FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 5 definitions, 2 theorems, 4 propositions, 2 lemmas, 3 corollaries and 2 remarks.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction / notation
For a symmetric form \(b\) on an \(R\)-module \(V\), the correlation map \(b_V:V\to V^*\) sends \(x\) to \([y\mapsto b(x,y)]\); non-degeneracy means injectivity and unimodularity means bijectivity.
Lemma C.1.1
Lemma C.1.1. Let \(\{ w _ { 1 } , \ldots , w _ { r } \}\) be a basis of \(W\) . Then \(\beta _ { W }\) is surjective if and only if there exists vectors \(v _ { 1 } , \dotsc , v _ { r } \in V\) with \(b ( w _ { i } , v _ { j } ) = \delta _ { i j }\) . If \(\beta _ { W } ( Z ) = W ^ { * }\) for some submodule \(Z \subset W\) , then we may assume that the vectors \(v _ { i }\) belong to \(Z\) .
Definition / construction / notation
A quadratic form \(q\) on \(V\) is non-degenerate precisely if its polar form \(b _ { q }\) is non-degenerate. Recall also that any vector \(x \ \in \ V\) with \(q ( x )\) a unit defines a reflection \(\sigma _ { x } : V \to V\) given by \(y \mapsto y - b _ { q } ( x , y ) q ( x ) ^ { - 1 } x\) . The quadratic form \(q\) on an \(R\) -module \(V\) induces the \(k\) -valued quadratic form \(\overline { { q } }\) on \({ \overline { { V } } } = V \otimes _ { R } k = V / { \mathfrak { m } } V\) defined by \(\overline { { { q } } } ( \overline { { { x } } } ) = q ( x )\) mod \(\mathbf { m }\) and \(q\) is unimodular if and only if the form \(\overline { { q } }\) is unimodular (see \(\ S\) 6.3.B, Example 6).
Proposition C.1.2
Proposition C.1.2. Let \(( V , q )\) be a quadratic inner product space over \(R\) of finite rank, and \(W , W ^ { \prime } , Z\) submodules. Let \(W , W ^ { \prime }\) be free and assume that \[ \beta _ { W } ( Z ) = W ^ { \ast } , \quad \beta _ { W ^ { \prime } } ( Z ) = W ^ { \prime \ast } . \] Suppose \(t : W \stackrel { \simeq } { \to } W ^ { \prime }\) is an isometry such that \[ t ( x ) - x \in Z \ f o r \ a l l \ x \in W . \] Then ?? extends to an isometry \(\tilde { t } : V \ \stackrel { \sim } { \to } \ V\) such that \(\tilde { t } \ =\) id on the orthogonal complement of \(Z\) . Moreover, \(\tilde { t }\) is a product of reflections in vectors of \(Z\) if we are in one of the following cases: \[ \begin{array} { r l } { k \neq \mathbb { F } _ { 2 } \quad } & { a n d \overline { { q } } ( \overline { { Z } } ) \neq 0 , } \\ { k = \mathbb { F } _ { 2 } \quad } & { a n d \overline { { q } } ( \overline { { Z } } ^ { \perp } ) \neq 0 . } \end{array} \] Taking \(Z = V\) gives an unconditional extension of \(t\) to \(V\) which generalizes Corollary 7.2.8:
Corollary C.1.3
Corollary C.1.3 (Witt’s extension theorem over local rings). Let \(( V , q )\) be a quadratic inner product space over \(R\) of finite rank, ??, ??′ primitive free submodules such that \(\beta _ { W }\) and \(\beta _ { W ^ { \prime } }\) are surjective (this is in particular the case for unimodular submodules \(W\) and \(W ^ { \prime }\) ) and let \(t : W \to W ^ { \prime }\) be an isometry. Then ?? extends to an isometry of \(V\) . If \(k \neq \mathbb { F } _ { 2 }\) , then this extension is a product of hyperplane reflections.
Remark: Remark
Remark. 1. As we have observed (see Remark 7.2.9.1), Witt’s extension theorem is equivalent to Witt’s cancelation theorem 7.2.7. The proof of this does not depend on 2 being invertible or not, and so Witt’s cancelation theorem likewise holds over any local ring. 2. We may take \(W \ = \ W ^ { \prime }\) which implies that in case \(k \neq \mathbb { F } _ { 2 }\) , every isometry is a product of hyperplane reflections. Below we discuss the case \(k = \mathbb { F } _ { 2 }\) . See Theorem C.1.4.
Theorem C.1.4
Theorem C.1.4 (Cartan–Dieudonn´e over local rings). Every isometry of a quadratic inner product space \(( V , q )\) over a local ring \(( R , \mathfrak { m } )\) is a product of hyperplane reflections, except if \(k = R / { \mathfrak { m } } = \mathbb { F } _ { 2 }\) , \(\operatorname { r a n k } ( V ) = 4\) with quadratic form isometric to \(x ^ { 2 } + x y + y ^ { 2 } + u ^ { 2 } + u v + v ^ { 2 }\) or if \(k = R / { \mathfrak { m } } = \mathbb { F } _ { 2 }\) , \(\operatorname { r a n k } ( V ) = 2\) .
Definition / construction / notation
For a non-degenerate quadratic torsion group \((G,q)\), the normalized Gauss sum is \(\gamma_{(G,q)}=|G|^{-1/2}\sum_{x\in G}\exp(2\pi i q(x))\in\mathbf C\).
Theorem C.2.1
Theorem C.2.1 (Milgram). Let \(L\) be a non-degenerate integral quadratic lattice with discriminant form \(( G , q )\) . Then \(\gamma _ { ( G , q ) } = \rho _ { 8 } ^ { \tau _ { 8 } ( q ) } , \quad \rho _ { 8 } = \exp ( 2 \pi i / 8 )\) . 1
Definition / construction
For a possibly degenerate quadratic torsion form \(q\) on \(G\) and a homomorphism \(f:\mathbf Q/\mathbf Z\to\mathbf Q/\mathbf Z\), the generalized Gauss sum is \(\gamma_{(G,q)}(f)=|G|^{-1/2}\sum_{x\in G}\exp(2\pi i\,f(q(x)))\).
Proposition C.2.2
Proposition C.2.2. (1) If \(\iota : ( G , q ) \simeq ( G ^ { \prime } , q ^ { \prime } )\) is an isometry of torsion quadratic forms and \(f , f ^ { \prime } : \mathbb { Q } / \mathbb { Z } \to \mathbb { Q } / \mathbb { Z }\) homomorphisms for which \(f ^ { \prime } { \circ } q ^ { \prime } { \circ } l ~ = ~ f { \circ } q\) , then \(\gamma _ { ( G ^ { \prime } , q ^ { \prime } ) } ( f ^ { \prime } ) = \gamma _ { ( G , q ) } ( f )\) . (2) Let \(\left( G , q \right) = \left( G ^ { \prime } , q ^ { \prime } \right) \oplus \left( G ^ { \prime \prime } , q ^ { \prime \prime } \right)\) , then for all homomorphisms \(f : \mathbb { Q } / \mathbb { Z } \to \mathbb { Q } / \mathbb { Z }\) one has \(\gamma _ { ( G , q ) } ( f ) = \gamma _ { ( G ^ { \prime } , q ^ { \prime } ) } ( f ) \cdot \gamma _ { ( G ^ { \prime \prime } , q ^ { \prime \prime } ) } ( f )\) . (3) If \(H\) is a totally isotropic subgroup of \(( G , q )\) then \(\begin{array} { r } { \gamma _ { ( G , q ) } = \sqrt { | H | } \frac { 1 } { \sqrt { | G / H ^ { \perp } | } } \cdot \gamma _ { ( H ^ { \perp } / H , q ) } . } \end{array}\) If, moreover, \(( G , q )\) is non-degenerate, then \(\gamma _ { ( G , q ) } = \gamma _ { ( H ^ { \perp } / H , q ) }\) . (4) The quotient group2 \(\overline { { { G } } } ~ = ~ G / \mathrm { r a d } ( q )\) with induced quadratic form \(\overline { { q } }\) satisfies \(\gamma _ { ( G , q ) } ( f ) = \sqrt { | \operatorname { r a d } ( q ) | } \cdot \gamma _ { ( \overline { G } , \overline { q } ) } ( f )\) for every homomorphism \(f : \mathbb { Q } / \mathbb { Z } \to \mathbb { Q } / \mathbb { Z }\) .. (5) Suppose that \(\operatorname { r a d } ( q ) \neq G ^ { \bot }\) , then \(\gamma _ { ( G , q ) } ( = \gamma _ { ( G , q ) } ( \mathrm { i d } ) ) = 0\) .
Proposition C.2.3
Proposition C.2.3. (1) Let \(p\) be a prime and \(G = \mathbb { Z } / p ^ { k } \mathbb { Z }\) , \(k \geq 1\) . Then \[ \gamma _ { ( G , \langle u \cdot p ^ { - k } \rangle ) } ( p ^ { \ell } ) = { \left\{ \begin{array} { l l } { p ^ { k / 2 } } & { i f \ell > k } \\ { p ^ { \ell / 2 } } & { i f \ell = k , \quad p \neq 2 } \\ { 0 } & { i f \ell = k , \quad p = 2 } \\ { p ^ { \ell / 2 } \cdot \gamma _ { ( G , \langle u \cdot p ^ { - ( k - \ell ) } \rangle ) } } & { i f \ell < k } \end{array} \right. } \] (2) For \(u _ { k } , v _ { k }\) , \(G = \mathbb { Z } / 2 ^ { k } \mathbb { Z } \oplus \mathbb { Z } / 2 ^ { k } \mathbb { Z }\) , we have \[ \gamma _ { ( G , u _ { k } ) } ( 2 ^ { \ell } ) = \left\{ \begin{array} { l l } { 2 ^ { k } } & { \mathrm { ~ } i f \ell \ge k } \\ { 2 ^ { \ell } \cdot \gamma _ { ( G , u _ { k - \ell } ) } } & { \mathrm { ~ } i f \ell < k } \end{array} \right. \mathrm { ~ } \mathrm { ~ } a n d \gamma _ { ( G , v _ { k } ) } ( 2 ^ { \ell } ) = \left\{ \begin{array} { l l } { 2 ^ { k } } & { \mathrm { ~ } i f \ell \ge k } \\ { 2 ^ { \ell } \cdot \gamma _ { ( G , v _ { k - \ell } ) } } & { \mathrm { ~ } i f \ell < k } \end{array} \right. \]
Corollary C.2.5
Corollary C.2.5. (1). Let \(p\) be a prime and \(G = \mathbb { Z } / p ^ { k } \mathbb { Z }\) . Then \[ \sigma _ { \ell } ( G , \langle u \cdot p ^ { - k } \rangle ) = { \left\{ \begin{array} { l l } { 1 } & { \ i f \ \ell > k } \\ { 1 } & { \ i f \ \ell = k , \quad p \neq 2 } \\ { 0 } & { \ i f \ \ell = k , \quad p = 2 } \\ { \gamma _ { ( G , \langle u \cdot p ^ { - ( k - \ell ) } \rangle ) } } & { \ i f \ \ell < k . } \end{array} \right. } \] (2) For \(u _ { k } , v _ { k }\) , \(G = \mathbb { Z } / 2 ^ { k } \mathbb { Z } \oplus \mathbb { Z } / 2 ^ { k } \mathbb { Z }\) , we have \[ \sigma _ { \ell } ( G , u _ { k } ) = \left\{ \begin{array} { l l } { 1 } & { \mathrm { ~ } i f \ell \geq k } \\ { \gamma _ { ( G , u _ { k - \ell } ) } } & { \mathrm { ~ } i f \ell < k } \end{array} \right. \quad a n d \sigma _ { \ell } ( G , v _ { k } ) = \left\{ \begin{array} { l l } { 1 } & { \mathrm { ~ } i f \ell \geq k } \\ { \gamma _ { ( G , v _ { k - \ell } ) } } & { \mathrm { ~ } i f \ell < k } \end{array} \right. \]
Proposition C.3.1
Proposition C.3.1. Let \(( G , q )\) be a homogeneous 2-primary quadratic torsion group of exponent \(k\) and length \(\ell ( G )\) . Then (1) \(q \simeq w _ { k } \oplus ^ { b } v _ { k } \oplus ^ { c } u _ { k }\) , where \(w _ { k }\) is an orthogonal direct sum of \(a\) cyclic groups, \(a \leq 2\) , \(b \leq 1\) , \(\begin{array} { r } { c = \frac { 1 } { 2 } ( \ell ( G ) - a ) - b } \end{array}\) , and where ?? is an isometry invariant of \(q\) . (2) If \(a = 2 , b = 0\) , then \(w _ { k }\) is isometric to one of the following: \[ \langle 2 ^ { - k } \rangle \mathbb { \ O } \langle 2 ^ { - k } \rangle , \quad \langle - 2 ^ { - k } \rangle \mathbb { \ O } \langle - 2 ^ { - k } \rangle , \ w i t h \ \delta ( q ) = 1 \] \[ \langle - 2 ^ { - k } \rangle \oplus \langle 3 \cdot 2 ^ { - k } \rangle , \quad \langle 2 ^ { - k } \rangle \oplus \langle - 3 \cdot 2 ^ { - k } \rangle \ w i t h \ \delta ( q ) = - \] (3) If \(a = 2 , b = 1\) , one may assume that \(w _ { k } \oplus v _ { k }\) is one of the following: \[ \begin{array} { r l r } & { } & { \langle 2 ^ { - k } \rangle \oplus \langle 3 \cdot 2 ^ { - k } \rangle \oplus v _ { k } \ w i t h \ \delta ( q ) = 1 } \\ & { } & { \langle 2 ^ { - k } \rangle \oplus \langle - 2 ^ { - k } \rangle \oplus v _ { k } \ w i t h \ \delta ( q ) = - 3 . } \end{array} \]
Definition / construction / notation
A normal form as in Proposition C.3.1 is called a reduced homogeneous normal form. Such normal forms are collected in the following table.
Corollary C.3.2
Corollary C.3.2. Every 2-primary homogeneous quadratic form has a unique reduced normal form given in Table C.3.1. For \(k \geq 2\) these forms are mutually non-isometric. For \(k = 1\) the forms with \(\delta = 1 , - 3\) with corresponding remaining invariants are isometric and the same holds for forms with \(\delta = - 1 , 3\) .
Lemma C.3.3
Lemma C.3.3. Let \(u , u ^ { \prime } , u ^ { \prime \prime }\) be units in \(\mathbb { Z } _ { 2 }\) . Then the following relations hold between non-homogeneous dyadic forms. \[ \langle u \cdot 2 ^ { k - 1 } \rangle \oplus U _ { k } \simeq \langle - 3 u \cdot 2 ^ { k - 1 } \rangle \oplus V _ { k } ~ f o r ~ k \geq 1 \] \[ U _ { k - 1 } \oplus \langle u \cdot 2 ^ { k } \rangle \simeq \langle 3 u \cdot 2 ^ { k } \rangle \oplus V _ { k - 1 } ~ f o r ~ k \geq 1 \] \[ \langle u \cdot 2 ^ { k - 1 } \rangle \mathbb { O } \langle u ^ { \prime } \cdot 2 ^ { k } \rangle \simeq \langle ( u + 2 u ^ { \prime } ) \cdot 2 ^ { k - 1 } \rangle \mathbb { O } \langle ( u ^ { \prime } + 2 u ) \cdot 2 ^ { k } \rangle f o r k \geq 1 \] \[ ( \langle u \rangle \mathbb { O } \langle u ^ { \prime } \rangle ) ( 2 ^ { k - 1 } ) \mathbb { O } \langle 2 ^ { k } \rangle \simeq ( \langle u + 2 \rangle \mathbb { O } \langle u ^ { \prime } - 2 \rangle ) ( 2 ^ { k - 1 } ) \mathbb { O } \langle - 3 \cdot 2 ^ { k } \rangle \] \[ \dot { u } \equiv u ^ { \prime } \bmod 4 , \quad k \geq 1 . \] \[ \langle u \cdot 2 ^ { k - 2 } \rangle \mathbb { O } \langle u ^ { \prime } \cdot 2 ^ { k } \rangle \simeq \langle - 3 u \cdot 2 ^ { k - 2 } \rangle \mathbb { O } \langle - 3 u ^ { \prime } \cdot 2 ^ { k } \rangle , f o r k \geq 3 . \] Similar relations hold for their quadratic torsion forms, that is, for the 2-primary quadratic torsion groups where the exponents \(2 ^ { j }\) are replaced by \(2 ^ { - j }\) , and for \(u _ { k }\) and \(v _ { k }\) instead of \(U _ { k }\) and \(V _ { k }\) , respectively.
Remark C.3.4
Remark C.3.4. The relations (V)–(IX) from Lemma C.3.3 involving different exponents give further reductions of non-homogeneous forms whose homogeneous summands are already in reduced normal form. These relations affect three consecutive exponents. Starting with the largest exponent and going down, this gives a new reduced normal form as is shown in [156, Ch.4.4]. Furthermore, as a consequence, every 2-primary quadratic torsion form and every quadratic dyadic lattice has a reduced normal form and no two such forms are isometric.