FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 9 definitions, 1 theorem, 4 propositions, 9 lemmas, 1 corollary, 3 examples and 1 construction.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Lemma 9.1.1
Lemma 9.1.1. The torsion module \(G\) is isomorphic to its dual \(G _ { Q ( R ) / R } ^ { * }\) . Let \[ G \simeq \oplus _ { j = 1 } ^ { r } R / d _ { j } R , \quad d _ { 1 } | d _ { 2 } | \cdot \cdot \cdot | d _ { r } \] be the invariant factor decomposition with corresponding generators \(e _ { 1 } , \ldots , e _ { r }\) . Then the isomorphism \(G \stackrel { \simeq } { \to } G _ { Q ( R ) / R } ^ { * }\) can be chosen such that \(e _ { j }\) maps to the function \(u _ { j } ^ { \ast } \in G _ { Q ( R ) / R } ^ { \ast }\) for which \(u _ { j } ^ { \ast } ( e _ { i } ) = d _ { j } ^ { - 1 } \delta _ { i j } \in Q ( R ) / R\) .
Definition 9.1.2
Definition 9.1.2. An ordered basis of \(G\) is a system of generators \(e _ { 1 } , \ldots , e _ { r }\) of \(G\) adapted to the elementary divisors \(\{ d _ { 1 } , \ldots , d _ { r } \}\) ordered in such a way that \(d _ { 1 } | d _ { 2 } | \cdots | d _ { r }\) as in the above lemma.
Lemma 9.1.3
Lemma 9.1.3. Let \(B ^ { \prime } = ( B _ { i j } )\) be a symmetric \(r \times r\) matrix with entries in the field \(Q ( R )\) , Then \(B = B ^ { \prime }\) mod \(R\) is the Gram matrix of a symmetric torsion form on \(G\) relative to an ordered basis for \(G\) adapted to the elementary divisors \(\{ d _ { 1 } , \ldots , d _ { r } \}\) if and only if \(d _ { i } B _ { i j } \in R\) and \(d _ { j } B _ { i j } \in R\) for \(i , j = 1 , \dots , r\) . In other words, using that \(d _ { 1 } | d _ { 2 } | \cdots | d _ { r }\) , this is the case if and only if \(B ^ { \prime }\) has the form \[ B ^ { \prime } = ( B _ { i j } ) = \left( \begin{array} { c c c c } { { \displaystyle \frac { A _ { 1 1 } } { d _ { 1 } } } } & { { \displaystyle \frac { A _ { 1 2 } } { d _ { 1 } } } } & { { \cdots } } & { { \displaystyle \frac { A _ { 1 r } } { d _ { 1 } } } } \\ { { \displaystyle \frac { A _ { 2 1 } } { d _ { 1 } } } } & { { \displaystyle \frac { A _ { 2 2 } } { d _ { 2 } } } } & { { \cdots } } & { { \displaystyle \frac { A _ { 2 r } } { d _ { 2 } } } } \\ { { \vdots } } & { { \ddots } } & { { \vdots } } & { { \vdots } } \\ { { \displaystyle \frac { A _ { r 1 } } { d _ { 1 } } } } & { { \displaystyle \frac { A _ { r 2 } } { d _ { 2 } } } } & { { \cdots } } & { { \displaystyle \frac { A _ { r r } } { d _ { r } } } } \end{array} \right) \qquad \boldsymbol { w i t h } \ A _ { i j } = A _ { j i } \in { \cal R } . \]
Corollary 9.1.4
Corollary 9.1.4. Let \(B ^ { \prime } = ( B _ { i j } )\) represent a Gram matrix of a symmetric \(R\) -torsion form \(b\) on \(G\) relative to an ordered basis corresponding to the elementary divisors \(d _ { 1 } , \ldots , d _ { r }\) of \(G\) as in Lemma 9.1.3. Then 1. \(d _ { 1 } \cdot \cdot \cdot d _ { r } \cdot \operatorname* { d e t } B \in R\) ; 2. if \(b\) is non-degenerate, then \(d _ { 1 } \cdot \cdot \cdot d _ { r } \cdot \operatorname* { d e t } B\) is relatively prime to \(d _ { 1 }\) .
Definition / construction
The reduced discriminant \(\delta(b)\) is the discriminant invariant attached to a non-degenerate symmetric torsion form.
Definition / construction
The reduced discriminant \(\delta(q)\) of a quadratic torsion form is the corresponding invariant of its polar form with the prescribed modulus.
Lemma 9.1.5
Lemma 9.1.5. Let \(( G , b )\) , \(( G ^ { \prime } , b ^ { \prime } )\) be two non-degenerate \(p\) -primary bilinear torsion forms 1 and let \(p ^ { a }\) , \(p ^ { a ^ { \prime } }\) be the first elementary divisors of \(G\) , respectively \(G ^ { \prime }\) . Then the reduced discriminant of \(G \oplus G ^ { \prime }\) is the product \(\delta ( b ) \cdot \delta ( b ^ { \prime } )\) considered in \(\mathsf { D } ( R / p ^ { c } R )\) , \(c = \operatorname* { m i n } ( a , a ^ { \prime } )\) , under the homomorphism which is induced by multiplication \[ R / p ^ { a } R \times R / p ^ { a ^ { \prime } } R \longrightarrow R / p ^ { c } R . \]
Example / comparison
Example 9.1.6. For a homogeneous \(p\)-primary discriminant group, the reduced discriminant is one of the two square classes when \(p\) is odd. For \(p=2\), the basic cyclic and exceptional blocks give the reduced-discriminant values recorded in Table 9.1.1.
Definition / construction / notation
From now on we assume that \(G\) is a finite abelian group. We shall write the group operation additively so that we consider \(G\) as a torsion \(\mathbb { Z }\) -module.
Proposition 9.2.1
Proposition 9.2.1. 1. There is a commutative diagram \[ \begin{array} { l } { G \times G \xrightarrow { b \quad \quad \quad } \displaystyle > { \mathbb Q } / { \mathbb Z } } \\ { \displaystyle \int _ { G _ { p } \times G _ { p } } ^ { \textstyle \int \quad \quad } \displaystyle \iint _ { 0 } ^ { } } \\ { G _ { p } \times G _ { p } \xrightarrow { b | _ { G _ { p } } \quad \quad > { \mathbb Q } ^ { ( p ) } / { \mathbb Z } } \xrightarrow { \quad \quad \quad } \displaystyle { \mathbb Q } _ { p } / { \mathbb Z } _ { p } . } \end{array} \] In particular, \(( G _ { p } , b | _ { G _ { p } } )\) is in a natural way a \(p\) -primary symmetric torsion group. 2. The Sylow decomposition is orthogonal, i.e. \(( G , b ) = \oplus _ { p } ( G _ { p } , b | _ { G _ { p } } )\) . Similar considerations hold for quadratic torsion forms \(( G , q )\) , where the polar form \(b _ { q }\) is used. Since \(q ( x ) = { \textstyle \frac { 1 } { 2 } } b _ { q } ( x , x )\) , the values \(q ( x )\) for \(x \in G _ { p }\) belong to \(\begin{array} { r } { \frac { 1 } { 2 } \mathbb { Q } ^ { ( p ) } / \mathbb { Z } \simeq \frac { 1 } { 2 } \mathbb { Q } _ { p } / \mathbb { Z } _ { p } } \end{array}\) which is isomorphic to \(\mathbb { Q } _ { p } / \mathbb { Z } _ { p }\) if \(p\) is odd, since then 2 is a unit in \(\mathbb { Z } _ { p }\) . For \(p = 2\) we have \(\begin{array} { r } { \frac { 1 } { 2 } \mathbb { Q } ^ { ( 2 ) } / \mathbb { Z } = \mathbb { Q } ^ { ( 2 ) } / \mathbb { Z } _ { 2 } \simeq \mathbb { Q } _ { 2 } / \mathbb { Z } _ { 2 } } \end{array}\) . So for quadratic forms the diagram of Proposition 9.2.1 becomes \[ \begin{array} { l } { G \xrightarrow { q } \mathbb { Q } / \mathbb { Z } } \\ { \displaystyle \int _ { G _ { p } } \omega \xrightarrow [ q ] { \mathcal { q } } \mathbb { \bigcup } _ { \mathbb { Q } ^ { ( p ) } / \mathbb { Z } \xrightarrow [ \infty ] { } \mathbb { Q } _ { p } / \mathbb { Z } _ { p } . } } \end{array} \] Hence also \(( G _ { p } , q | _ { G _ { p } } )\) is in a natural way a \(p\) -primary quadratic torsion group and one has an orthogonal Sylow decomposition \(( G , q ) = \oplus _ { p } ( G _ { p } , q | _ { G _ { p } } )\) .
Lemma 9.3.1
Lemma 9.3.1. 1. Let \(( H , b )\) be a symmetric form on a homogeneous \(p\) -primary torsion group of exponent \(k\) . Then the Gram matrix of \(^ { b }\) with respect to an ordered basis is represented by \[ B = p ^ { - k } A , \quad A = ( A _ { i j } ) , A _ { i j } = A _ { j i } \in \mathbb { Z } , \] and \(b\) is non-degenerate if and only if \(\operatorname* { d e t } ( A )\) is relatively prime to \(p\) . 2. Let \(( G , b )\) be a symmetric form on a \(p\) -primary group \(G\) with Jordan decomposition \(G \ : = \ : \oplus _ { k = 1 } ^ { s } H _ { k }\) . If \(^ { b }\) is non-degenerate, the restriction of \(^ { b }\) to each homogeneous summand \(H _ { k }\) is non-degenerate.
Example / comparison
Example 9.3.2. A non-degenerate exponent-one quadratic \(p\)-primary torsion form is an ordinary quadratic space over \(\mathbf F_p\) for odd \(p\), hence has two isometry types; for \(p=2\) one must distinguish type-I forms governed by the Arf invariant from type-II \(\mathbf Z/4\)-valued forms.
Definition / construction / notation
The reduced discriminant \(\delta ( b ) \in \mathsf { D } ( \mathbb { Z } / p ^ { k } \mathbb { Z } )\) is defined by means of formula (9.2). If \(p\) is odd, by (9.3), multiplication with \(p ^ { k - 1 }\) induces an isomorphism and identifies \(\delta ( b )\) with \(\delta ( \rho ( b ) ) \in \mathsf { D } ( \mathbb { Z } / p \mathbb { Z } )\) . This invariant can also be viewed as \(\mathrm { d i s c } ( \rho ( b ) ) \in\) \(\mathsf { D } ( \mathbb { F } _ { p } )\) provided we identify \(\mathbb { Z } / p \mathbb { Z }\) with the field \(\mathbb { F } _ { p }\) . The form \(\rho ( b )\) reflects the non-degeneracy of \(^ { b }\) :
Lemma 9.3.3
Lemma 9.3.3. If \(G\) is a homogeneous \(p\) -primary group of exponent \(k\) equipped with a symmetric \(p\) -primary torsion form \(^ { b }\) , then \(( G , b )\) is non-degenerate if and only if \(\rho ( b )\) is. More precisely, \(b \simeq p ^ { - k } \cdot b ^ { \prime }\) where \(b ^ { \prime }\) is unimodular and is (under suitable identifications) isometric to \(\rho ( b )\) . In that case, if \(H\) splits off orthogonally, then so does \(\rho ( H )\) and conversely.
Example / comparison
Example 9.3.4. For \(T\cong\mathbf Z/4\oplus\mathbf Z/4\oplus\mathbf Z/16\), the canonical filtration \(T_m=\{x:2^mx=0\}\) has \(\rho_4(T)\cong\mathbf Z/2\), \(\rho_2(T)\cong(\mathbf Z/2)^2\), and \(\rho_3(T)=\rho_1(T)=0\), recovering the homogeneous exponent-\(4\) and exponent-\(2\) layers.
Lemma 9.3.5
Lemma 9.3.5. Let \(G\) be a \(p\) -primary torsion group. 1. If \(G\) is a homogeneous group of exponent \(k\) , then \(\rho _ { k } ( G )\) coincides with \(\rho ( G )\) as in (9.5). 2. Every \(\rho _ { k }\) is additive on direct sums. 3. If \(G = \oplus H _ { m }\) with \(H _ { m }\) homogeneous of exponent ??, then \(\rho _ { k } ( G ) = \rho _ { k } ( H _ { k } ) =\) \(\rho ( H _ { k } )\) for every \(k\) .
Definition / construction / notation
analogous to the one in (9.5). Explicitly, if \([ x ]\) is the coset containing \(x \in G _ { k }\) we put for \(x , y \in G _ { k }\) \[ \rho _ { k } ( b ) ( [ x ] , [ y ] ) : = p ^ { k - 1 } b ( x , y ) . \]
Lemma 9.3.6
Lemma 9.3.6. Let \(( G , b )\) be a \(p\) -primary symmetric torsion group with Jordan decomposition \(G = \oplus H _ { k }\) . 1. If \(b\) is non-degenerate, then \(\rho _ { k } ( b )\) is non-degenerate for all \(k\) . 2. If \(\rho _ { k } ( b )\) is non-degenerate, then \(b | _ { H _ { k } }\) is non-degenerate.
Proposition 9.3.7
Proposition 9.3.7. A non-degenerate symmetric \(p\) -primary torsion group \(( G , b )\) admits a Jordan splitting, that is, an orthogonal Jordan decomposition2 \(G =\) \(\mathbb { O } _ { k = 1 } ^ { s } H _ { k }\) , where, we recall, each \(H _ { k }\) is a homogeneous \(p\) -primary torsion group of exponent \(k\) equipped with a non-degenerate symmetric form. Moreover, each of the symmetric torsion forms \(\rho _ { k } ( b )\) are non-degenerate and the reduced discriminant3 \(\delta ( b )\) is invertible.4
Definition 9.3.8
Definition 9.3.8. The basic invariants under isometry of a non-degenerate \(p\) -primary symmetric torsion group \(( G , b )\) are: • the number of elementary divisors of \(G\) equal to \(p ^ { k }\) \(=\) the length of \(H _ { k }\) in any Jordan splitting \(G = \mathbb { Q } H _ { k }\) of \(G\) = the rank of \(\rho _ { k } ( G )\) as a \(\mathbb { Z } / p \mathbb { Z }\) -module; • if \(p\) is odd, for each \(k\) as above, \(\mathrm { d i s c } ( \rho _ { k } ( b ) ) \in \mathsf { D } ( \mathbb { F } _ { p } )\) . For \(p\) -primary quadratic torsion groups \(( G , q )\) the invariants are those of the associated polar form \(b _ { q }\) .
Lemma 9.3.9
Lemma 9.3.9. Let \(p\) be an odd prime and let \(G\) be a \(p\) -primary torsion equipped with a quadratic form \(q\) . Define \(\rho _ { k } ( q )\) on \(\rho _ { k } ( G )\) by \[ \rho _ { k } ( q ) [ x ] : = \ c l a s s \ o f \ p ^ { k - 1 } q ( x ) \ i n \ \mathbb { Q } ^ { ( p ) } / \mathbb { Z } . \] This form is a well-defined \(p ^ { - 1 } \mathbb { Z } / \mathbb { Z }\) -valued quadratic form on \(\rho _ { k } ( G )\) . It is nondegenerate if and only if \(q\) is.
Definition 9.3.10
Definition 9.3.10. Let \(G\) be a 2-primary torsion group equipped with a symmetric form \(^ { b }\) ). Its halving , \(( \pmb { G } , \pmb { b } ) = ( \textstyle \frac { 1 } { 2 } \pmb { G } , \textstyle \frac { 1 } { 2 } \pmb { b } )\) , is the pair consisting of a 2-primary torsion group \(\textbf { G }\) of exponent \(\geq 2\) together with a symmetric form \(\pmb { b }\) on \(\textbf { G }\) constructed as follows: For each summand \(G _ { k }\) of exponent \(k\) in a chosen decomposition of \(G\) . choose generators, \(e _ { k 1 } , \ldots , e _ { k r }\) (with \(r\) depending on \(k\) ). Define \(\mathbf { G } _ { k }\) by generators \({ \pmb e } _ { k j }\) , \(j = 1 , \dotsc , r\) , of order precisely \(2 ^ { k + 1 }\) and set \(\pmb { G } = \bigoplus \pmb { G } _ { k }\) . The form \(\mathbf { b } = { \textstyle \frac { 1 } { 2 } } b\) is given by \[ { \pmb b } ( { \pmb e } _ { k i } , { \pmb e } _ { \ell j } ) = \frac { 1 } { 2 } b ( { \pmb e } _ { k i } , { \pmb e } _ { \ell j } ) \in { \mathbb { Q } } ^ { ( 2 ) } / { \mathbb { Z } } . \]
Definition / construction / notation
For a non-degenerate symmetric \(2\)-primary torsion group \((G,b)\) without order-\(2\) summands, the doubling construction puts a quadratic form on \(2G=G/G_1\) by \(q(\bar x)=b(x,x)\).
Construction / comparison
The doubling construction carries a non-degenerate symmetric \(2\)-primary torsion form without order-\(2\) summands to a non-degenerate quadratic form on \(2G\): its polar form satisfies \(b_q(\bar x,\bar y)=2b(x,y)\), so a vector in its radical lifts to \(G_1\).
Lemma 9.3.11
Lemma 9.3.11. Let \(( G , q )\) be a 2-primary quadratic torsion group. Then the halving \(( \pmb { G } , \pmb { b } )\) of \(( G , b _ { q } )\) gives a symmetric torsion group (without cyclic summands of order 2) such that we retrieve \(( G , q )\) via \(2 ( G , \pmb { b } )\) as above. Also \(( G , q )\) is non-degenerate if and only if \(( G , b )\) is non-degenerate.
Theorem 9.3.12
Theorem 9.3.12. A non-degenerate \(p\) -primary quadratic form \(( G , q )\) admits \(a\) Jordan splitting ?? = ⦹????=1????. Moreover, 1. For each \(k = 1 , \dots , s\) , the restriction \(q | _ { H _ { k } }\) is non-degenerate if and only if the quadratic torsion form \(\rho _ { k } ( q )\) is non-degenerate. Under suitable identifications \(q \vert H _ { k }\) is isometric to \(p ^ { - k } \rho _ { k } ( q )\) . 2. The reduced discriminant \(\delta ( b _ { q } )\) is invertible if and only if the quadratic torsion forms \(\rho _ { k } ( q )\) , \(k = 1 , \dots , s\) , are non-degenerate.
Proposition 9.4.1
Proposition 9.4.1. Let \(p\) be odd. \(A\) non-degenerate homogeneous \(p\) -primary symmetric torsion form \(^ { b }\) of exponent \(k\) and length \(r\) is isometric to \(\mathbb { O } ^ { r } \langle p ^ { - k } \rangle\) if \(\mathrm { d i s c } ( \rho _ { k } ( b ) ) = 1\) and to \({ \mathbb { O } ^ { r - 1 } \langle p ^ { - k } \rangle \mathbb { O } \langle \epsilon p ^ { - k } \rangle }\) if d \(\mathrm { l i s c } ( \rho _ { k } ( b ) ) = \epsilon\) , a non-square modulo \(p\) . A similar assertion holds for quadratic forms.
Proposition 9.4.2
Proposition 9.4.2. • A non-degenerate symmetric (respectively quadratic) form on a homogeneous 2-primary group of exponent \(k \geq 2\) and length \(r\) is isometric to \[ \begin{array} { r l } & { \quad \mathbb { O } _ { j = 1 } ^ { a } \langle u ^ { ( j ) } 2 ^ { - k } \rangle \mathbb { O } ^ { b } u _ { k } \mathbb { O } ^ { c } v _ { k } \quad a + 2 ( b + c ) = r , \quad u ^ { ( j ) } \in \mathsf { D } ( \mathbb { Z } / 2 ^ { k } \mathbb { Z } ) } \\ & { \quad \mathbb { O } _ { j = 1 } ^ { a } [ u ^ { ( j ) } 2 ^ { - k - 1 } ] \mathbb { O } ^ { b } u _ { k } \mathbb { O } ^ { c } v _ { k } \quad r e s p e c t i v e l y . } \end{array} \] • A non-degenerate symmetric form on a homogeneous 2-primary group of exponent 1 and length \(r\) is isometric to \[ \mathbb { O } _ { j = 1 } ^ { a } \langle u ^ { ( j ) } 2 ^ { - 1 } \rangle \mathbb { O } ^ { b } u _ { 1 } , \quad a + 2 b = r , \quad u ^ { ( j ) } \in \mathsf { D } ( \mathbb { Z } / 2 ^ { k } \mathbb { Z } ) . \] • A non-degenerate quadratic form on a homogeneous 2-primary group of exponent 1 and length \(r\) is isometric to \[ \begin{array} { r } { \mathbb { Q } ^ { a } [ u \cdot 2 ^ { - 2 } ] \ \mathbb { Q } ^ { b } \ u _ { 1 } \ \mathbb { Q } ^ { c } \ v _ { 1 } , \quad a + 2 b + 2 c = r , \quad u \in 1 , 3 = \mathsf { D } ( \mathbb { Z } / 4 \mathbb { Z } ) . } \end{array} \]