FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 4 theorems, 9 propositions, 1 lemma, 3 corollaries, 9 examples, 1 construction and 3 remarks.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Proposition 15.1.1
Proposition 15.1.1. Let ?? be a non-degenerate symmetric (respectively quadratic) lattice, \(S\) a primitive non-degenerate sublattice and \(T = S ^ { \perp }\) . With the notation of diagram (15.1), the following equivalent properties hold: 1. on \(\mathsf { H } _ { S } \subset \mathsf { d g } _ { S }\) the glueing criterion \(b _ { T } ^ { \# } ( \psi _ { S , T } - , \psi _ { S , T } - ) + b _ { S } ^ { \# } ( - , - ) = 0\) , respectively \(q _ { T } ^ { \# } \ L { \circ } \psi _ { S , T } + q _ { S } ^ { \# } = 0\) holds; 2. up to a sign the embedding \(\psi _ { S , T }\) is an isometry of torsion forms; 3. \(\mathsf { H } : = g r a p h\) of \(\psi _ { S , T } \subset \mathrm { d g } _ { S } \oplus \mathrm { d g } _ { T }\) is an isotropic subspace. If ?? is unimodular, then \(\mathsf { H } _ { S } = \mathsf { d } \mathrm { g } _ { S }\) , \(\psi _ { S , T }\) is an isomorphism and the glueing condition becomes \(b _ { S } ^ { \# } \simeq - b _ { T } ^ { \# }\) (respectively \(q _ { S } ^ { \# } \simeq - q _ { T } ^ { \# } ,\) .
Example / comparison
Example 15.1.2. An example illustrating Proposition 15.1.1 is the following. Let \(L = U \mathbb { \ Q } U\) be the sum of two copies of the hyperbolic plane \(U\) , the first one with the usual basis \(e , f\) (so \(b ( e , e ) = b ( f , f ) = b ( e , f ) - 1 = 0 ,\) ) and the second one with similar basis \(e ^ { \prime } , f ^ { \prime }\) . Take \(S\) to be the span of \(e + f , e ^ { \prime } + f ^ { \prime }\) in \(L\) , and let \(T\) be the span of \(e - f , e ^ { \prime } - f ^ { \prime }\) in \(L\) . Then \(L\) is an even unimodular overlattice of \(S \textsuperscript { \textregistered T }\) . The discriminant groups of \(S\) and \(T\) are \(\begin{array} { r } { \mathrm { d } \mathrm { g } _ { S } = \frac { 1 } { 2 } S / S = \langle u = \frac { 1 } { 2 } ( e + f ) + S , u ^ { \prime } = \frac { 1 } { 2 } ( e ^ { \prime } + f ^ { \prime } ) + S \rangle } \end{array}\) and \(\begin{array} { r } { \mathrm { d } \mathrm { g } _ { T } = \frac { 1 } { 2 } T / T = \langle v = \frac { 1 } { 2 } ( e - f ) + T , v ^ { \prime } = \frac { 1 } { 2 } ( e ^ { \prime } - f ^ { \prime } ) + T \rangle } \end{array}\) . The image of \(L / ( S \mathbb { Q } T )\) in \(\mathsf { d g } _ { S } \oplus \mathsf { d g } _ { T }\) equals \(\left\{ 0 , ( u , v ) , ( u ^ { \prime } , v ^ { \prime } ) , ( u + u ^ { \prime } , v + v ^ { \prime } ) \right\}\) , which is the graph of the isomorphism \(\psi _ { S , T } : \mathrm { d g } _ { S } \mathrm { d g } _ { T }\) determined by \(\psi _ { S , T } ( u ) = v\) and \(\psi _ { S , T } ( u ^ { \prime } ) = v ^ { \prime }\) . Note that \(\begin{array} { r } { b _ { S } ^ { \# } ( u , u ) = \frac { 1 } { 2 } + \mathbb { Z } = - b _ { T } ^ { \# } ( v , v ) } \end{array}\) , etc. Also, for the associated quadratic form \(\begin{array} { r } { q ( x ) = \frac { 1 } { 2 } b ( x , x ) } \end{array}\) we have \(\begin{array} { r } { q _ { S } ^ { \# } ( u ) = \frac { 1 } { 2 } \cdot \frac { 1 } { 2 } + \mathbb { Z } = \frac { 1 } { 4 } + \mathbb { Z } } \end{array}\) and \(\begin{array} { r } { q _ { T } ^ { \# } ( v ) = \frac { 1 } { 2 } \cdot - \frac { 1 } { 2 } + \mathbb { Z } = - \frac { 1 } { 4 } + \mathbb { Z } } \end{array}\) , etc.
Proposition 15.1.3
Proposition 15.1.3. Let \(S\) and \(T\) be non-degenerate symmetric (respectively quadratic) lattices, \(\mathsf { H } _ { S } \subset \mathsf { d g } _ { S }\) a subgroup, and \(\psi _ { S , T } : { \mathsf { H } } _ { S } \hookrightarrow { \mathsf { d g } } _ { T }\) an injective homomorphism. Suppose that the glueing criterion of Proposition (15.1.1) holds. Then, setting \(N : = S \textcircled { \perp } T\) , we have 1. The lattice \(L = \{ y \in N ^ { * } \mid y\) mod \(N \in \mathsf { H }\) }, where \(\mathsf { H }\) is the graph of \(\psi _ { S , T }\) , is \(a\) symmetric (respectively quadratic) overlattice of \(N\) satisfying \([ L : N ] = | \mathsf { H } _ { S } |\) ; 2. ?? and \(T\) embed primitively in \(L\) and \(T = S ^ { \perp }\) , \(S = T ^ { \perp }\) ; 3. If \(\mathsf { H } _ { S } = \mathsf { d } \mathrm { g } _ { S }\) and \(\psi _ { S , T }\) is an isomorphism, then \(L\) is unimodular.
Corollary 15.1.4
Corollary 15.1.4. Let \(L\) be a non-degenerate even lattice with one isometry class in its genus and suppose that the non-degenerate lattice \(S\) embeds primitively in \(L\) . Then any lattice \(S ^ { \prime }\) in the same genus as \(S\) embeds primitively in \(L\) .
Example / comparison
Example 15.1.5. Here is an example of two non-isometric lattices in the same genus that embed in the same even unimodular lattice. The lattices \(E _ { 8 } \ @ E _ { 8 }\) and \(\Gamma _ { 1 6 }\) are in the same genus, but are not isometric (see Examples 1.5.1 and 11.3.2). The indefinite unimodular lattices \(\{ \Phi ^ { k } U \ @ E _ { 8 } \ @ E _ { 8 }\) and \(\mathbb { \oplus } ^ { k } U \mathbb { \oplus } \Gamma _ { 1 6 }\) , \(k = 1 , 2 , \ldots\) , however, are both even and have the same signature, and so are isometric by Theorem 2.4.1. Note that Corollary 15.1.4 applies in this case, but is not needed.
Proposition 15.1.6
Proposition 15.1.6 (Extending isometries, II). Let \(S , S ^ { \prime } , T , T ^ { \prime }\) be non-degenerate quadratic lattices related by isometries \(\sigma : S \stackrel { \sim } { \to } S ^ { \prime }\) and \(\tau : T \stackrel { \sim } { } T ^ { \prime }\) . Suppose \(L\) , \(L ^ { \prime }\) are quadratic overlattices of \(S \oplus T\) , \(S ^ { \prime } \mathcal { \text O T } ^ { \prime }\) , respectively. Assume that \(S\) and \(T\) are primitively embedded in \(L\) , and similarly for \(S ^ { \prime }\) and \(T ^ { \prime }\) . Then the isometry \(\lambda = \sigma \oplus \tau\) extends to an isometry \(L \stackrel { \sim } { } L ^ { \prime }\) if and only if (the map induced by) \(\lambda\) sends the subgroup \(L / ( S \mathbb { Q } T )\) of \(\mathsf { d g } _ { S } \mathbb { O } \mathsf { d g } _ { T }\) to the subgroup \(L ^ { \prime } / ( S ^ { \prime } \mathbb { \oplus } T ^ { \prime } )\) of \(\mathsf { d g } _ { S ^ { \prime } } \mathbb { O } \mathsf { d g } _ { T ^ { \prime } }\) . In other words, using the notation of diagram (15.1), \(\lambda\) extends if and only if there is a commutative diagram \[ \begin{array} { r l } { \mathrm { d } \mathrm { g } _ { S } \longleftrightarrow } & { { } \longleftrightarrow \mathsf { H } _ { S } \longleftrightarrow \quad \quad \psi _ { S , T } \quad \quad \mathrm { ~ } \mathsf { \Gamma } \lambda } \\ { r _ { S , S ^ { \prime } } ( \sigma ) \underset { \mathrm { d } \mathrm { g } _ { S ^ { \prime } } \longleftrightarrow } { \downarrow } } & { { } \underset { H _ { S ^ { \prime } } \longleftrightarrow } { \downarrow } \underbrace { \vphantom { \sqrt { \Gamma _ { T , T ^ { \prime } } ( \tau ) } } \mathrm { d } r _ { T , T ^ { \prime } } ( \tau ) } } & { { } \mathrm { ~ } \forall r _ { T , T ^ { \prime } } ( \tau ) } \end{array} \] where \(r _ { S , S ^ { \prime } } ( \sigma )\) and \(r _ { T , T ^ { \prime } } ( \tau )\) are induced by \(\sigma\) and \(\tau\) , respectively. In particular, if \(L\) (and hence \(L ^ { \prime }\) ) is unimodular, using Propositions 15.1.1, 15.1.3, this reduces to the glueing criterion exhibited by the commutative diagram \[ \begin{array} { r l } { \mathrm { d g } _ { S } \cfrac { \simeq } { \psi _ { S , T } } } & { \to \mathrm { d g } _ { T } } \\ { r _ { S , S ^ { \prime } } ( \sigma ) \sqrt { \qquad } } & { \sqrt { r _ { T , T ^ { \prime } } ( \tau ) } } \\ { \mathrm { d g } _ { S ^ { \prime } } \cfrac { \simeq } { \psi _ { S ^ { \prime } , T ^ { \prime } } } } & { \to \mathrm { d g } _ { T ^ { \prime } } } \end{array} \]
Theorem 15.1.7
Theorem 15.1.7 (Witt’s extension theorem for lattices). 1. Let \(L\) be a nondegenerate even lattice and let \(i , i ^ { \prime } : S \hookrightarrow L\) be two primitive embeddings. Then these embeddings are equivalent if and only if an isometry \(T = i ( S ) ^ { \perp } \stackrel { \sim } { \to } T ^ { \prime } = i ^ { \prime } ( S ) ^ { \perp }\) exists inducing a commutative diagram like (15.2). 2. Suppose in addition that \(L\) is unimodular, then two primitive embeddings of ?? in \(L\) with isometric orthogonal complements are equivalent if the reduction homomorphism \(r _ { T } : { \cal O } ( T ) \to { \cal O } ( q _ { T } ^ { \# } )\) is surjective, where \(T\) is the orthogonal complement of an embedded copy of \(S\) .
Proposition 15.2.1
Proposition 15.2.1 ([171, Theorem 1.12.2 & Cor. 1.12.3]). Let \(( r _ { + } , r _ { - } )\) be a pair of non-negative integers for which \(r _ { + } - r _ { - } \equiv 0\) mod 8 and let \(S\) be a non-degenerate even lattice of signature \(( s _ { + } , s _ { - } )\) with \(s _ { + } \leq r _ { + }\) and \(s _ { - } \leq r _ { - }\) , discriminant group \({ \mathsf { d g } } _ { S }\) and discriminant quadratic form \(q ^ { \# }\) . Suppose that, moreover, \(( r _ { + } + r _ { - } ) - \mathrm { r a n k } ( S ) \geq\) \(\ell ( \mathrm { d g } _ { S } )\) . Then the following conditions are equivalent. 1. ?? can be primitively embedded in some even unimodular lattice \(L\) of signature \(( r _ { + } , r _ { - } )\) ( \(L\) is unique up to isometry in case \(r _ { + } > 0\) and \(r _ { - } > 0\) ); 2. A (non-degenerate) even lattice \(T\) of signature \(( r _ { + } - s _ { + } , r _ { - } - s _ { - } )\) with discriminant quadratic form \(- q ^ { \# }\) exists; 3. A (non-degenerate) even lattice \(T ^ { \prime }\) of signature \(( r _ { - } - s _ { - } , r _ { + } - s _ { + } )\) with discriminant quadratic form \(q ^ { \# }\) exists; 4. The equality2 \(\mathrm { d i s c } ( L _ { q _ { p } ^ { \# } } ) ~ = ~ ( - 1 ) ^ { r _ { - } - s _ { - } } | \ : \mathrm { d g } _ { S } \ : |\) in \(\mathsf { D } ( \mathbb { Z } _ { p } )\) holds in the following cases: • for any prime \(p \neq 2\) for which \(\mathrm { d g } _ { S _ { p } }\) , the \(p\) -primary part of \({ \mathsf { d g } } _ { S }\) , has length equal to \(( r _ { + } + r _ { - } ) - \mathrm { r a n k } ( S )\) ; • for \(p = 2\) in case \(\ell ( \mathrm { d } \mathrm { g } _ { S _ { 2 } } ) = ( r _ { + } + r _ { - } ) - \mathrm { r a n k } ( S )\) and \(\mathsf { d g } _ { S _ { 2 } }\) does not split off a cyclic order 2 module. In particular, all of these these conditions are satisfied if \(( r _ { + } + r _ { - } ) \mathrm { - r a n k } ( S ) > \ell ( \mathrm { d g } _ { S } )\) .
Remark 15.2.2
Remark 15.2.2. If \(S\) is definite, the preceding embedding result is less interesting since the uniqueness of \(L\) need not hold. For instance \(E _ { 8 }\) embeds trivially in \(E _ { 8 } \ @ E _ { 8 }\) but does not embed in \(\Gamma _ { 1 6 }\) .
Theorem 15.2.3
Theorem 15.2.3 (Universal Embeddability). Let \(( r _ { + } , r _ { - } )\) be a pair of non-negative integers for which \(r _ { + } { - } r _ { - } \equiv 0\) mod 8 and let \(( s _ { + } , s _ { - } )\) be a pair of non-negative integers with \(s _ { + } \leq r _ { + }\) and \(s _ { - } \leq r _ { - }\) . Then every even non-degenerate lattice \(S\) of signature \(( s _ { + } , s _ { - } )\) satisfing \(r _ { + } + r _ { - } -\) \(\operatorname { r a n k } ( S ) \geq \ell ( \mathsf { d g } _ { S } )\) can be primitively embedded in some even unimodular lattice \(L\) of signature \(( r _ { + } , r _ { - } )\) if \[ s _ { + } + s _ { - } \leq \frac 1 2 \mathrm { r a n k } ( L ) . \] In case \(r _ { + } > 0\) and \(r _ { - } > 0\) , the unimodular lattice \(L\) is unique up to isometry.
Lemma 15.2.4
Lemma 15.2.4. An even lattice \(S\) can be embedded in an indefinite even unimodular lattice \(L\) if the Witt index of \(L\) is \(\geq\) rank(??). In particular, \(S\) embeds in \(\mathbb { Q } ^ { s } U\) , \(s = \mathrm { r a n k } ( S )\) , with \(S ^ { \perp } \simeq S ( - 1 )\) .
Construction / comparison
For non-degenerate \(S\), Theorem 15.2.3 gives the primitive-embedding bound \(\operatorname{rank}S\le\frac12\operatorname{rank}L\), whereas Lemma 15.2.4 gives the stronger sufficient bound \(\operatorname{rank}S\le a\) for \(L=U^{\oplus a}\oplus E_8(\pm1)^{\oplus b}\).
Example / comparison
Example 15.2.5. Let \(T\) be an even non-degenerate lattice of rank \(a+e\), with \(e>0\). Then \(T\) embeds primitively in \(U^{\oplus a}\) exactly when \(U^{\oplus e}\) splits off orthogonally from \(T\).
Theorem 15.2.6
Theorem 15.2.6. Let \(L\) be an indefinite even unimodular lattice and \(( S , q )\) a nondegenerate quadratic lattice with \(\ell ( \mathbf { d g } _ { S } ) \leq \mathrm { r a n k } ( L ) - \mathrm { r a n k } ( S ) - 2\) . Let \(L\) and \(S\) have signature \(( r _ { + } , r _ { - } )\) , \(( s _ { + } , s _ { - } )\) , respectively, and assume that \(r _ { + } > s _ { + }\) , \(r _ { - } > s _ { - }\) . Then 1. the orthogonal complement of a primitively embedded \(S \hookrightarrow L\) has genus invariant \(( r _ { + } - s _ { + } , r _ { - } - s _ { - } , - [ q _ { S } ^ { \# } ] )\) ; 2. ?? admits a primitive embedding in \(L\) and such primitive embeddings are unique up to equivalence, i.e., \({ \sf O } ( L )\) acts transitively on primitive embeddings of \(S\) in \(L\) .
Corollary 15.2.7
Let \(L\) be an indefinite even unimodular lattice of \(\operatorname{rank} \geq 4\) and signature \((r_{+}, r_{-})\). A non-degenerate even lattice \(S\) of signature \((s_{+}, s_{-})\), with \(s_{\pm} \leq r_{\pm}\), admits a primitive embedding in \(L\), unique up to equivalence, if \(\operatorname{rank}(S) \leq \tfrac{1}{2}\operatorname{rank}(L) - 1\).
Example / comparison
Let \(L = \Lambda _ { { \sf K } 3 } = U \oplus U \oplus U \oplus E _ { 8 } ( - 1 ) \oplus E _ { 8 } ( - 1 )\) , the K3-lattice. Then any non-degenerate even lattice of rank \(\leq 2\) can be embedded uniquely as a primitive lattice in \(\Lambda _ { \mathsf { K 3 } }\) . This holds more generally for any non-degenerate even lattice of rank \(r \leq 1 0\) which has signature \(( 0 , r )\) , \(( 1 , r - 1 )\) or \(( 2 , r - 2 )\) . For the same reason, if \(L = U ^ { \oplus a } \oplus E _ { 8 } ( - 1 ) ^ { \oplus b }\) , then any non-degenerate lattice of rank \(\leq a - 1\) or of rank \(r \leq a + 4 b - 1\) and signature \(( s , r - s )\) , \(s \leq a - 1\) can be embedded uniquely in \(L\) as a primitive lattice.
Example / comparison
Again \(L = { \Lambda } _ { \mathsf { K 3 } }\) but now \(S = A _ { k _ { 1 } } ( - 1 ) \oplus A _ { k _ { 2 } } ( - 1 ) \oplus \dots \oplus A _ { k _ { \ell } } ( - 1 )\) . If \(\textstyle \sum k _ { j } \leq 1 0\) conditions 1 and 2 hold.
Proposition 15.2.9
Proposition 15.2.9. Let \(L\) be a quadratic unimodular lattice, \(S \subset L\) a primitive non-degenerate sublattice, \(T = S ^ { \perp }\) and \(\sigma \in \operatorname { O } ^ { \# } ( S )\) . The isometry \(\sigma\) ⊕id?? of \(S @ T\) extends to an isometry \(e _ { S } ( \sigma )\) of \(L\) . The assignment \(\sigma \mapsto e _ { S } ( \sigma )\) defines an injective homomorphism \(e _ { S } : { 0 } ^ { \# } ( S ) \to { 0 } ( L )\) with image contained in the subgroup \({ \mathsf { O } } ^ { \# } ( L ) _ { S }\) of the stabilizer \({ \sf O } ( L ) _ { S }\) of \(S\) in \({ \sf O } \left( L \right)\) given by \[ { \sf O } ^ { \# } ( L ) _ { S } : = \{ \gamma \in { \sf O } ( L ) _ { S } \ | \ r _ { S } ( \gamma ) = r _ { T } ( \gamma ) \} . \] In other words, there is an exact sequence \[ 1 \to 0 ^ { \# } ( S ) \stackrel { e _ { S } } { \longrightarrow } 0 ^ { \# } ( L ) _ { S } \stackrel { \rho _ { T } } { \longrightarrow } 0 ( T ) , \] where \(\rho _ { T }\) is the restriction map. If the reduction map \(r _ { S } : { \cal O } ( S ) { \cal O } ( q _ { S } ^ { \# } )\) is surjective, then also \(\rho _ { T }\) is surjective.
Proposition 15.2.10
Proposition 15.2.10 (Comparing lattice involutions). Let \(L\) and \(L ^ { \prime }\) be isometric indefinite quadratic unimodular lattices admitting lattice involutions \(i\) , respectively \(i ^ { \prime }\) . Suppose that • the corresponding invariant lattices \(S , S ^ { \prime }\) are isometric, • \(T = S ^ { \perp } , T ^ { \prime } = S ^ { \prime \perp }\) are indefinite and have rank \(\geq 4\) hen there is an isometry \(\lambda : L \ { \stackrel { \simeq } { \to } } \ L ^ { \prime }\) intertwining \(i\) and \(i ^ { \prime }\) , that is, there is a commutative diagram \[ \begin{array} { r l r } { L \longrightarrow } & { { } } & { \simeq \qquad \times L ^ { \prime } } \\ { i \Bigg \downarrow \Bigg \downarrow } & { { } } & { i ^ { \prime } \Bigg \downarrow } \\ { L \longrightarrow } & { { } } & { \simeq \qquad \times L ^ { \prime } . } \end{array} \]
Proposition 15.3.1
Proposition 15.3.1. Let \(S\) and \(M\) be non-degenerate quadratic lattices with genus invariants \(\mathsf { g } ( S ) = ( s _ { + } , s _ { - } , [ q _ { S } ^ { \# } ] )\) , \(\mathsf { g } ( M ) = ( m _ { + } , m _ { - } , [ q _ { M } ^ { \# } ] )\) , and satisfying \(s _ { \pm } ~ \leq ~ m _ { \pm }\) . Primitive embeddings \(S \hookrightarrow M\) are determined by quadruples \(( H _ { S } , \psi , K , \gamma _ { K } ^ { \psi } )\) , where • \(H _ { S } \subset \mathrm { d g } _ { S }\) is a subgroup, \(\psi : H _ { S } \hookrightarrow \mathrm { d g } _ { M }\) an embedding of quadratic torsion groups, that is, \(\psi ( H _ { S } )\) is a subgroup of \({ \mathsf { d g } } _ { M }\) and \(q _ { S } ^ { \# } ( x ) = q _ { M } ^ { \# } ( \psi ( x ) )\) for all \(x \in H _ { S }\) ; • \(K\) is a quadratic lattice with genus-invariant \(( m _ { + } - s _ { + } , m _ { - } - s _ { - } , [ - \kappa ^ { \psi } ] ) ,\) \(\kappa ^ { \psi } : =\) \(\Gamma _ { \psi } { ^ { \perp } } / \Gamma _ { \psi }\) , where \(\Gamma _ { \psi } \subset \mathsf { d g } _ { S } \oplus \mathsf { d g } _ { M }\) is the graph of \(\psi\) and \(\Gamma _ { \psi } ^ { \bot }\) its orthogonal complement with respect to the polar form of \(q _ { S } ^ { \# } \oplus q _ { M } ^ { \# }\) ; • \(\gamma _ { K } ^ { \psi } : q _ { K } ^ { \# } \{ \tilde { \to } - \kappa ^ { \psi }\) is an isometry. The orthogonal complement of \(S\) in \(M\) in this embedding is isometric to \(K\) . Two embeddings \(i : S \hookrightarrow M\) and \(i ^ { \prime } : S \hookrightarrow M\) given by \(( H _ { S } , \psi , K , \gamma )\) and \(( H _ { S } ^ { \prime } , \psi ^ { \prime } , K ^ { \prime } , \gamma ^ { \prime } )\) are equivalent if and only if the following two conditions hold simultaneously: • \(H _ { S } = H _ { S } ^ { \prime }\) and there exists an isometry \(\lambda\) of \(\mathsf { d g } _ { M }\) such that \(\lambda \circ \psi = \psi ^ { \prime }\) ; • there is an isometry \(\varphi : K { \stackrel { \sim } { \to } } K ^ { \prime }\) such that \(\mathrm { d g } _ { \varphi } \circ \gamma = \gamma ^ { \prime } \circ \lambda ^ { \prime }\) , where the isometry \(\lambda ^ { \prime } : - \kappa ^ { \psi } \stackrel { \sim } { } - \kappa ^ { \psi ^ { \prime } }\) is induced by \(\lambda\) .
Remark 15.3.2
Remark 15.3.2. 1. The case \(H _ { S } = 0\) is allowed and serves for instance to cover unimodular lattices \(M\) . Similarly, if \(M = L \oplus M ^ { \prime }\) , where \(L\) is unimodular and \(S\) embeds primitively in \(L\) , then \(H _ { S } = 0\) . 2. If other lattices genus equivalent to \(K\) exist, these give rise to non-isomorphic embeddings of \(S\) in \(M\) . Moreover, different subgroups of \({ \mathsf { d g } } _ { S }\) may embed in \({ \mathsf { d g } } _ { M }\) and a given subgroup \(H _ { S } \subset \mathrm { d g } _ { S }\) may embed in non-isomorphic ways in \({ \mathsf { d g } } _ { M }\) . These all give rise to non-isomorphic embeddings of \(S\) in \(M\) .
Corollary 15.3.3
Corollary 15.3.3. Let \(S , M\) be non-degenerate quadratic lattices and \(H\) a subgroup of \({ \mathsf { d g } } _ { S }\) . Primitive embeddings of \(S\) in \(M\) determined by isometric embeddings \(\psi , \psi ^ { \prime }\) : \(H \hookrightarrow \mathsf { d g } _ { M }\) are equivalent, i.e., they are related by an isometry of \(M\) , in case the following conditions hold simultaneously: 1. The isometric embeddings \(\psi , \psi ^ { \prime } : H \hookrightarrow \mathsf { d g } _ { M }\) are related by an isometry \(\lambda\) of \({ \mathsf { d g } } _ { M }\) ; 2. Up to isometry there is a unique lattice \(K\) with genus-invariant \(( m _ { + } - s _ { + } , m _ { - } -\) ?? \(- , [ - \Gamma _ { \psi } ^ { \bot } / \Gamma _ { \psi } ] )\) . 3. For every primitive embedding of \(S\) in \(M\) , \(S ^ { \perp }\) is isometric to \(K\) ; identifying \(S ^ { \perp }\) with \(K\) , the reduction map \(r _ { K } : { \mathsf { O } } ( K ) \to { \mathsf { O } } ( q _ { K } ^ { \# } )\) , \(K = S ^ { \perp }\) , is surjective. Conditions \(\mathcal { Q }\) and 3 hold if \(\operatorname { r a n k } ( K ) \ \geq \ 3\) , \(K\) is indefinite and the conditions of Theorem 14.5.5 hold, e.g., if \(\ell ( \mathrm { d } \mathrm { g } _ { K } ) \leq \mathrm { r a n k } ( K ) - 2 = \mathrm { r a n k } ( M ) - \mathrm { r a n k } ( S ) - 2\) .
Remark 15.3.4
Remark 15.3.4. Observe that for a non-degenerate quadratic lattice \(S\) , and \(L\) unimodular, taking \(H _ { S } = 0\) , Theorem 15.2.6 and Corollary 15.3.3 give identical conditions.
Example / comparison
Consider the action of \(\mathrm { O } ( M )\) on vectors \(x \in M\) of given length, say \(q ( x ) \ = \ p ^ { k }\) where \(p\) is an odd prime. In this case \(S ~ = ~ \langle { 2 p ^ { k } } \rangle\) and \(\mathsf { d g } _ { S } \simeq \mathbb { Z } / 2 \mathbb { Z } \oplus \mathbb { Z } / p ^ { k } \mathbb { Z }\) . If the Sylow-decomposition of \({ \mathsf { d g } } _ { M }\) does not contain 2- primary or \(p\) -primary components, the only possiblity for \(H _ { S }\) is the trivial group and then \(\mathsf { d g } _ { K } = - \mathsf { d g } _ { M }\) . The two conditions of Corollary 15.3.3 hold if for instance \(\ell ( \mathrm { d g } _ { M } ) \le \mathrm { r a n k } ( M ) - 3\) .
Example / comparison
Let \(M = L \oplus L ^ { \prime } ( - 2 )\) with \(L\) and \(L ^ { \prime }\) indefinite unimodular quadratic lattices. Consider the \(\mathrm { O } ( M )\) -orbits of vectors \(x \in M\) with \(q ( x ) = - 2\) . Then \(S = \langle - 4 \rangle\) and hence \(\mathrm { d g } _ { S } = \frac { 1 } { 4 } \mathbb { Z } / \mathbb { Z } \simeq \mathbb { Z } / 4 \mathbb { Z }\) . So \(H _ { S } = 0\) or \(H _ { S } = \mathbb { Z } / 2 \mathbb { Z }\) . By the classification of even indefinite unimodular forms as given in Section 2.4, the lattice \(M\) splits off \(U \oplus U ( - 2 )\) , and so the lattice \(K = S ^ { \perp }\) is indefinite of rank \(\geq 3\) . As follows from Example 1.6.8.2 and 11.2.5.3, the discriminant quadratic form of \(L ^ { \prime } ( 2 )\) is isometric to a number of copies of \(u _ { 1 }\) \(\mathbb { Q } ^ { m } u _ { 1 }\) \(2 m = \mathrm { r a n k } ( L ^ { \prime } )\) If ???? = 0, ??#?? \(q _ { K } ^ { \# } = q _ { L ^ { \prime } ( 2 ) } ^ { \# }\) hich off a co \(u _ { 1 }\) ence Theorem 14.5.5 shows that conditions 2 and 3 are satisfied. Condition 1 being trivially satisfied, the preceding theorem shows that there is a unique orbit of \(x\) of this type. Since \(L\) contains a copy of \(U\) , the orbit contains \(e \mathrm { - 2 } f\) , where \(\{ e , f \}\) is the standard basis of \(U\) . This deals with the case \(H _ { S } = 0\) . Assume now that \(H _ { S } = \mathbb { Z } / 2 \mathbb { Z }\) . We can view \({ \mathsf { d g } } _ { M }\) as a symplectic \(\mathbb { F } _ { 2 }\) -space and so by an argument as in the proof of Proposition A.5.2 we may assume that \(H _ { S }\) embeds in the first copy of \(u _ { 1 }\) as \(\langle { \bar { e } } + { \bar { f } } \rangle\) , where \(\{ \bar { e } , f \}\) is the standard basis of \(u _ { 1 }\) . Then \(q _ { K } ^ { \# } \simeq \langle 2 ^ { - 1 } \rangle \oplus ^ { m - 1 } u _ { 1 }\) . In this case \[ \ell ( \mathrm { d } \mathrm { g } _ { K } ) = \mathrm { r a n k } ( L ^ { \prime } ) - 1 = \mathrm { r a n k } ( M ) - \mathrm { r a n k } ( L ) - 1 \leq \mathrm { r a n k } ( M ) - 3 , \] and hence conditions 2 and 3 are satisfied. Note that the two types of orbits of \(x\) can be characterized as follows: for \(H _ { S } = 0\) one must have \(x \cdot M = \mathbb { Z }\) (since \(x\) can be assumed to be in the unimodular lattice \(L\) ). For \(H _ { S } = \mathbb { Z } / 2 \mathbb { Z }\) for similar reasons one must have \(x \cdot M = 2 \mathbb { Z }\) .
Proposition 15.4.1
Proposition 15.4.1. Let \(( r _ { + } , r _ { - } )\) be a pair of non-negative integers and let \(S\) be a non-degenerate odd lattice of signature \(( s _ { + } , s _ { - } )\) with \(s _ { + } ~ \leq ~ r _ { + }\) and \(s _ { - } ~ \leq ~ r _ { - }\) and discriminant bilinear form \(b ^ { \# }\) . The following conditions are equivalent. 1. ?? can be primitively embedded in some odd unimodular lattice \(L\) of signature \(( r _ { + } , r _ { - } )\) and such that \(T = S ^ { \perp }\) is odd, respectively even; 2. A (non-degenerate) odd, respectively even lattice \(T\) of signature \(( r _ { + } - s _ { + } , r _ { - } - s _ { - } )\) with discriminant form \(- b ^ { \# }\) ; 3. A (non-degenerate) odd respectively even lattice \(T ^ { \prime }\) of signature \(( r _ { - } - s _ { - } , r _ { + } - s _ { + } )\) with discriminant form \(b ^ { \# }\) exists. If, moreover, both \(r _ { + } > 0\) and \(r _ { - } ~ > ~ 0\) , the unimodular lattice \(L\) is unique up to isometry. These conditions hold in particular if \(\ell ( \mathrm { d g } _ { S } ) \leq \mathrm { r a n k } ( L ) - \mathrm { r a n k } ( S ) - 3\) .
Proposition 15.4.2
Proposition 15.4.2. Let \(( r _ { + } , r _ { - } )\) be a pair of non-negative integers and let \(S\) be a non-degenerate even lattice of signature \(( s _ { + } , s _ { - } )\) with \(s _ { + } \leq r _ { + }\) and \(s _ { - } \leq r _ { - }\) , discriminant quadratic form \(q ^ { \# }\) with polar form \(b ^ { \# }\) . The following conditions are equivalent. 1. \(S\) can be primitively embedded in some odd unimodular lattice \(L\) of signature \(( r _ { + } , r _ { - } )\) and such that \(T = S ^ { \perp }\) is even; 2. Either \(r _ { + } - r _ { - } \not \equiv 0\) mod 8, or else, \({ \mathsf { O } } ( q ^ { \# } )\) is properly included in \({ \mathsf { O } } ( b ^ { \# } )\) and a (non-degenerate) even lattice \(T\) of signature \(( r _ { + } - s _ { + } , r _ { - } - s _ { - } )\) with discriminant form \(- b ^ { \# }\) exists; 3. Either \(r _ { + } - r _ { - } \not \equiv 0\) mod 8, or else, \({ \mathsf { O } } ( q ^ { \# } )\) is properly included in \({ \mathsf { O } } ( b ^ { \# } )\) and a (non-degenerate) even lattice \(T ^ { \prime }\) of signature \(( r _ { - } - s _ { - } , r _ { + } - s _ { + } )\) with discriminant form \(b ^ { \# }\) exists. If, moreover, both \(r _ { + } > 0\) and \(r _ { - } ~ > ~ 0\) , the unimodular lattice \(L\) is unique up to isometry. These conditions hold if \(\ell ( \mathrm { d g } _ { S } ) \leq \mathrm { r a n k } ( L ) - \mathrm { r a n k } ( S ) - 3\) and either \(\tau ( L )\) is not divisible by 8 or \(0 ( q _ { s } ^ { \# } )\) is properly included in \(0 ( b _ { s } ^ { \# } )\)
Example / comparison
\(L = \textcircled { 9 } ^ { a + b } \langle 1 \rangle \textcircled { \textmd { 1 } } \textcircled { 9 } ^ { b + c } \langle - 1 \rangle\) , where \(a , b , c \ > \ 0\) and let \({ \boldsymbol { S } } \ =\) \({ \mathbb { O } } ^ { a } \langle 1 \rangle \oplus { \mathbb { O } } ^ { b } \langle A _ { 1 } ( - 2 ) \rangle\) . Then \(\operatorname { r a n k } ( L ) - \operatorname { r a n k } ( S ) = b + c = \ell ( \deg _ { S } ) + c\) and so, if \(c \geq 3\) \(S\) can be primitively embedded in \(L\) with \(S ^ { \bot }\) odd or even.
Example / comparison
Let \(L = \oplus ^ { a + b } \langle 1 \rangle \oplus \oplus ^ { b + c } \langle - 1 \rangle\) , \(a , b , c > 0\) with \(a \equiv c\) mod 8. Let \(S\) be an even unimodular lattice of signature \(( a , c )\) . Then \(S\) cannot be embedded in \(L\) with even orthogonal complement, but if \(a \mathrm { ~ + ~ } c \mathrm { ~ \geq ~ 3 ~ }\) the lattice \(S\) can be embedded in \(L\) with \(S ^ { \perp } = \oplus ^ { b } \langle 1 \rangle \oplus \oplus ^ { b } \langle - 1 \rangle\) . If instead \(S ~ = ~ \oplus ^ { b } U ( 2 )\) , then \(b _ { s } ^ { \# } \simeq \oplus ^ { b } u _ { 1 }\) is isometric to a symplectic form \(b _ { q }\) on an \(\mathbb { F } _ { 2 }\) -vector space of rank \(2 b\) and hence \({ \mathrm { O } } ( q )\) is strictly contained in \({ \mathrm { O } } ( b _ { q } )\) . Such \(S\) can be embedded in \(L\) with even orthogonal complement as soon as \(2 b \leq a + c - 3\) . Recall that using Theorem 14.5.5, the existence criterion Corollary 12.4.6 for even lattices yields Theorem 15.2.6, which is a version of Witt’s extension theorem for embeddings in even unimodular lattices. Similarly, using the above existence criterion for odd lattices one deduces a version of Witt’s theorem for embeddings of odd or even lattices \(S\) in indefinite odd unimodular lattices. We only state the version for \(S\) odd, leaving the even case to the reader:
Theorem 15.4.4
Theorem 15.4.4. Let \(L\) be an indefinite odd unimodular lattice and \(S\) a nondegenerate odd lattice with \(\ell ( { \mathrm { d g } } _ { S } ) \leq \mathrm { r a n k } ( L ) - \mathrm { r a n k } ( S ) - 3\) . Then \(S\) admits \(a\) primitive embedding into \(L\) and such primitive embeddings are unique up to equivalence, i.e., \(\mathrm { O } ( L )\) acts transitively on primitive embeddings of \(S\) in \(L\) . Furthermore, for each primitive embedding of \(S\) with \(T = S ^ { \perp }\) odd, respectively even, \(T\) has genus invariant \(( r _ { + } - s _ { + } , r _ { - } - s _ { - } , [ - b _ { S } ^ { \# } ] )\) , respectively \(( r _ { + } - s _ { + } , r _ { - } - s _ { - } , [ - q _ { S } ^ { \# } ] )\) , where \(( r _ { + } , r _ { - } )\) , \(( s _ { + } , s _ { - } )\) are the signatures of \(L\) and \(S\) , respectively.