14.2.5
Theorem 14.2.5 (Spinor equivalence \(=\) genus equivalence). Let \(( V , q )\) be a quadratic inner product space of dimension \(\geq 3\) over \(\mathbb { Q }\) and let \(L \subset V\) be a maximal rank sublattice with \(b _ { q } ( L , L ) \subset \mathbb { Z }\) . If for all primes \(p\) the coset in \(\mathsf { D } ( \mathbb { Q } _ { p } )\) of every \(p\) -adic unit is the spinor norm of some rotation of \(L _ { p }\) , all lattices in the genus of \(L\) are spinor equivalent. In other words, every lattice in the same genus as \(L\) is spinor equivalent to it if \[ \mathsf { S } _ { L _ { p } } = \mathsf { N m } _ { \mathrm { s p i n } } ( \mathsf { S O } ( L _ { p } ) ) \supset \mathbb { Z } _ { p } ^ { \times } \cdot ( \mathbb { Q } _ { p } ^ { \times } ) ^ { 2 } \ w i t h i n \ \mathsf { D } ( \mathbb { Q } _ { p } ) \ ( f o r \ a l l \ p r i m e s \ p ) . \] For a given prime \(p\) the inclusion (14.4) holds if a collection of \(p\) -adic units \(u _ { j }\) , and vectors \(x _ { j } \in L _ { p }\) , \(j \in J\) , exists such that (a) \(q ( x _ { j } ) = u _ { j } p ^ { k }\) for all \(j \in J\) , and some fixed \(k\) , and all reflections \(\sigma _ { x _ { j } }\) preserve the lattice \(L _ { p }\) ; (b) the set of pairwise products \(u _ { j _ { 1 } } u _ { j _ { 2 } }\) generate \(\mathbb { Z } _ { p } ^ { \times } \cdot ( \mathbb { Q } _ { p } ^ { \times } ) ^ { 2 }\) . This occurs for instance in the following cases: 1. In case \(p \neq 2\) : at least one homogeneous summand of exponent \(k\) and rank \(\geq 2\) splits off from \(L _ { p }\) . This holds if for example \(\ell ( { \mathrm { d g } _ { L _ { p } } } ) \le \mathrm { r a n k } ( L ) - 2\) . 2. In case \(p = 2\) : (a) If \(L\) (and hence \(L _ { 2 }\) ) is an even lattice, at least one homogeneous summand of exponent \(k\) and rank \(\geq 2\) splits off from \(L _ { 2 }\) , for example in case \(\ell ( \mathrm { d g } _ { L _ { 2 } } ) \leq\) \(\mathrm { r a n k } ( L ) - 2\) . (b) If ?? (and hence \(L _ { 2 }\) ) is an odd lattice and a summand of one of the following three types splits off from \(L _ { 2 }\) : • either one of the rank two lattices \(U _ { k }\) , \(V _ { k }\) ; • any homogeneous summand of exponent \(k\) and rank \(\geq 3\) ; • a rank 3 lattice of the form \({ \cal M } = \langle u ^ { ( 1 ) } \cdot 2 ^ { k } \rangle \oplus \langle u ^ { ( 2 ) } \cdot 2 ^ { k } \rangle \oplus \langle u ^ { ( 3 ) } \cdot 2 ^ { k + 1 } \rangle\) for some dyadic units \(\boldsymbol { u } ^ { ( i ) }\) , \(i = { 1 , 2 , 3 }\) . This is the case if for example \(\ell ( \mathrm { d g } _ { L _ { 2 } } ) \le \mathrm { r a n k } ( L ) - 3\) or if \(u _ { k }\) or \(v _ { k }\) , \(k \geq 1\) , splits off from \(q _ { L } ^ { \# }\) .
Corollary 14.2.6
Corollary 14.2.6. Let \(( W , q )\) be a quadratic inner product space over \(\mathbb { Q }\) of dimension \(n \geq 3\) and let \(L , L ^ { \prime }\) be maximal rank quadratic sublattices of \(W\) with the same discriminant \(d\) . Suppose that the factorization of \(d\) contains the prime 2 to the power \(< \lfloor \textstyle { \frac { 1 } { 2 } } ( n ^ { 2 } + 1 ) \rfloor\) and any odd prime to the power \(\begin{array} { r } { < \frac { 1 } { 2 } ( n ( n - 1 ) ) } \end{array}\) . Then \(L , L ^ { \prime }\) are spinor equivalent if and only if they are in the same genus. In the case of odd symmetric lattices, the estimate for the power of 2 dividing ?? can be replaced \(b y < \lfloor \frac { 1 } { 2 } ( ( n - 1 ) ^ { 2 } + 1 ) \rfloor\) .
Remark 14.2.7
Remark 14.2.7. The theory of binary forms (lattices of rank 2) cannot be treated with the above approach. See [36, Chapter 14 and Section 13.3] for a method to classify the latter. J. Cassels gives in [36, Section 11.3] a constructive way to calculate the number of spinor genera in a given genus as the order of a certain computable finite group \(G ( L )\) of exponent 2. The latter is defined as follows. Fix a finite set \(P\) of prime numbers such that \(2 \in P\) and \(\mathsf { N m } _ { \mathrm { s p i n } } ( \mathsf { S O } ( L _ { p } ) ) = \mathbb { Z } _ { p } ^ { \times } \cdot ( \mathbb { Q } _ { p } ^ { \times } ) ^ { 2 }\) for all \(p \notin P\) . Such sets exist: Take for \(P\) the set of primes dividing \(\mathrm { d i s c } ( L )\) and add 2 if \(\mathrm { d i s c } ( L )\) happens to be odd. This follows from Example 14.2.4.1 since \(L _ { p }\) is unimodular for all \(p \in P\) and \(p \neq 2\) . Making use of \(P\) , introduce the groups \[ \begin{array} { r l } & { S = \displaystyle \prod _ { p \in P } \mathsf { S } _ { L _ { p } } , \quad \mathsf { S } _ { L _ { p } } = \mathsf { N m } _ { \mathrm { s p i n } } ( \mathsf { S O } ( L _ { p } ) ) , \qquad R = \displaystyle \prod _ { p \in P } \mathsf { D } ( \mathbb { Q } _ { p } ) } \\ & { T = \{ ( t , \dots , t ) \in R \mid t \in \mathsf { D } ( \mathbb { Q } ) \mathrm { ~ s u c h ~ t h a t ~ } \left\{ t \in \mathsf { D } ( \mathbb { Z } _ { p } ) \mathrm { ~ s i m u l t a n e o u s l y ~ f o r ~ a l l ~ } p \notin P \right\} \quad } . \end{array} \] Here we make use of the map \(\mathsf { D } ( \mathbb { Q } ) \to \mathsf { D } ( \mathbb { Q } _ { p } )\) induced by the embedding \(\mathbb { Q } \hookrightarrow \mathbb { Q } _ { p }\) (which preserves units and squares). Finally set \(G ( L ) = R / S T\) . The result alluded to is as follows:
Theorem 14.2.8
Theorem 14.2.8 (Cassels, [36, §11.3]). The (finite commutative) group \(G ( L )\) does not depend on the choice of \(P\) . It has exponent 2. The set of spinor genera in the genus of \(( L , b )\) forms a principal space under \(G ( L )\) and so its number is equal to the order of the group \(G ( L )\) and hence a power of 2.
Example / comparison
Example 14.2.9. We enlarge the lattice of Example 14.2.4.2 to a rank 3 even lattice as follows. The Gram matrix \[ \left( { \begin{array} { c c c } { 2 } & { 1 } & { 0 } \\ { 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 1 8 } \end{array} } \right) \] defines a quadratic lattice isometric to \(M = L \textcircled { \ ! } \langle 2 { \cdot } 3 ^ { 2 } \rangle\) , \(L \simeq A _ { 2 }\) , (with quadratic form \(q ( x , y , z ) = x ^ { 2 } + x y + y ^ { 2 } + 3 ^ { 2 } z ^ { 2 } )\) . It has discriminant \(2 { \cdot } 3 ^ { 3 }\) . We can take \(P = \{ 2 , 3 \}\) . The local lattice \(M _ { 3 }\) is isometric to \(\left. 2 \right. \mathbb { O } \left. 2 \cdot 3 \right. \mathbb { O } \left. 2 \cdot 3 ^ { 2 } \right.\) . The corresponding reflections in \(e _ { 1 } , e _ { 2 } , e _ { 3 }\) have spinor norms \(1 , 3 , 1\) . Note that \(q ( x , y , z ) = x ^ { 2 } + 3 y ^ { 2 } + 3 ^ { 2 } z ^ { 2 }\) does not represent \(- 1 \cdot ( \mathbb { Q } ^ { \times } ) ^ { 2 }\) since squares modulo 3 are 1. So \(\mathsf { S } _ { M _ { 3 } }\) is the group generated by the class \(3 \cdot ( \mathbb { Q } _ { 3 } ^ { \times } ) ^ { 2 }\) . The local lattice \(M _ { 2 }\) is isometric to \(L _ { 2 } \textsuperscript { \textregistered } \left. 2 \right.\) and so, by Theorem 14.2.5, \(\mathsf { S } _ { M _ { 2 } }\) contains all units. In fact, it is not larger: any reflection has to preserve the indecomposable summands and we have seen in Example 14.2.4.2 that reflections from \(L _ { \mathrm { 2 } }\) have spinor norm a unit, while the reflection \(e _ { 3 } \ \to \ - e _ { 3 }\) has spinor norm 1. There is a subtle interaction between \(S\) and \(T\) in this example. Here \(R = \mathsf { D } ( \mathbb { Q } _ { 2 } ) \times \mathsf { D } ( \mathbb { Q } _ { 3 } )\) contains 32 elements and the group \(S T\) contains 16 elements: the 8 elements \(( \pm 1 , 1 ) , ( \pm 3 , 1 ) , ( \pm 1 , 3 ) , ( \pm 3 , 3 )\) and (via the diagonal action of \(\{ ( 2 , 2 ) , ( 3 , 3 ) \} ~ \subset ~ T\) by coordinate-wise multiplication) we obtain the supplementary set \(\{ ( \pm 2 , - 1 ) , ( \pm 6 , - 1 ) , ( \pm 2 , - 3 ) , ( \pm 6 , - 3 ) \}\) also consisting of 8 elements. Hence \(G ( M ) = R / S T\) is cyclic of order two and so there are two spinor genera. A representing quadratic form for the second spinor genus is given by \(q ^ { \prime } ( x , y , z ) = z ^ { 2 } + 3 ( x ^ { 2 } + x y + y ^ { 2 } )\) . To show this, observe first of all that \(q ^ { \prime }\) and \(q\) are in the same genus (they are positive definite and have the same localizations at 2 and 3). On the other hand \(q\) and \(q ^ { \prime }\) are not isometric. This can be seen using the unique splitting of positive definite forms into indecomposable forms (Eichler’s theorem 1.12.3). Indeed, the form \(x ^ { 2 } + x y + y ^ { 2 }\) is indecomposable, since this is the case for its localization \(L _ { \mathrm { 2 } }\) (see Section 10.1) and so is \(3 ( x ^ { 2 } + x y + y ^ { 2 } )\) . So the forms \(q\) and \(q ^ { \prime }\) are already given as sums of indecomposable summands, and these are different.
Theorem 14.3.1
Theorem 14.3.1 (Strong approximation for the spin group). Let \(( V , q )\) be a quadratic inner product space over \(\mathbb { Q }\) of dimension \(\geq 3\) with \(q\) indefinite and let \(L \subset V\) be a maximal rank integral lattice in \(V\) . For all prime numbers \(p\) let \(U _ { p } \subset \mathsf { S p i n } ( V _ { p } , q _ { p } )\) be a non-empty open subset such that Then there exists \(u \in \mathsf { S p i n } ( V , q )\) such that its localization \(u _ { p }\) belongs to \(U _ { p }\) for all primes \(p\) .
Corollary 14.3.2
Corollary 14.3.2. Under the assumptions on \(( V , q )\) and \(L\) of Theorem 14.3.1, suppose that for all primes \(p\) we have a non-empty open subset \(W _ { p } \subset \mathsf { S O } ^ { + } ( V _ { p } , q _ { p } )\) with the property that Then there is a ?? ∈ SO \(^ { + } ( V , q )\) such that \(\delta\) belongs to \(W _ { p }\) for all primes \(p\) .
Remark 14.3.3
Remark 14.3.3. 1. There is no analog of Corollary 14.3.2 for the larger group \({ \mathsf { S O } } ( V , q )\) . This is explained in [36] on page 187. 2. In subsequent sections we shall use the above corollary to show that spinor equivalence and equivalence coincide in the indefinite case.
Definition / construction / notation
In this section integral sublattices \(L\subset(V,q)\) are taken in the strong sense \(b_q(L,L)\subset\mathbf Z\); the polar form is abbreviated by \(b\).
Theorem 14.4.1
Theorem 14.4.1 (Spinor equivalence \(=\) proper equivalence). Let \(( V , q )\) be an indefinite quadratic inner product space over \(\mathbb { Q }\) of dimension \(\geq 3\) . Then spinor equivalence for integer valued lattices in \(V\) is the same as proper equivalence.
Theorem 14.4.2
Theorem 14.4.2 (criterion for class number \(1\)). Let \((L,b)\) be an indefinite non-degenerate symmetric lattice of rank at least \(3\). If for every prime \(p\), \(\operatorname{Nm}_{\rm spin}(SO(L_p))\supset \mathbf Z_p^\times(\mathbf Q_p^\times)^2\), then \(L\) has class number \(1\).
Criterion / construction
At a prime \(p\), the local spinor-norm inclusion used in the class-number-one criterion follows if there are vectors \(x_j\in L_p\) of values \(q(x_j)=u_jp^k\) whose reflections preserve \(L_p\) and whose pairwise unit products generate \(D(\mathbf Z_p)\).
Construction / comparison
Under the preceding local criterion, every lattice isometric to \(L\) is properly isometric to it and \(SO(L)\) has index two in \(O(L)\).
Corollary 14.4.3
Corollary 14.4.3. Let ?? be an even non-degenerate indefinite lattice of rank \(r \geq 3\) . Assume that the minimal number of generators of the discriminant group \({ \mathsf { d g } } _ { L } =\) \(L ^ { * } / L\) is at most \(r - 2\) . Then \(L\) has class number 1. In other words: if \(\ell ( \mathrm { d g } _ { L } ) \leq r - 2\) , form then the isometry class of ?? is determined by \(q _ { L } ^ { \# }\) . \(r\) , the index \(\tau ( L )\) and the discriminant If \(L\) is odd, a similar statement is true provided \(\ell ( { \mathsf { d g } } _ { L } ) \leq r - 3\) and we replace ??#?? with \(b _ { L } ^ { \# }\) .
Corollary 14.4.4
Corollary 14.4.4. Let \(L , L ^ { \prime }\) be non-degenerate indefinite lattices of rank \(r \ \geq 3\) with the same discriminant. Suppose that \(\operatorname { d i s c } ( L ) \in \mathbb { Z }\) contains the prime 2 to the \(p o w e r < \lfloor \frac { 1 } { 2 } r ^ { 2 } + 1 \rfloor\) if \(L\) is even, or, if \(L\) is odd, to the power \(< \lfloor \frac { 1 } { 2 } ( r - 1 ) ^ { 2 } + 1 \rfloor\) , and any odd prime to the power \(< \frac { 1 } { 2 } r ( r - 1 )\) . Then \(L\) and \(L ^ { \prime }\) are isometric if and only if they are in the same genus.
Definition / construction
For an even integral lattice \((L,q)\), \(r_L:O(L,q)\to O(q_L^\#)\) is the reduction homomorphism sending a lattice isometry to its induced isometry of the discriminant quadratic form.
Proposition 14.5.1
Proposition 14.5.1. The local reduction maps \(r _ { L _ { p } }\) are surjective for every prime \(p\) , that is, every isometry of the discriminant quadratic form of \(L _ { p }\) can be lifted to an isometry of \(L _ { p }\) .
Definition / construction
The stable orthogonal group is the kernel of the reduction homomorphism from the lattice isometry group to the isometry group of its discriminant form.
Definition / construction / notation
Similarly, we use the local versions \(0 ^ { \# } ( L _ { p } , q _ { p } )\) and \(\mathsf { S O } ^ { \# } ( L _ { p } , q _ { p } )\) , the kernels of the local reduction maps. We often drop the \(q\) or \(q _ { p }\) in the notation. Standard examples of isometries inducing the identity on the discriminant group are the reflections in \(- 2\) -roots (Cf. Lemma 7.1.1.3). Example 13.4.1 shows that the image of the spinor norm evaluated on these smaller (local) groups already contains all units:
Lemma 14.5.2
Lemma 14.5.2. Let \(L _ { p }\) be a non-degenerate \(p\) -adic lattice, \(p \neq 2\) . Suppose that a homogeneous rank 2 sublattice splits off. Then the image of the spinor norm restricted to SO \({ } ^ { \# } ( L _ { p } )\) contains \(\mathbb { Z } _ { p } ^ { \times } \cdot ( \mathbb { Q } ^ { \times } ) ^ { 2 }\) . The same holds if \(p = 2\) and \(L _ { \mathrm { 2 } }\) is even. If no homogeneous rank 2 lattice splits off, assume instead that a homogeneous rank 3 sublattice splits off.
Proposition 14.5.3
Proposition 14.5.3. Let \(( L , q )\) be a non-degenerate indefinite quadratic integral lattice of rank \(\geq 3\) . Suppose that \[ \mathrm { N m } _ { \mathrm { s p i n } } ( \mathsf { S O } ^ { \# } ( L _ { p } ) ) \supset \mathbb { Z } _ { p } ^ { \times } \cdot ( \mathbb { Q } _ { p } ^ { \times } ) ^ { 2 } f o r a l l p r i m e s p . \] Then the reduction map \(r _ { L } ^ { \prime } : \mathsf { S O } ( L , q ) \to \mathsf { O } ( q _ { L } ^ { \# } )\) is surjective. Moreover, the map \(r _ { L } ^ { \prime }\) is then already surjective on the subgroup of \({ \mathsf { S O } } ( L , q )\) consisting of rotations with real spinor norm 1.
Remark 14.5.4
Remark 14.5.4. Of course, the same conditions as stated in the previous proposition imply surjectivity of the reduction homomorphisms for the full orthogonal groups. A priori these conditions can be weakened but this leads to very technical considerations. See [156, Ch. VIII, 5] where the failure of surjectivity of the reduction map is captured in an explicit group.
Theorem 14.5.5
Theorem 14.5.5. 1. Let \(L\) be a non-degenerate indefinite quadratic lattice of rank \(\geq 3\) . Suppose that for all primes \(p\) a homogeneous rank 2 sublattice splits off from \(L _ { p }\) which is the case if for instance \(\ell ( { \mathrm { d g } } _ { L } ) \leq { \mathrm { r a n k } } ( L ) - 2\) . Then • ?? has class number 1; • the reduction homomorphism \(r _ { L } ^ { \prime } : \mathsf { S O } ( L , q ) \to \mathsf { O } ( q _ { L } ^ { \# } )\) is surjective, as well as its restriction to the subgroup \({ \mathsf { S O } } ^ { + } ( L , q )\) of \({ \mathsf { S O } } ( L , q )\) consisting of isometries having real spinor norm 1. 2. For odd non-degenerate indefinite lattices a similar assertion holds provided one sharpens the condition for the prime \(p = 2\) to demand that a homogeneous rank 3 sublattice splits off from \(L _ { 2 }\) . All conditions are satisfied in this case if e.g. \(\ell ( { \mathrm { d g } } _ { L } ) \leq { \mathrm { r a n k } } ( L ) - 3\) .
Example / comparison
\(L = \oplus ^ { k } U ( 2 )\) , \(k \geq 3\) , satisfies the conditions of the above theorem.
Example / comparison
Sometimes the condition \(\ell ( { \mathrm { d g } } _ { L } ) \ \leq \ { \mathrm { r a n k } } ( L ) - 2\) is not satisfied but can be remedied by adding a non-trivial even unimodular lattice.
Lemma 14.6.1
Lemma 14.6.1. Let \(( L , b )\) be a non-degenerate \(p\) -elementary lattice. Then 1. \(\mathrm { d i s c } ( L ) = \pm p ^ { w }\) , \(w = \dim W _ { L }\) . If \(b\) is an even form and \(p\) is odd, \(\mathrm { r a n k } ( L )\) is even; 2. There is a splitting \(L _ { p } = L ^ { ( 0 ) } \oplus L ^ { ( 1 ) } ( p )\) , where \(L ^ { ( 0 ) }\) and \({ \cal L } ^ { ( 1 ) }\) are unimodular, \(\mathrm { r a n k } ( L ^ { ( 0 ) } ) = \mathrm { d i m } V _ { L }\) and \(\mathrm { r a n k } ( L ^ { ( 1 ) } ) = \dim W _ { L }\) .
Corollary 14.6.2
Corollary 14.6.2. Let \(L\) be an even indefinite \(p\) -elementary lattice of rank \(\geq 4\) . Then the class number of \(L\) is at most one. Up to isometry \(L\) is thus determined by its signature and its discriminant quadratic form.
Construction / comparison
For odd \(p\), an even \(p\)-elementary discriminant quadratic form has normal form \(\langle up^{-1}\rangle\oplus\langle p^{-1}\rangle^{\oplus(w-1)}\), and its index modulo \(8\) is determined by \(p\bmod8\), the length \(w\), and the unit square class as recorded in Table 14.6.1.
Construction / comparison
For \(p=2\), the type-I/type-II normal forms and the mod-\(8\) indices of their basic blocks give the possibilities in Table 14.6.2; relation (IV) identifies the starred duplicate rows with the corresponding unstarred normal forms.
Proposition 14.6.3
Proposition 14.6.3. Let \(( L , b )\) be an indefinite, even \(p\) -elementary lattice of rank \(\geq 4\) and \(\mathrm { d i s c } ( L ) = \pm p ^ { w }\) (so that \(w = \ell ( \mathrm { d g } _ { L } )\) , the length of \({ \mathsf { d g } } _ { L }\) ). 1. For \(p \neq 2\) the isometry class of \(L\) is uniquely determined by its signature and \(w\) . 2. For \(p = 2\) the class of \(L\) is determined by its type, ?? and the signature. Moreover, in case \(L\) is of type \(I I\) , \(w\) must be even and two types occur according to whether the Arf invariant of \(q ^ { \# }\) equals 0 or 1. In the first case, \(q ^ { \# } \simeq \oplus ^ { \frac { 1 } { 2 } w } u _ { 1 }\) with \(\tau _ { 8 } \equiv 0\) mod 8, and in the second case \(q ^ { \# } \simeq \oplus ^ { \frac { 1 } { 2 } ( w - 2 ) } u _ { 1 } \oplus v _ { 1 }\) with \(\tau _ { 8 } \equiv 4\) mod 8.
Theorem 14.6.4
Theorem 14.6.4. Given positive \(r_+,r_-\), \(w\ge0\), \(\tau_8\in\mathbf Z/8\mathbf Z\), and for \(p=2\) a type \(I\) or \(II\), an indefinite \(p\)-elementary quadratic lattice of rank \(r=r_++r_-\ge4\), signature \((r_+,r_-)\) and \(\dim L^{(1)}=w\) exists exactly under the stated conditions: \(w\le r\) and \(r_+-r_-\equiv\tau_8\pmod8\); for odd \(p\), \(r\) is even, \(\tau_8\) has one of the two values in Table 14.6.1, and if \(w=r\) then \(u=(-1)^{r_-}\); for \(p=2\), the prescribed type is \(I\) or \(II\), with \(\tau_8\equiv0,4\pmod8\) if \(w\ne r\) and \(\tau_8\equiv0\pmod8\) if \(w=r\).