FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 2 definitions, 3 propositions, 7 lemmas, 3 corollaries, 3 examples and 3 remarks.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Example / comparison
The dyadic rank-two lattices \(U_k=U_{\mathbf Z_2}(2^k)\) are indecomposable; \(U_0\) is the symmetric hyperbolic lattice and its associated quadratic form is \((x_1,x_2)\mapsto x_1x_2\).
Example / comparison
The dyadic rank-two lattices \(V_k=V(2^k)\), with \(V\) having Gram matrix \(\begin{pmatrix}2&1\\1&2\end{pmatrix}\), are indecomposable and are not isometric to \(U_k\); \(\operatorname{disc}(V_k)=3\cdot2^{2k}\) modulo unit squares.
Lemma 10.1.1
Lemma 10.1.1. The lattices \(U _ { k }\) are the only indecomposable rank two symmetric dyadic lattices which represent zero. There are no indecomposable symmetric rank 2 dyadic lattices other than \(U _ { k }\) and \(V _ { k }\) .
Lemma 10.1.2
Lemma 10.1.2 (Recognising binary dyadic forms). For a Gram matrix \(\binom { \alpha } { \beta } \begin{array} { l } { \beta } \\ { \gamma } \end{array}\) of a unimodular dyadic symmetric form \(b\) , setting \(d = \operatorname { d i s c } ( b )\) , the following possibilities occur: 1. The form is decomposable if and only if either \(\alpha\) or \(\gamma\) is odd, i.e., a unit. More precisely, (a) if \(\alpha\) is odd, \(b \simeq \left. \alpha \right. \textcircled { \pm } \left. \alpha \cdot d \right.\) . (b) if ?? is odd, \(b \simeq \langle \gamma \rangle \oplus \langle \gamma \cdot d \rangle\) . 2. If ?? and ?? are both even (hence \(b\) is even), then \(\beta\) is odd, ?? is indecomposable, and (a) if \({ \scriptstyle { \frac { 1 } { 4 } } } \alpha \gamma\) is even, \(b \simeq U\) (b) and if \({ \scriptstyle { \frac { 1 } { 4 } } } \alpha \gamma\) is odd, \(b \simeq V\) . For a unimodular binary dyadic quadratic lattice, only case 2 occurs.
Lemma 10.1.3
Lemma 10.1.3. A \(p\) -adic unimodular quadratic form of rank two represents all \(p\) -adic units.
Corollary 10.1.4
Corollary 10.1.4. Let \(( L , q )\) be a unimodular binary quadratic \(p\) -adic lattice and ?? a \(p\) -adic unit. Then ?? admits a reflection \(\sigma _ { x }\) in a vector \(x \in L\) with \(q ( x ) = u\) .
Definition / construction
For a \(p\)-adic lattice \(L\), \((q(L))\) and \((b(L,L))\) denote the ideals generated by its quadratic and bilinear values.
Lemma 10.2.1
Lemma 10.2.1. Let \(( L , q )\) be a non-degenerate \(p\) -adic quadratic lattice. Write \(b\) for the polar form of \(q\) . Let \(( q ( L ) )\) , \(( b ( L , L ) )\) and \(x _ { 0 } \in L\) be as above. Suppose that \(2 ( q ( L ) ) = ( b ( L , L ) )\) (which is the case for \(p\) odd, but not necessarily for \(p = 2\) ). Then \(\mathbb { Z } _ { p } \cdot x _ { 0 }\) splits off orthogonally. Consequently, if there is no rank one sublattice of \(L\) which splits off, then \(p = 2\) and \(\mathfrak { Z } ( q ( L ) ) \subsetneq ( b ( L , L ) ) \subset \mathbb { Z } _ { 2 }\) .
Proposition 10.2.2
Proposition 10.2.2 (Classification (I)). 1. For \(p\) odd, any (non-degenerate) symmetric (quadratic) \(p\) -adic lattice is isometric to an orthogonal direct sum of rank one \(p\) -adic lattices of the form \(\langle u p ^ { k } \rangle\) , where \(u \in \mathbb { Z } _ { p } ^ { \times }\) and \(k \geq 0\) . 2. Dyadic symmetric, respectively quadratic lattices split into an orthogonal direct sum of rank one lattices of the form \(\langle u \cdot 2 ^ { k } \rangle\) , \(u \in \{ \pm 1 , \pm 3 \}\) , where \(k \geq 0\) , respectively \(k \geq 1\) , together with copies of \(U _ { k }\) and of \(V _ { k }\) , \(k \geq 0\) .
Example / comparison
Example 10.2.3. Since \(\mathrm { d i s c } ( U ) = - 1\) , \(\operatorname { d i s c } ( V ) = 3\) , the lattice \(E _ { 8 } \otimes \mathbb { Z } _ { 2 }\) , being even with discriminant 1, is isometric to either \(\mathbb { O } ^ { 4 } U\) , \(\mathbb { O } ^ { 4 } V\) or to \(\mathbb { O } ^ { 2 } U \mathbb { O } \mathbb { O } ^ { 2 } V\) . We shall see (Lemma 11.2.1) that these 3 dyadic lattices are isometric.
Corollary 10.2.4
Corollary 10.2.4. A \(p\) -adic unimodular quadratic form of rank \(\geq 2\) represents all \(p\) -adic units. For \(p \neq 2\) the lattice admits isotropic vectors if its rank is at least 3, for \(p = 2\) this is the case if the rank is at least 4.
Remark 10.2.5
Remark 10.2.5. The existence of isotropic vectors is used in the proof of Corollary A.3.7. This result in particular states that the occurrence of one root in an indefinite lattice of rank \(\geq 4\) implies that there are infinitely many. This, in turn, is a crucial ingredient in the proof of Theorem 17.2.11 which roughly states that for ”most” lattices with Witt index \(\geq 2\) the Weyl group is as big as possible.
Definition / construction / notation
A Jordan splitting of a \(p\)-adic lattice is an orthogonal decomposition into summands \(L(p^k)\) with \(L\) unimodular; such a summand is homogeneous of exponent \(k\).
Proposition 10.2.6
Proposition 10.2.6. A non-degenerate symmetric \(p\) -adic lattice \(( L , b )\) has a Jordan splitting, i.e., \[ L = L _ { 0 } \oplus L _ { 1 } ( p ) \oplus \cdots \oplus L _ { r } ( p ^ { r } ) , \] where each \(L _ { j }\) is unimodular. The rank of \(L _ { j }\) as well as its discriminant, \(\operatorname { d i s c } ( L _ { j } ) \in\) \(\mathsf { D } ( \mathbb { Z } _ { p } )\) , is uniquely determined by \(( L , b )\) .
Remark 10.2.7
Remark 10.2.7. For \(p = 2\) a Jordan splitting need not be unique. We discuss this extensively in Appendix C. See in particular Lemma C.3.3 where relation (V) as well as (VI) exhibit two distinct Jordan splittings for a non-degenerate rank 3 dyadic lattice. The summands on the left and on the right side of the equivalence are indeed non-isometric since these have different discriminants. For uniqueness for odd \(p\) , see Proposition 11.1.3.
Corollary 10.2.8
Corollary 10.2.8 (Suggested by M. Sch¨utt). An even integral lattice of odd rank has even discriminant.
Lemma 10.3.1
Lemma 10.3.1. Let \(( G , q )\) be a non-degenerate \(p\) -adic torsion quadratic group. Write ?? for the polar form of \(q\) . Let \(( q ( G ) ) \subset \mathbb { Q } _ { p } / \mathbb { Z } _ { p }\) , \(( b ( G , G ) ) \subset \mathbb { Q } _ { p } / \mathbb { Z } _ { p }\) be the \(\mathbb { Z } _ { p }\) -submodule generated by the values of \(q ( x ) , x \in G\) , respectively \(b ( x , y ) , x , y \in G\) , and let \(x _ { 0 } \in G\) such that \(q ( x _ { \mathrm { 0 } } )\) generates \(( q ( G ) )\) . Suppose that \(2 ( q ( G ) ) = ( b ( G , G ) )\) (which is the case for \(p\) odd). Then the cyclic group generated by \(x _ { 0 }\) splits off orthogonally. Consequently, if there is no non-degenerate cyclic torsion subgroup of \(G\) which splits off, then \(p = 2\) and \(2 ( q ( G ) ) \subsetneq ( b ( G , G ) ) \subset \mathbb { Q } _ { 2 } / \mathbb { Z } _ { 2 }\) .
Proposition 10.3.2
Proposition 10.3.2 (Compatibility of splittings). 1. Let \(L\) be a non-degenerate symmetric or quadratic \(p\) -adic lattice with Jordan splitting \[ L = L _ { 0 } \oplus L _ { 1 } ( p ) \oplus \cdots \oplus L _ { n } ( p ^ { n } ) . \] Then its discriminant form admits a Jordan splitting \[ \begin{array} { r } { \deg _ { L } = \deg _ { L _ { 1 } ( p ) } \oplus \deg _ { L _ { 2 } ( p ^ { 2 } ) } \oplus \cdot \cdot \cdot \oplus \deg _ { L _ { n } ( p ^ { n } ) } . } \end{array} \] 2. Given a basis for the homogeneous components of the discriminant form of \(L\) , there is a corresponding basis for the homogeneous components of \(L\) . More precisely, \(p\) odd. For all \(j \geq 1\) , let \(\{ g _ { 1 } ^ { ( j ) } , \ldots , g _ { s } ^ { ( j ) } \}\) be an orthogonal basis for \(\mathsf { d g } _ { L _ { j } ( p ^ { j } ) }\) with sis \(b _ { j } ^ { \# } ( g _ { k } ^ { ( j ) } , g _ { k } ^ { ( j ) } ) = u _ { k , j } p ^ { - j }\) \(\{ e _ { 1 } ^ { ( j ) } , \ldots , e _ { s } ^ { ( j ) } \}\) for the summand , a \(p\) -adic unit. Then there is an orthogonal \(L _ { j } ( p ^ { j } )\) of \(L\) with \(b _ { j } ( e _ { k } ^ { ( j ) } , e _ { k } ^ { ( j ) } ) = u _ { k , j } p ^ { j }\) . \(A\) similar result is true for quadratic \(p\) -adic lattices. \(p = 2\) . For all \(j \geq 1\) , let \[ \begin{array} { r } { \small { \mathrm { d } } \mathrm { g } _ { L _ { j } ( 2 ^ { j } ) } = \mathbb { G } _ { k = 1 } ^ { a _ { j } } \langle u _ { j , k } \cdot 2 ^ { - j } \rangle \ \mathbb { G } ^ { b _ { j } } v _ { j } \ \mathbb { G } ^ { c _ { j } } u _ { j } , \quad u _ { j , k } \in \mathrm { D } ( \mathbb { Z } / 2 ^ { j + 1 } \mathbb { Z } ) , } \end{array} \] be an orthogonal direct sum splitting. Then there exists a compatible orthogonal direct sum splitting \[ L _ { j } ( 2 ^ { j } ) = \mathbb { \oplus } _ { k = 1 } ^ { a _ { j } } \langle \widetilde { u } _ { j , k } \cdot 2 ^ { j } \rangle \ \mathbb { \oplus } ^ { b _ { j } } V _ { j } \ \mathbb { \oplus } ^ { c _ { j } } U _ { j } , \quad \widetilde { u } _ { j , k } \in \mathsf { D } ( \mathbb { Z } _ { 2 } ) . \] Here \(\widetilde { u } _ { j , k } = u _ { j , k } \in \{ \pm 1 , \pm 3 \}\) if \(j \geq 2\) , while for \(j = 1\) the representative \(\widetilde { u } _ { 1 , k } \in\) \(\{ \pm 1 , \pm 3 \}\) coincides up to multiplication by \(^ { - 3 }\) mod 8 with the representative \(u _ { 1 , k } \in \{ 1 , 3 \}\) . In other words, such non-isometric rank one lattices have the same discriminant form. For \(j \geq 2\) , \(a _ { j } = 0\) if and only if the ideal in \(\mathbb { Z } _ { 2 }\) generated by \(b _ { q } ( L _ { j } , L _ { j } )\) differs from the ideal generated by \(2 \cdot q _ { j } ( L _ { j } )\) – or – equivalently, if and only the submodule of \(\mathbb { Q } _ { 2 } / \mathbb { Z } _ { 2 }\) generated by \(b _ { q } ( G _ { j } , G _ { j } )\) , \(G _ { j } = \mathsf { d g } _ { L _ { j } ( 2 ^ { j } ) }\) , differs from the submodule generated by \(2 \cdot q _ { j } ( G _ { j } )\) . In particular, the vanishing or not of \(a _ { j }\) is an intrinsic property of the lattice.
Remark 10.3.3
Remark 10.3.3. In Chapter 11 we return to the nature of the discriminant form map on lattices whose rank equals the length of its discriminant quadratic form. See Proposition 11.1.3 and Proposition 11.2.4.
Lemma 10.4.1
Lemma 10.4.1. Let \(f , g\) be two non-degenerate quadratic forms over \(\mathbb { Q } _ { p }\) . Then \[ \varepsilon _ { p } ( f \oplus g ) = \varepsilon _ { p } ( f ) \cdot \varepsilon _ { p } ( g ) \cdot ( \operatorname { d i s c } ( f ) , \operatorname { d i s c } ( g ) ) _ { p } . \]
Lemma 10.4.2
Lemma 10.4.2. 1. Let \(p\) be an odd prime and suppose that \(f = p ^ { k } \cdot g\) with \(g\) unimodular of rank \(r\) . Then we have \[ \varepsilon _ { p } ( f ) = ( - 1 ) ^ { \frac { 1 } { 2 } r ( r - 1 ) k \cdot \varepsilon ( p ) } \left( { \frac { \operatorname { d i s c } ( g ) } { p } } ^ { k ( r - 1 ) } \right) , \quad \varepsilon ( p ) = \left\{ { \begin{array} { l l } { 0 } & { { \mathrm { ~ } } i f \ p \equiv 1 { \bmod { 4 } } } \\ { 1 } & { { \mathrm { ~ } } i f \ p \equiv 3 { \bmod { 4 } } . } \end{array} } \right. \] 2. Let \(f\) be a unimodular dyadic quadratic lattice of rank \(r _ { f }\) and discriminant \(d _ { f }\) , and set \[ \mu ( f ) = - r _ { f } - d _ { f } + 1 , \quad \delta _ { f } = \frac 1 2 ( d _ { f } - 1 ) . \] Then \(\mu ( f )\) is divisible by 4 and \[ \varepsilon _ { 2 } ( f ) = ( - 1 ) ^ { { \frac { 1 } { 4 } } \mu ( f ) } = ( - 1 ) ^ { { \frac { 1 } { 4 } } ( - r _ { f } - d _ { f } + 1 ) } . \] If \(g\) is also a unimodular form, then \(\mu ( f \circledast g ) = \mu ( f ) + \mu ( g ) - 4 \delta _ { f } \cdot \delta _ { g }\) .