FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 23 definitions, 10 propositions, 10 lemmas, 6 corollaries, 14 examples, 3 constructions and 1 remark.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction / notation
5.1.A Lattices obtained from codes. A linear code is a linear subspace of \(\mathbb { F } _ { q } ^ { n }\) where \(q = p ^ { s }\) , a power of a prime number \(p\) . If \(p = 2\) the code is called a binary code. A vector \(x\) of a code \(c\) is also called a word . It has weight \[ w ( x ) = \mathrm { n u m b e r ~ o f ~ n o n \mathrm { - } z e r o ~ c o o r d i n a t e s ~ o f } \ x , \quad x = ( x _ { 1 } , \dots , x _ { n } ) \in C \subset \mathbb { F } _ { q } ^ { n } . \]
Definition / construction / notation
The dot-product \(x \cdot y \ = \ \sum x _ { j } y _ { j } \ \in \ \mathbb { F } _ { q }\) of two vectors \(x \ : = \ : ( x _ { 1 } , \ldots , x _ { n } )\) and \(y \ =\) \(( y _ { 1 } , \ldots , y _ { n } )\) in \(\mathbb { F } _ { q } ^ { n }\) defines a non-degenerate symmetric bilinear form and a code is called isotropic if \(c\) is an isotropic subspace of \(\mathbb { F } _ { q } ^ { n }\) with respect to the dot-product, i.e., if \(C \subset C ^ { \perp }\) , and self-dual if \(C = C ^ { \perp }\) .
Remark 5.1.1
Remark 5.1.1. In Chapter 8 we answer the obvious question: how to place the dotproduct on vector spaces over \(\mathbb { F } _ { 2 }\) in the isometry classification? It turns out that Corollary 8.3.4 implies that in odd characteristic \(b\) is isometric to the dot-product if and only if \(\mathrm { d i s c } ( b ) = 1\) As a consequence of Proposition 8.2.1 if \(p = 2\) the odd form \(^ { b }\) is isometric to the dot-product if and only if \(\mathrm { d i s c } ( b ) = 1\) and \(^ { b }\) is totally anisotropic, i.e., there are no isotropic vectors.
Definition / construction
For a code \(C\subset\mathbf F_p^n\), let \(\rho:\mathbf Z^n\to\mathbf F_p^n\) be reduction modulo \(p\) and equip \(\mathbf Z^n\) with the Euclidean form scaled by \(p^{-1}\). The Construction-A lattice is \(\Gamma_C=\rho^{-1}(C)\subset\mathbf Z^n(p^{-1})\); it is positive definite of rank \(n\) and contains \(p\mathbf Z^n\).
Definition / construction / notation
a code \(c\) lifts to a submodule \(\rho ^ { - 1 } C\) of \(\mathbb { Z } ^ { n }\) which contains the submodule \(p \cdot \mathbb { Z } ^ { n } =\) \(\rho ^ { - 1 } ( 0 )\) and hence is of finite index in \(\mathbb { Z } ^ { n }\) . It inherits the structure of a lattice from \(\mathbb { Z } ^ { n }\) equipped with its dot-product. However, it turns out to be more convenient to use a different lattice structure, namely \(\mathbb { Z } ^ { n }\) equipped with the standard euclidean form scaled by \(p ^ { - 1 }\) which we denote \[ \mathbb { Z } ^ { n } ( p ^ { - 1 } ) = \mathbb { O } ^ { n } \langle p ^ { - 1 } \rangle . \]
Lemma 5.1.2
Lemma 5.1.2. 1. The sublattice \(p \mathbb { Z } ^ { n }\) of \(\Gamma _ { C }\) is isometric to \(\mathbb { O } ^ { n } \langle p \rangle\) . In particular, if \(p = 2\) , the root lattice \({ \mathfrak { Q } } ^ { n } A _ { 1 }\) is contained in \(\Gamma _ { C }\) . 2. \(\Gamma _ { C }\) is an integral lattice if and only if \(c\) is isotropic. 3. \(\Gamma _ { C } ^ { * } \subset \mathbb { Z } ^ { n } ( p ^ { - 1 } )\) . Moreover, if \(c\) is isotropic, the integral lattice \(\Gamma _ { C }\) is a \(p\) - elementary lattice, that is, we recall (cf. Subsection 1.7.B) \[ p \cdot \Gamma _ { C } ^ { * } \subset \Gamma _ { C } . \]
Corollary 5.1.3
Corollary 5.1.3. Isotropic codes in \(\mathbb { F } _ { p } ^ { n }\) are in a natural way in one-to-one correspondence with (integral) overlattices of \(\mathbb { Q } ^ { n } \langle p \rangle\) contained in \(\mathbb { Z } ^ { n } ( p ^ { - 1 } )\) .
Proposition 5.1.4
Proposition 5.1.4. Let \(C \subset \mathbb { F } _ { p } ^ { n }\) be an isotropic code and \(\Gamma _ { C } \subset \mathbb { Z } ^ { n } ( p ^ { - 1 } )\) the corresponding integral lattice. Then \(\Gamma _ { C }\) has the following properties: 1. \(\mathrm { d i s c } ( \Gamma _ { C } ) = p ^ { n - 2 m } , \quad m = \mathrm { d i m } { \cal C } .\) 2. The lattice \(\Gamma _ { C }\) is unimodular if and only if \(C\) is self-dual. 3. \(\Gamma _ { C } ^ { * } = \rho ^ { - 1 } C ^ { \perp }\) 4. If \(p = 2\) , the lattice \(\Gamma _ { C }\) is even if and only if all of the weights of \(C\) are divisible by 4.
Definition / construction / notation
In view of the assertion 4 we introduce an appropriate terminology for codes giving even lattices: we say that a binary code is doubly even if all its code words have weights divisible by 4.
Proposition 5.1.5
Proposition 5.1.5. Let \(C \subset \mathbb { F } _ { p } ^ { n }\) be an isotropic code of dimension ??. The discriminant group of the lattice \(\Gamma _ { C }\) is isomorphic to \(( \mathbb { Z } / p \mathbb { Z } ) ^ { n - 2 m }\) . If \(p = 2\) and \(C\) is doubly even, the discriminant quadratic form takes values in \({ \frac { 1 } { 4 } } \mathbb { Z }\) . If, moreover, all words in \(C ^ { \bot }\) have even weights, the values are taken in \({ \frac { 1 } { 2 } } \mathbb { Z }\) . In that case \(\Gamma _ { C }\) is \(a\) 2-elementary lattice of type \(I I\) .
Definition / construction / notation
The Hamming code \(H \subset \mathbb { F } _ { 2 } ^ { \prime }\) . This is the linear code of dimension 4 defined in so-called standard form as \[ H = \ker A , \quad A = { \left( \begin{array} { l l l l l l l l } { 1 } & { 0 } & { 1 } & { 1 } & { 1 } & { 0 } & { 0 } \\ { 1 } & { 1 } & { 0 } & { 1 } & { 0 } & { 1 } & { 0 } \\ { 0 } & { 1 } & { 1 } & { 1 } & { 0 } & { 0 } & { 1 } \end{array} \right) } . \]
Definition / construction / notation
Reed-Muller codes. The definition of this class of codes runs as follows. Let \(I\) be a finite set of size \(N\) . Then the functions \(I \to \mathbb { F } _ { 2 }\) form the \(\mathbb { F } _ { 2 }\) -vector space \(\mathbb { F } _ { 2 } ^ { I }\) Suppose that itself consists of the points of an \(\mathbb { F } _ { 2 }\) -vector space \(W\) of dimension \(m\) so that \(N = 2 ^ { m }\) . We order these points as follows. Let \(\{ e _ { 0 } , \dotsc , e _ { m - 1 } \}\) be a basis for \(W\) and identify a point \(\begin{array} { r } { x = \sum _ { j = 0 } ^ { m - 1 } x _ { j } e _ { j } } \end{array}\) with the binary expansion \(\begin{array} { r } { n _ { x } = \sum _ { j = 0 } ^ { m - 1 } x _ { j } 2 ^ { j } } \end{array}\) of an integer between 0 and \(2 ^ { m } - 1\) . The natural order gives an ordering of the points of \(W\) . An ?? \(^ 2\) -valued function \(f\) on \(W\) determines a vector \(( f _ { 0 } , \dotsc , f _ { N - 1 } ) \in \mathbb { F } _ { 2 } ^ { N }\) as follows. First note that a point \(x \in W\) determines the unique integer \(j = n _ { x } \ \in\) \(\lfloor 0 , \ldots , N - 1 \rfloor\) and then we set \(f _ { j } = f ( x )\) . This way, the polynomial functions of degree \(k\) on \(W\) together with the zero function define a subspace of \(\mathbb { F } _ { 2 } ^ { N }\) and this is also the case for polynomial functions of degree \(\leq k\) . For \(0 \leq k < m\) , the latter define the \(k\) -th order Reed–Muller code \(S ^ { \le k } ( W ) \subset \mathbb { F } _ { 2 } ^ { N }\) , \(N = 2 ^ { m }\) . For an extensive treatment of these codes we refer to [140, Sect. 4.5.].
Example / comparison
Example 5.1.6. Take \(m = 4\) and \(k = 1\) . Then \(N = 2 ^ { 4 } = 1 6\) and \(S ^ { \leq 1 } ( \mathbb { F } _ { 2 } ^ { 4 } ) \subset \mathbb { F } _ { 2 } ^ { 1 6 }\) \(4 \times 1 6\) together with the vector with all coordinates equal to 1 arising from the constant function 1. The columns correspond to the binary expansions of the numbers \(0 , . . . , 1 5\) and the rows correspond to the coordinate functions \(x _ { 0 } , x _ { 1 } , x _ { 2 } , x _ { 3 }\) : \[ \begin{array} { r } { \small { \left( \begin{array} { l l l l l l l l l l l l l l l l l l } { 0 } & { 1 } & { 0 } & { 1 } & { 0 } & { 1 } & { 0 } & { 1 } & { 0 } & { 1 } & { 0 } & { 1 } & { 0 } & { 1 } & { 0 } & { 1 } & { 0 } & { 1 } \\ { 0 } & { 0 } & { 1 } & { 1 } & { 0 } & { 0 } & { 1 } & { 1 } & { 0 } & { 0 } & { 1 } & { 1 } & { 0 } & { 0 } & { 1 } & { 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 1 } & { 1 } & { 1 } & { 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 1 } & { 1 } & { 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 1 } & { 1 } & { 1 } & { 1 } & { 1 } & { 1 } \end{array} \right) . } } \end{array} \]
Proposition 5.1.7
Proposition 5.1.7. Let \(W\) be an \(\mathbb { F } _ { 2 }\) -vector space of dimension ??. The Reed– Muller codes \(S ^ { \le k } ( W ) \subset \mathbb { F } _ { 2 } ^ { W } \simeq \mathbb { F } _ { 2 } ^ { N }\) , \(N = 2 ^ { m } = | W |\) , have the following properties. 1. \(S ^ { \leq k } ( W )\) has even weights. 2. Suppose \(0 \leq k \leq m - 1\) . With respect to the standard dot-product on \(\mathbb { F } _ { 2 } ^ { N }\) , the code orthogonal to \(S ^ { \leq k } ( W )\) is \(S ^ { \leq m - k - 1 } ( W )\) . 3. The code \(S ^ { \leq k } ( W )\) is isotropic if \(2 k \leq m - 1\) and self-dual if and only if equality holds.
Corollary 5.1.8
Corollary 5.1.8. Assume \(m \geq 4\) . The lattices \(\Gamma _ { \mathsf { C } _ { m } }\) and \(\Gamma _ { \mathsf { D } _ { m } }\) have discriminant group \(\oplus ^ { 2 ^ { m } - 2 m - 1 } \mathbb { Z } / 2 \mathbb { Z }\) , respectively \(\Phi ^ { 2 ^ { m - 1 } - 2 m } \mathbb { Z } / 2 \mathbb { Z }\) . The discriminant quadratic form in both cases is \({ \frac { 1 } { 2 } } \mathbb { Z }\) -valued, that is, \(\Gamma _ { { \sf C } _ { m } }\) and \(\Gamma _ { \mathsf { D } _ { m } }\) are 2-elementary lattices of type \(I I\) .2
Lemma 5.1.9
Lemma 5.1.9 ([17, §4]). 1. Let \(C \subset \mathbb { F } _ { 2 } ^ { n }\) be a code of dimension ?? and with non-zero weights \(> { \frac { 1 } { 2 } } n\) . Then \(n \geq 2 ^ { m } - 1\) and, if equality holds, \(C \simeq C _ { m }\) . 2. Let \(C \subset \mathbb { F } _ { 2 } ^ { n }\) be a code of dimension ?? and with non-zero weights \(\geq { \frac { 1 } { 2 } } n\) . Then \(n \geq 2 ^ { m - 1 }\) and equality holds if and only if \(C \simeq \mathsf { D } _ { m }\) , in which case it only has non-zero weights 1 ?? = 2??−2 and ?? = 2??−1.
Definition / construction / notation
5.2.A Even sets of nodes. Let \(Y\) be a (connected compact and smooth) complex surface which is the minimal resolution of a singular surface \(Y ^ { \prime }\) having a finite set of ordinary double points (and no other singularities). The example to have in mind is the (singular) Kummer surface (see Appendix B.3). As in that example, each double point is resolved by a nodal curve, by definition a smooth rational curve with self-intersection \(- 2\) . Conversely, such a curve can be contracted to give a surface with an ordinary double point. A special role is played by subsets \(\{ E _ { j } \} _ { j \in J }\) of disjoint nodal curves for which \(\begin{array} { r } { B _ { J } = \sum _ { j \in J } E _ { j } \in 2 \mathsf { N S } ( Y ) } \end{array}\) . Such a set is called an even set of nodal curves.3 Its role is highlighted by the following geometric construction. Evenness implies that there is a double covering \(^ 4\) of \(Y\) , exactly ramified over \(B _ { J }\) . The resulting surface \(X _ { J }\) is not minimal since the curves \(E _ { j }\) , \(j \in J\) , lift to exceptional curves which can be blown down to points \(p _ { j }\) , yielding a smooth surface, say \(X _ { J } ^ { \prime }\) . The situation can be summarized in the following commutative diagram \[ \begin{array} { r l } { ( X _ { J } , q ^ { - 1 } B _ { J } ) \xrightarrow { } } & { { } \qquad \quad \langle X _ { J } ^ { \prime } , \bigcup _ { j \in J } p _ { j } ) } \\ { q \Biggl \downarrow } & { { } \qquad q ^ { \prime } \Biggr \downarrow } \\ { ( Y , B _ { J } ) \xrightarrow { } } & { { } \qquad \quad \langle Y ^ { \prime } , \bigcup _ { j \in J } q ^ { \prime } ( p _ { j } ) \rangle } \end{array} \]
Proposition 5.2.1
Proposition 5.2.1. Suppose \(Y\) is a complex surface such that \(H ^ { 2 } ( Y , \mathbb { Z } )\) has no torsion and let \(\mathcal { E }\) be a set of disjoint nodal curves on \(Y\) . Non-zero code words in \(C _ { \mathcal { E } }\) correspond to even subsets of nodal curves, and conversely. More precisely, with \(e _ { 1 } , \ldots , e _ { n }\) the standard basis of \(\mathbb { F } _ { 2 } ^ { n }\) , the sum \(\textstyle \sum _ { i \in J } e _ { i }\) , where \(J \subset \{ 1 , \ldots , n \}\) , belongs to the code \(C _ { 8 }\) if and only if \(\textstyle \sum _ { i \in J } E _ { i }\) is even in NS(??). The weight of \(a\) word in \(C _ { \mathcal { E } }\) is the cardinality of the corresponding set.
Construction / comparison
For the code attached to disjoint nodal curves on a surface with torsion-free \(H^2\), primitivity of the Néron–Severi lattice and isotropy modulo \(2\) give \(\dim C_{\mathcal E}\ge n-\frac12 b_2(Y)\).
Lemma 5.2.2
Lemma 5.2.2. Let \(Y\) be a complex \(K \mathcal { 3 }\) surface containing an even set of \(k > 0\) disjoint nodal curves. Then \(k = 8\) or 16. If \(k = 8\) the associated double cover is \(a\) \(K \mathcal { 3 }\) surface and if \(k = 1 6\) it is a complex torus.
Corollary 5.2.3
Corollary 5.2.3. For a set of disjoint nodal curves on a complex K3 surface the weights of the associated code are \(0 , 8\) or 16.
Definition / construction / notation
For a complex torus \(A\), the Kummer surface \(\operatorname{Km}(A)\) is the minimal resolution of the quotient \(A/\langle -1\rangle\).
Definition / construction
The Kummer lattice is the primitive lattice generated by the exceptional curves of the Kummer resolution.
Lemma 5.2.4
Lemma 5.2.4. For any set \(\mathcal { E }\) of 16 nodal curves on a complex K3 surface, the associated code \(C _ { \mathcal { E } }\) is isomorphic to \({ \sf D } _ { 5 }\) , the Reed–Muller code (5.3).
Definition 5.2.5
Definition 5.2.5. The lattice \(\Lambda _ { \mathrm { K u m } } = \Gamma _ { \mathsf { D } _ { 5 } } ( - 1 )\) is the abstract Kummer lattice.
Lemma 5.2.6
Lemma 5.2.6. The abstract Kummer lattice is an even lattice of rank 16 and discriminant \(2 ^ { 6 }\) . It is \(a\) 2-elementary lattice of type II with discriminant group isomorphic to \(\oplus ^ { 6 } \mathbb { Z } / 2 \mathbb { Z }\) .
Proposition 5.2.7
Proposition 5.2.7. Let \(Y\) be a complex \(K \mathcal { 3 }\) surface containing a set \(\mathcal { E }\) of 16 disjoint nodal curves. Then \(\mathcal { E }\) is an even set, \(Y\) is a Kummer surface with \(\mathcal { E }\) the canonical set of 16 nodal curves. The primitive closure \(N _ { \mathcal { E } }\) of its span is isometric to the abstract Kummer lattice \(\Lambda _ { \mathrm { K u m } }\) .
Corollary 5.2.8
Corollary 5.2.8. Every Kummer surface admits a doubly covering K3 surface blown up in 8 points.
Definition / construction / notation
A Nikulin involution on a K3 surface is an involution preserving a non-zero holomorphic \(2\)-form.
Definition 5.2.9
Definition 5.2.9. The Nikulin lattice is the rank 8 even negative definite lattice \[ \Lambda _ { \mathrm { N i k } } = \Gamma _ { \mathcal { E } } ( - 1 ) \subset \frac { 1 } { 2 } \mathbb { Z } ^ { 8 } ( - 2 ) \] spanned by the standard basis vectors \(e _ { j }\) of \(\mathbb { Z } ^ { \mathrm { s } } ( - 2 )\) together with \(\textstyle { \frac { 1 } { 2 } } \sum _ { j = 1 } ^ { 8 } e _ { j }\)
Proposition 5.2.10
Proposition 5.2.10. Let \(Y\) be a \(K \mathcal { 3 }\) surface with an even set \(\mathcal { E }\) of 8 nodal curves. Then the lattice \(\Gamma _ { C _ { \mathcal { E } } } ( - 1 )\) is the Nikulin lattice, a 2-elementary type \(I I\) quadratic lattice with discriminant group \(\oplus ^ { 6 } \mathbb { Z } / 2 \mathbb { Z }\) .
Lemma 5.2.11
Lemma 5.2.11 ([17]). The code \(C _ { \mathcal { E } } \subset \mathbb { F } _ { 2 } ^ { n }\) , \(n = \# \mathcal { E }\) , has non-zero weights 16 and 20.
Corollary 5.2.12
Corollary 5.2.12. If \(\# \mathcal { E } = 3 1\) , we have \(C _ { \mathcal { E } } = C _ { 5 }\) , and so in this case NS(??) contains the lattice \(\Gamma _ { \mathsf { C } _ { 5 } } ( - 1 )\) . The surface \(\hat { X }\) has \(\leq 3 1\) double points.
Definition / construction
For a finite-dimensional algebra, multiplication by an element defines its trace and norm; the associated trace pairing is the corresponding symmetric bilinear form.
Lemma 5.3.1
Lemma 5.3.1. The trace form is non-degenerate and so \(( K , \mathsf { T r } _ { K / \mathbb { Q } } )\) is a \(\mathbb { Q }\) -inner product space.
Example / comparison
Example 5.3.2 (Quadratic number fields). Any quadratic extension of \(\mathbb { Q }\) can be written as √ \(\underline { { { K } } } = \mathbb { Q } ( \sqrt { d } )\) where \(d \in \mathbb { Z }\) is square free. If \(x = x _ { 1 } + x _ { 2 } \cdot { \sqrt { d } }\) and \(y = y _ { 1 } + y _ { 2 } \cdot { \sqrt { d } }\) , we have \({ \sf T r } _ { K / \mathbb { Q } } ( x y ) = 2 ( x _ { 1 } y _ { 1 } + d x _ { 2 } y _ { 2 } )\) . The discriminant of the trace form is (up to squares) \(d \in \mathsf { D } ( \mathbb { Q } )\) . Note also that the index of the trace form is \(2\) if \(d > 0\) and \(0\) if \(d < 0\) . The relation with the norm is as follows. √ \(N ( x ) = x _ { 1 } ^ { 2 } { - } d x _ { 2 } ^ { 2 }\) whose polarization equals \({ \sf T r } _ { K / \mathbb { Q } } ( x y ^ { \prime } )\) where \(y ^ { \prime } = y _ { 1 } - y _ { 2 } \cdot { \sqrt { d } }\) is the conjugate of \(y\) . So the trace form is not the polarization of the quadratic form defined by the norm. The norm form plays a role later on. See \(\ S \ 6 . 3 . \mathrm { B }\) , example 5 and 8.3.1.
Construction / comparison
If a number field has \(r\) real embeddings and \(s\) conjugate pairs of complex embeddings, its trace form has signature \((r+s,s)\).
Lemma 5.3.3
Lemma 5.3.3. Fix a non-real embedding \(K \hookrightarrow \mathbb { C }\) and let \(x \mapsto { \bar { x } }\) be the ensuing complex conjugation. The hermitian trace \[ ( x , y ) \mapsto x \cdot y : = { \mathsf { T r } } _ { K / \mathbb { Q } } ( x { \bar { y } } ) \] is a positive definite \(\mathbb { Q }\) -inner product on \(K\) . It is independent of the chosen non-real embedding.
Definition 5.3.4
Definition 5.3.4. The ring of integers of \(K\) is defined as \(\mathfrak { D } _ { K } = \{ x \in K \mid p _ { x } ( X ) \in \mathbb { Z } [ X ] \}\) , where \(p _ { x } ( X )\) is the monic minimal polynomial of \(x\) This is a free \(\mathbb { Z }\) -module of rank \(d = d _ { K }\) on which the trace is integer valued and hence the pairing \(( x , y ) \mapsto { \mathsf { T r } } _ { K / \mathbb { Q } } ( x y )\) , \(x , y \in \mathfrak { D } _ { K }\) , is integer valued. In other words, \(( \mathfrak { D } _ { K } , \mathsf { T r } _ { K / \mathbb { Q } } )\) is an integral \(\mathbb { Z }\) -lattice and so its discriminant, discriminant of \(\mathfrak { D } _ { K }\) – denoted \(\mathfrak { d } _ { K }\) – is an integer.
Example / comparison
Example 5.3.5. For \(K=\mathbf Q(\sqrt d)\) with square-free \(d\), \(\mathcal O_K=\mathbf Z\oplus\sqrt d\,\mathbf Z\) for \(d\equiv2,3\pmod4\) and \(\mathcal O_K=\mathbf Z\oplus\frac12(1+\sqrt d)\mathbf Z\) for \(d\equiv1\pmod4\). The trace lattice has Gram matrix \(\operatorname{diag}(2,2d)\) in the first case and \(\begin{pmatrix}2&1\\1&(d+1)/2\end{pmatrix}\) in the second, and discriminant \(d\).
Definition / construction / notation
5.3.B More lattices from codes. To make the link with codes, we consider the quotient of \(\mathfrak { O } _ { K }\) by a prime ideal \(\mathfrak { p }\) . This is a finite field, say \(\mathbb { F } _ { q }\) , \(q = p ^ { f }\) , with \(p\) prime, and where \(f\) is called the inertia index of \(\mathfrak { p }\) . To simplify matters we assume that \(f = 1\), so \(q = p\); that \(p\) and \(d_{K}\) are relatively prime; and that \(\operatorname{Tr}_{K/\mathbb{Q}}(\mathfrak{p}) \subset (p)\), which does not hold in general.
Lemma 5.3.6
Lemma 5.3.6. 1. \(\Gamma _ { C }\) is an integral lattice if and only if \(C \subset \mathbb { F } _ { p } ^ { n }\) is isotropic with respect to the standard dot product. 2. The discriminant of the lattice \(\Gamma _ { C }\) is given by \[ \operatorname { d i s c } ( \Gamma _ { C } ) = p ^ { 2 n - 2 m } \cdot \left( \mathfrak { d } _ { K } / p ^ { d _ { K } } \right) ^ { n } = \mathfrak { d } _ { K } ^ { n } \cdot p ^ { n ( 2 - d _ { K } ) - 2 m } , \quad m = \operatorname { d i m } C . \]
Example / comparison
Example 5.3.7. For \(K=\mathbf Q(\zeta_p)\) and the zero code in \(\mathbf F_p\), the Construction-A lattice \(\Gamma_0\) with the scaled Hermitian trace form is the root lattice \(A_{p-1}\); hence every isotropic code construction contains \(A_{p-1}^{\oplus n}\).
Lemma 5.3.8
Lemma 5.3.8. Let \(K \ = \ \mathbb { Q } ( \zeta )\) , \({ \mathfrak { O } } _ { K }\) and \(\rho _ { p } : \mathfrak { D } _ { K } ^ { n } \to \mathbb { F } _ { p } ^ { n }\) be as above. An ??- dimensional isotropic code \(C \subset \mathbb { F } _ { p } ^ { n }\) gives a positive definite even lattice \(\Gamma _ { C } = \rho _ { p } ^ { - 1 } C\) of rank \(n ( p - 1 )\) and discriminant \(p ^ { n - 2 m }\) . It is unimodular if and only if \(n = 2 m\) , that is, if \(C\) is self-dual. It contains \(A _ { p - 1 } ^ { \ircledast n }\) and if the minimal weight of \(C \ i s > p\) , then \(A _ { p - 1 } ^ { \oplus n }\) is the root sublattice of \(\Gamma _ { C }\) .
Example / comparison
Take \(p=3\) and \(C=\mathbf F_3(1,1,1)\subset\mathbf F_3^3\). The associated rank-\(6\), discriminant-\(3\) lattice is \(E_6\).
Example / comparison
Take \(p=5\) and the line in \(\mathbf F_5^2\) spanned by \((1,2)\). It is self-dual, so the associated positive-definite even unimodular rank-\(8\) lattice is \(E_8\).
Example / comparison
For self-dual codes with \(n(p-1)=24\), the construction gives Niemeier lattices; the cases \((p,n)=(3,12),(5,6),(7,4),(13,2)\) yield the listed root systems.
Example / comparison
For the extended ternary Golay code in \(\mathbf F_3^{12}\), the associated Niemeier lattice has root sublattice \(A_2^{\oplus12}\); another self-dual ternary code gives the Niemeier root system \(E_6^{\oplus4}\).
Example / comparison
The displayed self-dual code in \(\mathbf F_5^6\) gives a Niemeier lattice whose root sublattice is \(A_4^{\oplus6}\).
Example / comparison
The displayed self-dual code in \(\mathbf F_7^4\) gives a Niemeier lattice whose root sublattice is \(A_6^{\oplus4}\).
Example / comparison
The self-dual code generated by \((1,5)\) in \(\mathbf F_{13}^2\) gives the Niemeier lattice with root sublattice \(A_{12}^{\oplus2}\).
Definition 5.4.1
Definition 5.4.1. A quaternion algebra over \(k\) is a \(k\) -algebra \(D\) with the following properties: • \(D\) is central, i.e., its center is \(k\) ; • \(D = K + K j\) , a skew field of dimension 2 over a quadratic extension algebra \(K = k [ \pmb { i } ]\) of \(k\) , where \(\mathbf { i } ^ { 2 } \in k ^ { \times }\) ; • for all \(z = x + i y \in K\) , one has \(\mathbf { \vec { \mu } } _ { j z } = z ^ { * } \mathbf { \vec { \jmath } }\) , where \(z ^ { * } = x - i y\) .
Definition / construction / notation
From this it follows that in particular, \(i j = - j i\) . Moreover, if \(\pmb { i } ^ { 2 } : = \pmb { a } \in k ^ { \times }\) , \(\pmb { j } ^ { 2 } : = \pmb { b } \in k ^ { \times }\) , setting \(k = i j\) , then \(\pmb { k } ^ { 2 } = - a b\) . We denote such a quaternion algebra by \[ D = \left( { \frac { a , b } { k } } \right) . \]
Definition / construction / notation
Two quaternion algebras \(D , D ^ { \prime }\) over the same field \(k\) built from the quadratic extension agebras \(K / k , K ^ { \prime } / k\) , respectively, are said to be isomorphic if a \(k\) -algebra isomorphism \(f : D \to D ^ { \prime }\) exists with \(f ( K ) = K ^ { \prime }\) . For instance, we may take \(K = K ^ { \prime }\) but choose a different generator for the extension \(D / K\) , or we may exchange \(\textbf { \em i }\) and \(j\) so that \(K ^ { \prime } = k ( j )\) and \(D ^ { \prime } = K ^ { \prime } ( { \bf i } )\) . This leads to the isomorphisms \[ \begin{array} { r l } { \displaystyle \left( \frac { a , b } { k } \right) \simeq \left( \frac { b , a } { k } \right) , \ } & { ( i , { \pmb j } ) \mapsto ( { \pmb j } , { \pmb i } ) , } \\ { \displaystyle \left( \frac { a t ^ { 2 } , b } { k } \right) \simeq \left( \frac { a , b } { k } \right) , \ } & { ( i , { \pmb j } ) \mapsto ( t ^ { - 1 } { \pmb i } , { \pmb j } ) , t \in k ^ { \times } . } \end{array} \]
Example / comparison
The split quaternion algebra is \(M_2(k)\simeq(1,b)_k\). Over an algebraically closed field every quaternion algebra is split, so \((a,b)_{\bar k}\simeq M_2(\bar k)\).
Definition / construction
The reduced trace and reduced norm of a quaternion are the trace and norm with respect to quaternionic conjugation.
Definition / construction
Quaternionic conjugation gives reduced norm \(N(x)=xx^*\) and reduced trace \(T(x)=x+x^*\). In the standard basis, \(N(x)=x_0^2-ax_1^2-bx_2^2+abx_3^2\), its polar form is \(T(xy^*)\), and the norm form has discriminant \(1\) modulo squares.
Example / comparison
Example 5.4.2. The split quaternion algebra \((1,1)_k\) is isomorphic to \(M_2(k)\); under this identification the canonical anti-involution is the adjugate involution, the reduced trace is the ordinary matrix trace, and the reduced norm is the determinant.
Construction / comparison
A quaternion algebra is a division algebra exactly when its reduced norm is anisotropic: if \(N(x)\ne0\) then \(x^{-1}=x^*/N(x)\), while a non-trivial isotropic norm vector witnesses failure of the division property. The split case is characterized in Proposition 5.4.3.
Proposition 5.4.3
Proposition 5.4.3. A quaternion algebra is split, i.e. isomorphic to \(M _ { 2 } ( k )\) , if and only if the norm admits an isotropic vector.
Proposition 5.4.4
Let \(D, D'\) be quaternion algebras over \(k\). The following are equivalent: (1) \(D\) and \(D'\) are isomorphic; (2) the associated norm forms on the \(k\)-vector spaces \(D\) and \(D'\) are isometric; (3) the associated norm forms on \(D^{0}\) and \((D')^{0}\) are isometric. The quaternion algebra \(M_{2}(k)\) corresponds to the isotropic ternary form \(-x_{1}^{2} - x_{2}^{2} + x_{3}^{2}\), which is unique up to isometry.
Corollary 5.4.5
Corollary 5.4.5. The map which sends \(D\) to the quadratic form on \(D ^ { 0 }\) induced by the norm defines a bijection between the set of isomorphism classes of quaternion \(k\) -algebras and the equivalence classes of \(k\) -valued ternary forms with discriminant 1.
Definition 5.4.6
Definition 5.4.6. A \(\mathbb { Q }\) -quaternion algebra \(D\) is non-split or ramified at the place \(v \in \mathcal { P }\) if its localization \(D _ { v }\) at \(v\) is a skew field.
Proposition 5.4.7
Proposition 5.4.7. A quaternion algebra over \(\mathbb { Q }\) is ramified at an even number of places. Given an even number of places, there is a quaternion algebra ramified at exactly those places. In particular there is a unique isomorphism class of quaternion algebras split everywhere except at a given prime and at \(\infty\) .
Definition / construction / notation
5.4.D Ternary integral quadratic forms. To construct lattices in quaternion algebras \(D = D ( q )\) over \(\mathbb { Q }\) , one uses orders, i.e., subrings of \(D\) that are integral \(\mathbb { Z }\) -modules of rank 4. These exist: simply take \(\mathbb { Z } + \mathbb { Z } { \pmb { i } } + \mathbb { Z } { \pmb { j } } + \mathbb { Z } { \pmb { k } }\) . By definition, the discriminant of an order \(L \subset D\) is the discriminant of \(q | _ { L }\) . Every order of \(D\) is contained in a maximal order. These need not be unique, but we have:
Proposition 5.4.8
Proposition 5.4.8. The maximal orders of \(D = D ( q )\) are the orders with discriminant \(d ^ { 2 }\) where \(\boldsymbol { d } = \mathrm { d i s c } ( \boldsymbol { D } )\) , the discriminant of \(D\) . Any maximal order in \(D\) gives a positive definite rank 4 quadratic lattice with discriminant \(d ^ { 2 }\) . Such maximal orders exist.
Example / comparison
Example 5.4.9. If \(p\equiv3\pmod4\), the unique rational quaternion algebra ramified exactly at \(p\) and \(\infty\) is \((-1,-p)_{\mathbf Q}\), with ternary norm form \(x^2+py^2+pz^2\). The displayed maximal order has rank \(4\) and discriminant \(p^2\), and its trace form splits as two copies of the indicated binary form of discriminant \(p\).