FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 8 definitions, 8 theorems, 5 propositions, 4 lemmas, 5 corollaries, 8 examples, 1 construction and 4 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 non-degenerate quadratic torsion form \(q^\#\), define \(\tau_8(q^\#)\) as the index modulo \(8\) of any even integral lattice whose discriminant quadratic form is \(q^\#\).
Theorem 12.1.1
Theorem 12.1.1. Every non-degenerate quadratic torsion form is the discriminant quadratic form of a non-degenerate quadratic lattice.
Remark 12.1.2
Remark 12.1.2. The reader may wonder why one cannot find a rank 2 integral lattice in order to realize \(v _ { k }\) of length two. To explain this, if there would have been a rank two integral lattice with discriminant quadratic form \(v _ { k }\) , by Theorem 12.4.4, we would have \(\operatorname { d i s c } ( V _ { k } ) = \pm 2 ^ { 2 k }\) in \(\mathsf { D } ( \mathbb { Z } _ { 2 } )\) which is not the case since \(\mathrm { d i s c } ( V _ { k } ) = 3 { \cdot } 2 ^ { 2 k }\) .
Definition 12.2.1
Definition 12.2.1. Two non-degenerate symmetric lattices \(L _ { 1 }\) and \(L _ { \mathrm { 2 } }\) are stably equivalent if there are unimodular lattices \(U _ { 1 }\) and \(U _ { 2 }\) such that \(L _ { 1 } \odot U _ { 1 }\) and \(L _ { 2 } \odot U _ { 2 }\) are isometric. If \(L _ { 1 }\) and \(L _ { \mathrm { 2 } }\) are even and \(U _ { 1 }\) and \(U _ { 2 }\) can be taken even as well, then we say that \(L _ { 1 }\) and \(L _ { \mathrm { 2 } }\) are evenly stably equivalent.
Theorem 12.2.2
Theorem 12.2.2. Two non-degenerate symmetric integral lattices with isometric discriminant symmetric forms are stably equivalent. Two non-degenerate quadratic lattices with isometric discriminant quadratic forms are evenly stably equivalent.
Definition 12.2.3
Definition 12.2.3. The index modulo 8 of a non-degenerate quadratic torsion group \(( G , q )\) , denoted \(\tau _ { 8 } ( q )\) , is the modulo 8 integer \(\tau ( L )\) mod 8 of any nondegenerate quadratic lattice \(L\) whose discriminant form is \(q\) .
Corollary 12.2.4
Corollary 12.2.4. Let \(( G , q )\) be a non-degenerate quadratic torsion group and \(L\) a non-degenerate quadratic lattice with \(q _ { L } ^ { \# } = q\) . Then \[ \tau _ { 8 } ( q ) \equiv \tau ( L ) \bmod 8 \] is independent of the choice of \(L\) . The index modulo 8 is additive in \(\mathbb { Z } / 8 \mathbb { Z }\) , that is \(\tau _ { 8 } ( q _ { 1 } \oplus q _ { 2 } ) = \tau _ { 8 } ( q _ { 1 } ) + \tau _ { 8 } ( q _ { 2 } )\)
Example / comparison
1 we find for instance that \(\begin{array} { r } { \tau _ { 8 } [ \frac { n } { 2 ( n + 1 ) } ] = n } \end{array}\) mod 8. This follows since the quadratic torsion group \(\textstyle { \left[ { \frac { n } { 2 ( n + 1 ) } } \right] }\) is the discriminant form of the positive definite lattice \(A _ { n }\) of rank \(n\) .
Example / comparison
We have \(\tau _ { 8 } ( u _ { k } ) = 0\) since \(\tau ( U ( k ) ) = 0\) .
Example / comparison
Recall the notation \(\widetilde { V } _ { k } ^ { - 1 }\) for the even integral lattice (12.2) representing \(v _ { k }\) . For \(k = 6\) we find the quadratic form \(W _ { 6 } = 2 ^ { - 5 } ( x _ { 1 } ^ { 2 } + x _ { 1 } x _ { 2 } + x _ { 2 } ^ { 2 } ) + x _ { 2 } x _ { 3 } +\) \(2 1 x _ { 3 } ^ { 2 } + x _ { 3 } x _ { 4 } - x _ { 4 } ^ { 2 }\) on its dual whose signature is the same as that of \(\widetilde { V } _ { 6 } ^ { - 1 }\) . Since \(\mathrm { d e t } \Big ( \widetilde { V } _ { 6 } ^ { - 1 } \Big ) > 0\) and the value of the quadratic form on the fourth basis vector is negative, the signature must be \(( 2 , 2 )\) and so \(\tau _ { 8 } ( v _ { 6 } ) = 0\) . We give the argument for all \(k\) when proving Proposition 12.3.3.
Example / comparison
We construct an even lattice with discriminant form \(\langle 4 \cdot\) 5−2⟩. Let ?? = 4, ?? = 52. Since 25 = 6 · 4 − (−1), we get ??0 = 6 1 we g \(A _ { 1 } =\) \({ \binom { - 4 } { 1 } } \quad { \binom { - 1 } { 0 } } .\) \(a _ { 0 } = 6\) \(Q ^ { - 1 } = { \binom { 4 \cdot 5 ^ { - 2 } \quad 1 } { 1 \qquad 6 } }\) \(Q = { \binom { - 1 5 0 } { 2 5 } } \quad 2 5 \atop - 4 \quad\) \(\operatorname* { d e t } Q = - 2 5 < 0\) \(Q\) of \(\langle 4 \cdot 5 ^ { - 2 } \rangle\) is also zero.
Example / comparison
Consider now \(\big < - \frac { 2 } { 3 } \big >\) . Here \(- 3 = 2 \cdot ( - 2 ) - ( - 1 )\) and hence \(Q ^ { - 1 } = \left( \begin{array} { c c } { { - \frac { 2 } { 3 } } } & { { 1 } } \\ { { 1 } } & { { - 2 } } \end{array} \right)\) with inverse \(Q = \left( { \begin{array} { c c } { - 6 } & { - 3 } \\ { - 3 } & { - 2 } \end{array} } \right)\) which is negative definite, as one easily verifies. Hence \(\tau _ { 8 } ( \langle \langle \frac { - 2 } { 3 } \rangle ) = - 2\) (one can also use \(A _ { 2 } ( - 1 )\) ).
Example / comparison
The even form with matrix \(Q\) and discriminant form \(\big < \frac { 2 } { 5 } \big >\) we found in Example 4.3.11 shows that the signature is \(( 1 , 1 )\) and so its index mod 8 is zero.
Example / comparison
For \(\Big \langle \frac { 5 } { 8 } \Big \rangle\) we find \(a _ { 0 } = a _ { 1 } = 2\) and \(a _ { 2 } = - 2\) so that \[ Q ^ { - 1 } = \left( \begin{array} { l l l } { 5 \cdot 2 ^ { - 3 } } & { 1 } & { 0 } \\ { 1 } & { 2 } & { 1 } \\ { 0 } & { 1 } & { 2 } \end{array} \right) \mathrm { ~ a n d ~ } Q = \left( \begin{array} { l l l } { - 2 4 } & { 1 6 } & { - 8 } \\ { 1 6 } & { - 1 0 } & { 5 } \\ { - 8 } & { 5 } & { - 2 } \end{array} \right) . \] The signature of \(Q\) is found to be \(( 2 , 1 )\) and hence the index mod 8 of \(\Big \langle \frac { 5 } { 8 } \Big \rangle\) is 1.
Proposition 12.3.2
Proposition 12.3.2. The values of \(\tau _ { 8 }\) for \(\langle \frac { s } { p ^ { k } } \rangle\) , \(p\) odd, are as follows. 
Proposition 12.3.3
Proposition 12.3.3. The mod 8 indices of the basic building blocks for \(p = 2\) are as follows. • Forms of length 1. Here we have \(\begin{array} { r } { \tau _ { 8 } \left( \big \langle \frac { s } { 2 ^ { k } } \big \rangle \right) \equiv s + 4 k \omega ( s ) } \end{array}\) mod 8 which yields the table 
• The form \(u _ { k }\) . Here \(\tau _ { 8 } ( u _ { k } ) = 0\) for \(k = 1 , 2 \dots\) . • The form \(v _ { k }\) . Depending on the parity of \(k\) we find \[ \tau _ { 8 } ( v _ { k } ) \equiv \left\{ \begin{array} { l l } { { 0 \mathrm { m o d } 8 } } & { { k e v e n , } } \\ { { 4 \mathrm { m o d } 8 } } & { { k o d d . } } \end{array} \right. \]
Remark 12.3.4
Remark 12.3.4. For a \(2\)-primary quadratic torsion form \(q\), \(\tau_8(q(\tfrac12))=\tau_8(q)+4\omega(\delta(q))\pmod8\).
Theorem ([204, IV.3.3])
Theorem (existence over \(\mathbf Q\)). Given \(d\in D(\mathbf Q)\), non-negative \(r_+,r_-\) with \(r_++r_-\ge3\), and \(\varepsilon_v\in\{\pm1\}\) for every place \(v\), there is a rational quadratic form with discriminant \(d\), signature \((r_+,r_-)\) and Hasse invariants \(\varepsilon_v\) iff almost all \(\varepsilon_v=1\), \(\prod_v\varepsilon_v=1\), \(d_\infty=(-1)^{r_-}\), and \(\varepsilon_\infty=(-1)^s\) with \(s=\frac12r_-(r_--1)\).
Theorem 12.4.1
Theorem 12.4.1. Let \(L\) be a \(\mathbb { Q }\) -vector space of dimension \(\geq 2\) and suppose that for every \(v \in \mathcal { P }\) a non-degenerate quadratic form \(q _ { v }\) on \(L _ { v }\) is given. Let ?? be a non-zero rational number. Suppose that 1. \(\mathrm { d i s c } ( q _ { v } ) = d\) for all \(v \in { \mathcal { P } }\) (where we consider \(d \in \mathsf { D } ( \mathbb { Q } _ { v } )\) under the natural homomorphism \(\mathsf { D } ( \mathbb { Q } ) \to \mathsf { D } ( \mathbb { Q } _ { v } )\) induced by \(\mathbb { Q } \hookrightarrow \mathbb { Q } _ { v }\) ); 2. \(\varepsilon _ { v } ( q _ { v } ) = 1\) for almost all \(v \in { \mathcal { P } }\) and the product formula \(\begin{array} { r } { \prod _ { v \in \mathcal { P } } \varepsilon _ { v } ( q _ { v } ) = 1 } \end{array}\) holds. Then there exists a \(\mathbb { Q }\) -valued (non-degenerate) quadratic form \(q\) on \(L\) such that \(\operatorname { d i s c } ( q ) = d\) (up to squares) and the localization of \(q\) at \(v\) is isometric to \(q _ { v }\) for all \(v \in \mathcal { S } ^ { \mathsf { \Delta } }\) .
Theorem 12.4.2
Theorem 12.4.2 (Hasse principle for lattices). Let \(n \geq 1\) , \(d \ne 0\) be integers. Suppose that for every prime \(p\) a non-degenerate \(p\) -adic integral quadratic form \(q _ { p }\) of rank ?? and discriminant \(d\) is given, as well as a real valued form \(q _ { \infty }\) of rank \(n\) and discriminant \(d\) . If there exists a rational quadratic form \(g\) such that \(g _ { v } \simeq q _ { v } \otimes \mathbb { U } _ { v }\) for all \(v \in \mathcal { P }\) , then there exists an integral quadratic form \(q\) whose localization at \(v\) is equivalent (over \(\mathbb { Z } _ { v }\) ) to \(q _ { v }\) for all places \(v \in \mathcal { P }\) (with the convention that \(\mathbb { Z } _ { \infty } = \mathbb { R }\) ).
Corollary 12.4.3
Corollary 12.4.3. Let \(L\) be a free \(\mathbb { Z }\) -module of finite rank \(r \geq 2\) . Suppose that a pair \(( r _ { + } , r _ { - } )\) of non-negative integers is given with \(r = r _ { + } + r _ { - }\) , and for every prime \(p\) a non-degenerate \(p\) -adic integral quadratic form \(q _ { p }\) on \(L _ { p }\) . There exists a non-degenerate integral quadratic form \(q\) on \(L\) such that \(q \otimes \mathbb { Z } _ { p } \simeq\) \(q _ { p }\) for all primes \(p\) and with signature \(( r _ { + } , r _ { - } )\) (corresponding to a real form \(q _ { \infty }\) with this signature) if and only if the following conditions hold: 1. \(L _ { p }\) is unimodular for almost all primes \(p\) ; 2. Setting2 \(d = ( - 1 ) ^ { r _ { - } } \prod _ { p }\) prime \(| \mathrm { d g } _ { L _ { p } }\) |, then up to a \(p\) -adic unit \(\mathrm { d i s c } ( q _ { p } ) = d\) for all primes \(p\) ; 3. The product formula \(\begin{array} { r } { \prod _ { v \in \mathcal { P } } \varepsilon _ { v } ( q _ { v } ) = 1 } \end{array}\) holds.
Theorem 12.4.4
Theorem 12.4.4 (Existence of even lattices, [171, Thm. 1.10.1]). Let \(( r _ { + } , r _ { - } )\) be \(a\) pair of non-negative integers and let \(( G , q ^ { \# } )\) be a non-degenerate quadratic torsion form. There exists a non-degenerate quadratic lattice \(L\) with \(\mathsf { g } ( L ) = ( r _ { + } , r _ { - } , [ q ^ { \# } ] )\) (and hence of rank \(r = r _ { + } + r .\) −) if and only if all of the following conditions are fulfilled. 1. \(r _ { + } - r _ { - } \equiv \tau _ { 8 } ( q ^ { \# } )\) mod 8; 2. \(r \geq \ell ( G )\) ; 3. For all primes \(p \neq 2\) for which \(G _ { p }\) has length \(r\) and for \(p = 2\) in case \(G _ { 2 }\) has length \(r\) but does not split off an order two cyclic summand, one has3 \[ \operatorname { d i s c } ( L _ { q _ { p } ^ { \# } } ) = ( - 1 ) ^ { r _ { - } } | G | \ i n \ \mathsf { D } ( \mathbb { Z } _ { p } ) . \]
Remark 12.4.5
Remark 12.4.5. 1. Existence of a genus does not imply uniqueness of the lattice itself. We have seen that for instance non-isometric definite lattices exist in the same genus (see e.g. Table 1.12.1). . Let us illustrate what limitations condition 3 puts on a genus. • First, assume \(p\) is an odd prime. For example, assume that \(p \equiv - 1\) mod 8. Then \(^ { - 1 }\) is a non-square in \(\mathbb { Z } _ { p }\) (see e.g. [204, Ch. I, Theorem 5]). Suppose that \(G\) is \(p\) -primary of length \(r\) and homogeneous of odd degree \(k\) carrying the torsion form \(q ^ { \# }\) of the lattice \(L _ { q \# } = L ( p ^ { k } )\) , \(L = \langle - 1 \rangle \oplus \oplus ^ { r - 1 } \langle 1 \rangle\) . From the table of Proposition 12.3.2, we see that \(\tau _ { 8 } = - 2 + 2 ( r - 1 ) = 2 r - 4\) mod 8. But \(2 r _ { - } ( L ) = r - \tau ( L ) \equiv - r + 4\) mod 8 and so \(r\) has to be even and the parity of \(r _ { - } ( L )\) is the parity of \({ \frac { 1 } { 2 } } r\) . So, if \({ \frac { 1 } { 2 } } r\) is even, \(( - 1 ) ^ { r _ { - } ( L ) } = 1\) . This shows that condition 3 does not hold unless \({ \frac { 1 } { 2 } } r\) is odd. So one cannot weaken this condition to \(\operatorname { d i s c } ( L _ { q \# } ) = \pm | G |\) . • For the prime 2 this is different. It turns out that if \(\operatorname { d i s c } ( L _ { q ^ { \# } } ) = \pm | G |\) and \(q ^ { \# }\) does not have cyclic order 2 factors, the sign is automatically equal to \(( - 1 ) ^ { r _ { - } }\) . For this reason V. Nikulin in [171, Thm. 1.10.1] stated that in this situation the condition \(\operatorname { d i s c } ( L _ { q ^ { \# } } ) = \pm | G |\) is necessary and sufficient. To indicate why this is the case, it is helpful to distinguish two types of normal forms: Let us say that we have type \(I\) if the discriminant equals \(\pm | G |\) and type \(I I\) in case the discriminant is \(\pm 3 | G |\) . For type \(I\) one calculates as before that \(\begin{array} { r } { r _ { - } = \frac { 1 } { 2 } ( r - \tau ) } \end{array}\) mod 4, from which the parity of \(r _ { - }\) follows. Comparing with the sign of \(\mathrm { d i s c } ( L _ { q ^ { \# } } )\) then shows that this sign equals \(( - 1 ) ^ { r _ { - } }\) . For type \(I I\) one gets the opposite sign while \(\operatorname { d i s c } ( L _ { q \# } ) = \pm 3 \cdot | G |\) . Since the latter lattice cannot be isometric to the 2-adic localization of an integral lattice of the same rank, this shows that in the homogeneous situation the signs match. It also shows that in general an even number of type \(I I\) homogeneous summands must be present. Since the parities of \(r _ { - }\) for the various homogeneous summands add, while the discriminants multiply (an even number giving \(\pm 1\) ), one easily checks that the signs match in general. For example, consider first a homogeneous lattice in normal form, \(L _ { q ^ { \# } } =\) \(L ( 2 ^ { k } )\) , \(k\) odd, with \(L = \left. 3 \right. \mathbb { \oplus \oplus ^ { c - 1 } } U \oplus V\) , a lattice of rank \(2 c + 1\) . From the table of Proposition 12.3.3 we see that \(\tau _ { 8 } ( q ^ { \# } ) = 3\) . Since \(\mathrm { d i s c } ( L ) = \pm 1\) we have a type \(I\) lattice and so \(r _ { - } \equiv c + 1\) mod 2. In fact, \(\mathrm { d i s c } ( L ) = ( - 1 ) ^ { c + 1 }\) , and so the signs match. To give an example of an inhomogeneous normal form, consider \(M _ { k } \oplus M _ { k + 2 }\) with \(k\) odd and \(M = \left. 1 \right. \mathbb { \oplus } \mathbb { \oplus } ^ { c - 1 } U \oplus V\) so that \(M _ { k }\) and \(M _ { k + 2 }\) are both of type \(I I\) . One has \(r ( M ) = 2 c + 1\) , and from the table of Proposition 12.3.3, one sees that \(\tau _ { 8 } ( q ^ { \# } ( M _ { k } ) ) = \tau _ { 8 } ( q ^ { \# } ( M _ { k + 2 } ) ) = - 3\) . One calculates \(r _ { - } ( M _ { k } ) = r _ { - } ( M _ { k + 1 } ) \equiv\) \(c - 1\) mod 2. Then \(\operatorname { d i s c } ( L ) = 3 \cdot 3 | G | = | G |\) since \(\mathrm { d i s c } ( M ) = 3 ( - 1 ) ^ { c }\) , while \(r _ { - } \equiv 2 ( c - 1 ) \equiv 0\) mod 2. So, again, the signs match. 3. The theorem implies that unimodular even lattices of index \(( r _ { + } , r _ { - } )\) exist if and only if \(r _ { + } - r _ { - }\) is divisible by 8. We saw this already in Chapters 1 and 2: That even unimodular lattices have index divisible by 8 is part of the statement of Theorem 2.4.2, existence in the indefinite case is Corollary 2.4.3, while the definite case follows from the existence of \(E _ { 8 }\) . 4. Observe that the existence proof is not constructive at all. Consider for instance the genus given by signature \(( 8 , 0 )\) and zero torsion form. The proof starts out with \(L ^ { \prime } = 0\) and enlarges it to \(L ^ { \prime } = \mathbb { \Phi } ^ { \mathrm { s } } U\) . Then \(L _ { p } ^ { \prime }\) is replaced by a trivial rank 8 lattice for all odd \(p\) as well as for the place \(\infty\) , while \(L _ { 2 } ^ { \prime }\) is replaced by \(\mathbb { O } ^ { 4 } U\) . Then existence is deduced from the modified Hasse principle. Of course we know the required lattice exists: \(L = E _ { 8 }\) , but it never comes up during the proof.
Corollary 12.4.6
Corollary 12.4.6. Let \(( r _ { + } , r _ { - } )\) be a pair of non-negative integers and let \(( G , q )\) be a non-degenerate quadratic torsion group. There exists a non-degenerate even lattice \(L\) of rank \(r = r _ { + } + r _ { - }\) , signature \(( r _ { + } , r _ { - } )\) and with discriminant form \(( G , q )\) if the following two conditions are fulfilled. 1. \(r _ { + } - r _ { - } \equiv \tau _ { 8 } ( q )\) mod 8, 2. \(r > \ell ( G )\) .
Lemma 12.5.1
Lemma 12.5.1. The following pairs of 2-primary quadratic forms have isometric polar forms: 1. \(\langle 2 ^ { - 1 } \rangle\) and \(\langle 3 \cdot 2 ^ { - 1 } \rangle\) have polar form \(\langle 2 ^ { - 1 } \rangle\) , 2. \(\langle 2 ^ { - 2 } \rangle\) and \(\langle - 3 \cdot 2 ^ { - 2 } \rangle\) have polar form \(\langle 2 ^ { - 2 } \rangle\) , \(\langle 3 \cdot 2 ^ { - 2 } \rangle\) and \(\left. - 1 \cdot 2 ^ { - 2 } \right.\) ⟩ have polar form \(\langle - 1 \cdot 2 ^ { - 2 } \rangle\) , 3. \(u _ { 1 }\) and \(v _ { 1 }\) have polar form \(\left. u _ { 1 } \right.\) . If the normal form of a 2-primary quadratic torsion form does not contain either one of the above forms, then it is the polar form of a unique quadratic torsion form.
Proposition 12.5.2
Proposition 12.5.2. Two non-degenerate 2-primary quadratic torsion forms with isometric polar forms are themselves isometric if and only if they have the same index modulo 8.
Definition 12.5.3
Definition 12.5.3. The index set of a non-degenerate 2-primary symmetric torsion form \(^ { b }\) is defined as the following subset of \(\mathbb { Z } / 8 \mathbb { Z }\) \(\mathcal { T } _ { 8 } ( b ) = \{ \tau _ { 8 } ( q ) \in \mathbb { Z } / 8 \mathbb { Z } \mid q\) a non-degenerate quadratic torsion form with \(b _ { q } \simeq b \}\) .
Definition / construction / notation
If \(\mathcal { T } _ { 8 } ( b ) \subset \{ 0 , \pm 2 , 4 \}\) , respectively \(\mathcal T _ { 8 } ( b ) \subset \{ \pm 1 , \pm 3 \}\) , one calls \(\mathcal { T } _ { 8 } ( b )\) an even-index set, respectively an odd–index set. If equality holds, we say that \(\mathcal { T } _ { 8 } ( b )\) is a complete even-index set, respectively a complete odd-index set.
Definition / construction
An index set \(\mathcal T_8(b)\) is called even-index if it is contained in \(\{0,\pm2,4\}\) and odd-index if it is contained in \(\{\pm1,\pm3\}\); it is complete when the corresponding containment is equality.
Example / comparison
Example 12.5.4. The polar forms of Lemma 12.5.1 have the following index sets \(\mathcal { T } _ { 8 } ( \langle 2 ^ { - 1 } \rangle ) ~ = ~ \{ \bar { 1 } , - \bar { 1 } \}\) , \(\mathcal { T } _ { 8 } ( \langle 2 ^ { - 2 } \rangle ) = \{ \bar { 1 } , - \bar { 3 } \}\) , \(\mathcal { T } _ { 8 } ( \langle - 2 ^ { - 2 } \rangle ) = \{ - \bar { 1 } , \bar { 3 } \}\) , \(\mathcal { T } _ { 8 } ( u _ { 1 } ) ~ = ~ \{ 0 , 4 \}\) . Moreover, Lemma 12.5.1 also implies that if the normal form of \(^ { b }\) does not contain any of such forms, then \(b\) is the polar form of a unique quadratic form \(q\) and \(\mathcal { T } _ { 8 } ( b ) = \{ \tau _ { 8 } ( q ) \}\) .
Construction / comparison
Index sets of non-degenerate \(2\)-primary symmetric torsion forms add by Minkowski sum in \(\mathbf Z/8\mathbf Z\), and \(\mathcal T_8(b)+\mathcal T_8(b\prime)\subseteq\mathcal T_8(b\oplus b\prime)\). The parity of an index set is additive modulo \(2\).
Lemma 12.5.5
Lemma 12.5.5. Let \(( G , b )\) be a non-degenerate 2-primary symmetric torsion form. If \(\ell ( G )\) is even (odd), then \(\mathcal { T } _ { 8 } ( b )\) is an even-index (odd-index) set.
Lemma 12.5.6
Lemma 12.5.6. Let \(u , u ^ { \prime } , u ^ { \prime \prime }\) be units in \(\mathbb { Z } _ { 2 }\) . Then the following relations hold between non-homogeneous dyadic lattices. \[ \langle u \cdot 2 ^ { k - 1 } \rangle \oplus U _ { k } \simeq \langle - 3 u \cdot 2 ^ { k - 1 } \rangle \oplus V _ { k } ~ f o r ~ k \geq 1 \] \[ U _ { k - 1 } \odot \langle u \cdot 2 ^ { k } \rangle \simeq \langle - 3 u \cdot 2 ^ { k } \rangle \oplus V _ { k - 1 } ~ f o r ~ k \geq 1 \] \[ \langle u \cdot 2 ^ { k - 1 } \rangle \mathbb { O } \langle u ^ { \prime } \cdot 2 ^ { k } \rangle \simeq \langle ( u + 2 u ^ { \prime } ) \cdot 2 ^ { k - 1 } \rangle \mathbb { O } \langle ( u ^ { \prime } + 2 u ) \cdot 2 ^ { k } \rangle f o r k \geq 1 \] \[ ( \langle u \rangle \mathbb { O } \langle u ^ { \prime } \rangle ) ( 2 ^ { k - 1 } ) \mathbb { O } \langle 2 ^ { k } \rangle \simeq ( \langle u + 2 \rangle \mathbb { O } \langle u ^ { \prime } - 2 \rangle ) ( 2 ^ { k - 1 } ) \mathbb { O } \langle - 3 \cdot 2 ^ { k } \rangle \] \[ u \equiv u ^ { \prime } { \bmod { 4 } } \quad k \geq 1 \] \[ \langle u \cdot 2 ^ { k - 2 } \rangle \mathbb { O } \langle u ^ { \prime } \cdot 2 ^ { k } \rangle \simeq \langle - 3 u \cdot 2 ^ { k - 2 } \rangle \mathbb { O } \langle - 3 u ^ { \prime } \cdot 2 ^ { k } \rangle , f o r k \geq 3 . \] 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 \(V _ { k }\) . The proofs of these relations have been placed in Appendix C.3.B.
Corollary 12.5.7
Corollary 12.5.7. If \(^ { b }\) is the symmetric discriminant form of some non-degenerate dyadic lattice, there are \(\# \mathcal { I } _ { 8 } ( b )\) non-isometric non-degenerate dyadic lattices of rank equal to the length of the discriminant group for which ?? appears as discriminant form, unless \(\langle 2 ^ { - 1 }\) ⟩ splits off from \(b\) . In that case the number is twice as big.
Proposition 12.5.8
Proposition 12.5.8. A non-degenerate lattice \(( L , b )\) is odd if and only if \(L = L ( 2 )\) is an even non-degenerate lattice having the properties 1. \(\mathbf { L } _ { 2 }\) splits off a summand \(\langle u \cdot 2 \rangle\) , \(u\) a unit; 2. \(\mathrm { { d g } } _ { L _ { 2 } }\) has length equal to the rank of \(L\) , or, equivalently, with \({ L } _ { 2 } ^ { ( 0 ) }\) the exponent 0 summand in the Jordan splitting of \(L _ { 2 }\) , one has \[ \mathrm { r a n k } ( L ) - \mathrm { r a n k } ( L _ { 2 } ^ { ( 0 ) } ) = \ell ( { \tt d } { \tt d } _ { L _ { 2 } } ) . \]
Definition / construction / notation
The above observation motivates the following notation for a non-degenerate symmetric lattice \(( L , b )\) : \[ \begin{array} { l l l } { { { \cal L } = { \cal L } ( 2 ) , } } & { { { \bf q } _ { 2 } : = b _ { L _ { 2 } } ^ { \# } , } } & { { { \bf q } _ { 2 } : = q _ { L _ { 2 } } ^ { \# } , } } \\ { { { \cal L } _ { 2 } = { \cal L } _ { 2 } ^ { ( 0 ) } \oplus { \cal L } _ { 2 } ^ { \geq 1 } , } } & { { b _ { 2 } : = b _ { L _ { 2 } } ^ { \# } , } } & { { } } \\ { { { \cal L } _ { 2 } = { \cal L } _ { 2 } ^ { ( 1 ) } \oplus { \cal L } _ { 2 } ^ { \geq 2 } , } } & { { { \cal L } _ { 2 } ^ { ( 1 ) } = { \cal L } _ { 2 } ^ { ( 0 ) } ( 2 ) , } } & { { { \cal L } _ { 2 } ^ { \geq 2 } = { \cal L } _ { 2 } ^ { \geq 1 } ( 2 ) , } } \\ { { { b } _ { 2 } = b _ { 2 } ^ { ( 1 ) } \oplus b _ { 2 } ^ { \geq 2 } , } } & { { b _ { 2 } ^ { ( 1 ) } = b _ { L _ { 2 } ^ { ( 0 ) } ( 2 ) } ^ { \# } , } } & { { b _ { 2 } ^ { \geq 2 } = \frac 1 2 b _ { 2 } , } } \end{array} \]
Theorem 12.5.9
Theorem 12.5.9 ([171, Cor. 1.16.3]). Let \(( L , b )\) be a non-degenerate odd symmetric integral lattice. Then the genus \({ \mathfrak { g } } ( L )\) is determined by the discriminant symmetric form of ?? together with the signature of \(L\) .
Remark 12.5.10
Remark 12.5.10. For an odd lattice \(L\), its genus invariant is the triple consisting of its signature and the isometry class of its discriminant bilinear form; the genus is determined by this invariant.
Lemma 12.5.11
Lemma 12.5.11. 1. For a non-degenerate 2-primary quadratic torsion form ?? one has \(\tau _ { 8 } ( q ( 2 ^ { - 1 } ) ) \equiv \tau _ { 8 } ( q ) + 4 \omega ( \delta ( q ) )\) mod 8. 2. Suppose \(^ { b }\) is a (not necessarily 2-primary) non-degenerate symmetric torsion form which does not split off \(\langle 2 ^ { - 1 } \rangle\) and let \(q\) be a quadratic torsion form ?? with polar form \(^ { b }\) . Then the quantity \(\tau _ { 8 } ( q ) + 4 \omega ( \delta ( q _ { 2 } ) )\) mod 8 only depends on ?? and so \[ t _ { 8 } ( b ) : = \tau _ { 8 } ( q ) + 4 \omega ( \delta ( q _ { 2 } ) ) \bmod 8 \] is well defined. Moreover, for any such choice of \(q\) and any prime \(p\) one has \(t _ { 8 } ( b _ { p } ) = \tau _ { 8 } ( q _ { p } ( 2 ^ { - 1 } ) )\) .
Definition / construction / notation
For a non-degenerate \(2\)-primary symmetric torsion form \(b_2\), \(\mathcal T_8(b_2)\) denotes the collection of quadratic torsion forms having \(b_2\) as polar form, organized by their mod-\(8\) index data.
Proposition 12.5.12
Proposition 12.5.12. Let \(\pmb { b } _ { 2 }\) be a non-degenerate torsion symmetric form on a 2-primary group \(\mathbf { G } _ { 2 }\) such that \(\pmb { b } _ { 2 } ^ { ( 1 ) }\) splits off \(\langle 2 ^ { - 1 } \rangle\) . 1. For \(\pmb { b } _ { 2 } ^ { ( 1 ) }\) the following normal forms occur: (a) If \(b _ { 2 } ^ { ( 1 ) } = \langle 2 ^ { - 1 } \rangle\) , then \(\mathcal { T } _ { 8 } ( \pmb { b } _ { 2 } ^ { ( 1 ) } ) = \{ 1 , - 1 \}\) . (b) If \(\pmb { b } _ { 2 } ^ { ( 1 ) } = \mathbb { \oplus ^ { 2 } \langle 2 ^ { - 1 } \rangle }\) , then \(\mathcal { T } _ { 8 } ( \pmb { b } _ { 2 } ^ { ( 1 ) } ) = \{ 0 , 2 , - 2 \}\) . (c) If \(\pmb { b } _ { 2 } ^ { ( 1 ) } = \langle 2 ^ { - 1 } \rangle \oplus \oplus ^ { b } u _ { 1 }\) , \(b \geq 1\) , then \(\mathcal { T } _ { 8 } ( \pmb { b } _ { 2 } ^ { ( 1 ) } ) = \{ 1 , - 1 , 3 , - 3 \}\) , a complete set of odd parity5. (d) If \(\pmb { b } _ { 2 } ^ { ( 1 ) } = \oplus ^ { 2 } \langle 2 ^ { - 1 } \rangle \oplus ^ { b } u _ { 1 }\) , \(b \geq 1\) , then \(\mathcal { T } _ { 8 } ( \pmb { b } _ { 2 } ^ { ( 1 ) } ) = \{ 0 , 2 , - 2 , 4 \}\) , a complete set of even parity. 2. If \(b _ { 2 } ^ { \geq 2 } = b _ { 2 } ( 2 ^ { - 1 } )\) for some 2-primary torsion symmetric form \(b _ { 2 }\) which does not split off \(\langle 2 ^ { - 1 } \rangle\) , then \(t _ { 8 } ( b _ { 2 } ) = \tau _ { 8 } ( \pmb { q } _ { 2 } ^ { \geq 2 } )\) , where \(\pmb q _ { 2 } ^ { \geq 2 }\) is a quadratic form with polar form ??≥2.
Theorem 12.5.13
Theorem 12.5.13 (Existence for odd lattices [171, Thm. 1.16.5]). Let there be given two non-negative integers \(( r _ { + } , r _ { - } )\) and a non-degenerate symmetric torsion group \(( G , b ^ { \# } )\) . There exists a non-degenerate odd lattice \(( L , b )\) of rank \(r = r _ { + } + r _ { - }\) , signature \(( r _ { + } , r _ { - } )\) and with discriminant bilinear form \(( G , b ^ { \# } )\) if and only if all of the following conditions are fulfilled. a) \(r \geq \ell ( G _ { p } )\) for odd primes \(p\) , \(r \geq \ell ( G _ { 2 } ) + 1\) (recall that \(G _ { p }\) is the \(p\) -primary part of \(G\) ). b) For every odd prime \(p\) with \(r ~ = ~ \ell ( G _ { p } )\) we have \(( - 1 ) ^ { r _ { - } } | G _ { p } | ~ = ~ \operatorname { d i s c } ( b _ { p } ^ { \# } )\) · \(( \mathbb { Z } _ { p } ^ { \times } ) ^ { 2 }\) in \(\mathsf { D } ( \mathbb { Z } _ { p } )\) . c) Suppose that \(r ~ = ~ \ell ( G _ { 2 } ) + 1\) and no non-degenerate cyclic rank 2 torsion quadratic form splits off from \(b _ { 2 } ^ { \# }\) . Then \(r _ { + } - r _ { - } \equiv t _ { 8 } ( b ^ { \# } ) + \delta\) mod 8, \(\delta \in\) \(\{ 1 , - 1 \}\) . d) Suppose that \(r ~ = ~ \ell ( G _ { 2 } ) + 2\) and no non-degenerate cyclic rank 2 torsion quadratic form splits off from \(b _ { 2 } ^ { \# }\) . Then \(r _ { + } - r _ { - } \equiv t _ { 8 } ( b ^ { \# } ) + \delta\) mod 8, \(\delta \in\) \(\{ 0 , 2 , - 2 \}\) .
Corollary 12.5.14
Corollary 12.5.14. Let \(( r _ { + } , r _ { - } )\) be a pair of non-negative integers and let \(( G , b )\) be a non-degenerate symmetric torsion group. There exists a non-degenerate odd lattice \(L\) of rank \(r\) , signature \(( r _ { + } , r _ { - } )\) and with discriminant form \(( G , b )\) if \(r \_ { \geq }\) \(\ell ( G ) + 3\) .