FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 23 definitions, 1 theorem, 1 proposition, 4 lemmas, 1 corollary, 6 examples, 2 constructions, 1 remark, 2 conventions and 1 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction / notation
Let \(( V , b )\) be a real inner product space and write \(q ( x ) = { \textstyle { \frac { 1 } { 2 } } } b ( x , x )\) so that \(b\) is the polar form of \(q\) . Recall (cf. Section 7.1) that by definition rotations have determinant 1 and constitute the group SO(??), while an isometry \(g \in \mathsf { O } ( V )\) with \(\operatorname* { d e t } ( g ) = - 1\) is called a reflection. Special examples of reflections are the hyperplane reflections \(\sigma _ { x }\) given by \[ v \stackrel { \sigma _ { x } } { \longrightarrow } v - \frac { b ( x , v ) } { q ( x ) } x , \quad q ( x ) \not = 0 . \]
Definition / construction
The reduced real orthogonal group is the kernel of the real spinor norm.
Lemma 16.1.1
Lemma 16.1.1. Let \(( V , q )\) be a quadratic vector space and let \(x \ \in \ V\) and \(\ u \in \mathbf { \Sigma } \in\) \(\{ 1 , - 1 \}\) be such that the sign of \(q ( x )\) equals \(\epsilon\) . Then \(\begin{array} { r } { \mathsf { N m } _ { \mathrm { s p i n } } ^ { \epsilon } ( \sigma _ { x } ) = 1 } \end{array}\) .
Definition / construction
The signed-spinor subgroup \(O^\varepsilon(V)\) is the kernel of the signed spinor norm \(\operatorname{Nm}_{\rm spin}^\varepsilon:O(V)\to\{\pm1\}\).
Convention / notation
The signed spinor notation agrees with the ordinary spinor norm for the positive sign; on an \(\varepsilon\)-definite form the corresponding signed spinor norm is trivial, and changing \(q\) to \(-q\) interchanges the positive and negative versions.
Definition / construction / notation
For an indefinite real quadratic space, a \(q^\pm\)-orientation (sign structure) is an orientation of a maximal positive, respectively negative, subspace.
Definition / construction / notation
For an indefinite real quadratic space, a \(q^+\)-orientation is a coherent choice of orientation on all maximal positive-definite subspaces, and a \(q^-\)-orientation is defined analogously using maximal negative-definite subspaces.
Construction / comparison
Choosing both a positive and a negative sign structure orients the real quadratic space; the four components of \(O(n_+,n_-)\) are distinguished by signed spinor norm and determinant, with the rotation subgroup comprising two components.
Lemma 16.1.2
Lemma 16.1.2. Let \(n _ { \pm }\) be positive (i.e., the form is indefinite). 1. The connected component of \({ \sf O } \left( n _ { + } , n _ { - } \right)\) is the group \({ \mathsf { S O } } ^ { + } ( n _ { + } , n _ { - } )\) . It is the subgroup of \({ \sf O } ( n _ { + } , n _ { - } )\) preserving a given \(q ^ { + }\) -orientation as well as a given \(q ^ { - }\) -orientation. 2. The group \(0 ^ { + } ( n _ { + } , n _ { - } )\) of spinor norm 1 elements is the subgroup of \({ \sf O } ( n _ { + } , n _ { - } )\) preserving a given \(q ^ { - }\) -orientation. It is the union of the components \(I\) and \(I I\) in Figure 16.1.1. 3. The subgroup of elements with \(\mathsf { N } m _ { \mathrm { s p i n } } ^ { - } = 1\) is the subgroup of \({ \sf O } ( n _ { + } , n _ { - } )\) preserving a given \(q ^ { + }\) -orientation. It is the union of the components \(I\) and III.
Definition / construction / notation
We call \(V\) with this inner product a hyperbolic vector space.
Lemma 16.1.3
Lemma 16.1.3. Let \(V\) be a hyperbolic vector space. The “light cone” \(\{ x \in V \mid\) \(x \cdot x > 0 \}\) is a disjoint union \({ \mathsf { C } } _ { V } \cup { \mathsf { - C } } _ { V }\) of two convex cones where \(C _ { V }\) is contained in the half space \(x _ { 0 } > 0\) . For points \(x , y\) in the light cone we have: \[ x \cdot y > 0 \iff x , y \in \mathsf C _ { V } \ o r \ x , y \in - \mathsf C _ { V } . \] If \(x \neq 0\) and \(x \in { \overline { { C _ { V } } } }\) , \(y \in { \mathsf { C } } _ { V }\) , then we still have \(x \cdot y > 0\) .
Definition / construction
For a hyperbolic real vector space with chosen light-cone component \(C_V\), \(O^-(V)=\{g\in O(V):g(C_V)=C_V\}\) is the subgroup preserving that component, equivalently the corresponding orientation of positive lines.
Definition 16.2.1
Definition 16.2.1. Let \(s\) be a symplectic transformation whose matrix with respect to \(\pmb { { \cal E } }\) is given by \[ \begin{array} { r l } { \left( \begin{array} { l l } { ( a _ { i j } ) _ { 1 \leq i , j \leq n } } & { ( c _ { i j } ) _ { 1 \leq i , j \leq n } } \\ { ( b _ { i j } ) _ { 1 \leq i , j \leq n } } & { ( d _ { i j } ) _ { 1 \leq i , j \leq n } } \end{array} \right) \in \mathsf { S p } ( n ) . } \end{array} \] Then the Dickson invariant of the triple \(( q , s , \pmb { { E } } )\) is the element in the field \(k\) given by \[ D ( q , s , E ) = z _ { s E } - z _ { E } = p ( s ) = \sum _ { 1 \leq i , j \leq n } a _ { j } a _ { i j } c _ { i j } + \sum _ { 1 \leq i , j \leq n } a _ { j + n } b _ { i j } d _ { i j } + \sum _ { 1 \leq i , j \leq n } b _ { i j } c _ { i j } . \]
Corollary 16.2.2
Corollary 16.2.2. An isometry of the quadratic form \(q\) has Dickson invariant 0 or 1.
Example / comparison
Example 16.2.3. For a transvection \(t_{u,a}\) in characteristic \(2\), the displayed Dickson-invariant formula shows that \(t_{u,a}\) preserves \(q\) exactly when \(q(u)=a^{-1}\); in that case it is an orthogonal reflection and has Dickson invariant \(1\).
Construction / comparison
The Dickson invariant satisfies the stated cocycle identity on the symplectic group. Restricted to \(O(q)\) it is additive, hence defines a homomorphism \(O(q)\to\mathbf F_2\) whose kernel is the rotation subgroup.
Definition / construction / notation
So on transformations that preserve \(q\) the Dickson invariant gives an additive homomorphism \({ \mathsf { O } } ( q ) \to \{ 0 , 1 \} = \mathbb { F } _ { 2 }\) . The kernel of this homomorphism then is a subgroup of \({ \mathrm { O } } ( q )\) of index 2, by definition the subgroup of rotations:
Definition 16.2.4
Definition 16.2.4. Let \(( V , q )\) be a quadratic space over a field of characteristic 2. A rotation is an orthogonal transformation with Dickson invariant 0.
Example / comparison
Since an orthogonal transvection is just a reflection and has Dickson invariant 1, an even product of reflections is a rotation.
Example / comparison
Consider the hyperbolic plane \(U\) . We have seen in Example 6.5.5.2 that the only isometries are \(i _ { a }\) and \(j _ { b }\) . These have Dickson invariant 0, respectively 1. Since \(j _ { 1 ^ { \circ } } j _ { b } = i _ { b }\) , a rotation is a product of two reflections.
Example / comparison
Over \(\mathbf F_2\), the norm form of the quadratic extension gives the exceptional binary quadratic space; its rotations have Dickson invariant \(0\) and the other orthogonal transformations have invariant \(1\).
Definition / construction / notation
We set apart the values for \(s _ { 1 } ( V )\) and \(s _ { 0 } ( V )\) for the case \(k = \mathbb { F } _ { 2 }\) in Table 16.3.2.
Convention / notation
Over \(\mathbf F_2\), the Arf invariant is also called the democratic invariant because it is the value taken by the quadratic form on the majority of vectors.
Definition / construction
A divisor on a curve is a finite integral linear combination of points; its degree is the sum of coefficients, and linear equivalence is defined by principal divisors.
Definition / construction / notation
For a rational function \(f\) on \(c\) with support disjoint from the support of \(D\) we define \[ f ( D ) : = \prod _ { x \in { \mathrm { ~ s u p p o r t ~ } } D } f ( x ) ^ { n _ { x } } \in \mathbb { C } ^ { \times } . \]
Definition / construction / notation
16.4.B The jacobian. The Picard variety is an abelian variety. Let us sketch how this can be shown. One first introduces \(J ( C )\) , the jacobian of \(c\) , which is defined in a Hodge theoretic manner: One sets \[ J ( C ) = H ^ { 0 } ( C , \Omega _ { C } ^ { 1 } ) ^ { * } / \iota ( H _ { 1 } ( C , \mathbb { Z } ) ) , \]
Definition / construction / notation
Divisors on tori such as \(J ( C )\) come from holomorphic functions \(f\) on \(\mathbb { C } ^ { g }\) which are quasi-periodic with respect to the lattice \(\Gamma\) defining the torus, that is, for all \(\gamma \in \Gamma\) and \(z \in \mathbb { C } ^ { g }\) , one has \(f ( \pmb { z } + \gamma ) = \varphi _ { \gamma } ( \pmb { z } ) f ( \pmb { z } )\) for some non-vanishing holomorphic function \(\varphi _ { \gamma }\) . The classical theta functions give an abundance of such functions. The simplest is Riemann’s theta function \(\theta _ { Z }\) , a quasi-periodic function on \(\mathbb { C } ^ { g }\) defined by the everywhere convergent series \[ \theta _ { Z } ( z ) = \sum _ { \pmb { x } \in \mathbb { Z } ^ { g } } \exp 2 \pi i \left[ \pmb { x } \cdot ( \pmb { z } + \frac { 1 } { 2 } Z \pmb { x } ) \right] , \quad \pmb { z } \in \mathbb { C } ^ { g } , \]
Properties 16.4.1
Properties 16.4.1. 1. \(\theta _ { Z } \left[ \varepsilon _ { 1 } \right]\) is the translate of Riemann’s theta function by the vector \(Z \varepsilon _ { 1 } + \varepsilon _ { 2 }\) up to a multiplicative factor which is a function of \(z\) . \(k _ { 1 } , k _ { 2 } \in \mathbb { Z } ^ { g }\) , then \(\begin{array} { r } { \hat { \theta } _ { Z } \big [ _ { \varepsilon _ { 2 } + k _ { 2 } } ^ { \varepsilon _ { 1 } + k _ { 1 } } \big ] = \theta _ { Z } \big [ _ { \varepsilon _ { 2 } } ^ { \varepsilon _ { 1 } } \big ] } \end{array}\) 3. One has the symmetry relation \(\theta _ { Z } \left. { \varepsilon _ { 1 } } \right. ( - z ) = \exp { 4 \pi i [ \varepsilon _ { 1 } \cdot \varepsilon _ { 2 } ] } \cdot \theta _ { Z } \left. { \varepsilon _ { 1 } } \right. ( z )\) . If \(\begin{array} { r } { \varepsilon _ { 1 } , \varepsilon _ { 2 } \in \frac { 1 } { 2 } \mathbb { Z } ^ { g } } \end{array}\) , then \(\exp { 4 \pi } \pmb { i } [ \pmb { \varepsilon } _ { 1 } \cdot \pmb { \varepsilon } _ { 2 } ] \in \{ 1 , - 1 \}\) . In particular, \(\theta _ { Z } ( \pmb { z } ) = \theta _ { Z } ( - \pmb { z } )\) .
Definition / construction / notation
Fixing \(x_0\in C\), the Abel–Jacobi map \(\alpha:C\to J(C)\) sends \(x\) to the functional \(\omega\mapsto\int_{x_0}^x\omega\) modulo periods.
Definition / construction / notation
Here \([ K _ { C } ]\) is the class of a canonical divisor \(K _ { C }\) on \(c\) , i.e., the divisor of a holomorphic 1-form on \(C\) . Recall that this divisor has degree \(2 g - 2\) and so \(\boldsymbol { \kappa } = \alpha ( [ \boldsymbol { \vartheta } _ { 0 } ] )\) , where \(\vartheta _ { 0 }\) is a degree \(g - 1\) divisor such that \(2 \vartheta _ { 0 }\) is linearly equivalent to \(K _ { C }\) , also called a half-canonical divisor or a theta characteristic. The particular \(\kappa\) we found here is called the Riemann constant.
Definition / construction / notation
A theta characteristic \(\vartheta\) is called even or odd according as \(h^0(C,\vartheta)\) is even or odd.
Definition / construction / notation
A theta characteristic can be given as \(\vartheta = \vartheta _ { 0 } + \varepsilon\) where \(\alpha ( \varepsilon ) \in J _ { 2 } ( C )\) , the group of 2-torsion points of \(J ( C )\) . In terms of the period matrix \(Z\) , two-torsion points on \(J ( C )\) can be written as \(\varepsilon = \textstyle { \frac { 1 } { 2 } } \left( \varepsilon ^ { \prime \prime } + Z \varepsilon ^ { \prime } \right)\) , \(\varepsilon ^ { \prime } , \varepsilon ^ { \prime \prime } \in \mathbb { Z } ^ { g } / 2 \mathbb { Z } ^ { g } \simeq \mathbb { F } _ { 2 } ^ { g }\) . Using Property 16.4.1.1, the theta divisor belonging to the theta function with characteristics \(\begin{array} { r } { \varepsilon _ { 1 } = \frac { 1 } { 2 } \varepsilon ^ { \prime } } \end{array}\) , \(\begin{array} { r } { \varepsilon _ { 2 } = \frac { 1 } { 2 } \varepsilon ^ { \prime \prime } } \end{array}\) is a translate of the Riemann theta divisor by the two-torsion point \(\varepsilon\) which we denote \(t _ { \varepsilon } ^ { * } \Theta\) . Hence, applying Riemann’s singularity theorem, we find for the multiplicity at \(0\) of the translated theta-divisor \[ \mu _ { 0 } ( t _ { \varepsilon } ^ { * } \Theta ) = \mu _ { 0 } \mathrm { d i v } \left( \theta _ { Z } \left[ \varepsilon _ { \varepsilon ^ { \prime \prime } } ^ { \prime } \right] \right) = h ^ { 0 } ( \mathbb { O } _ { C } ( \vartheta ) ) . \]
Lemma 16.4.2
Lemma 16.4.2. Let \(\vartheta _ { 0 }\) be the theta characteristic corresponding to the Riemann constant. Using the dot product on \(\mathbb { F } _ { 2 } ^ { g }\) , the theta characteristic \(\vartheta = \vartheta _ { 0 } + \varepsilon\) has parity \(\varepsilon ^ { \prime } \cdot \varepsilon ^ { \prime \prime }\) , where we write \(\begin{array} { r } { \varepsilon = { \frac { 1 } { 2 } } \varepsilon ^ { \prime \prime } + { \frac { 1 } { 2 } } Z \varepsilon ^ { \prime } } \end{array}\) , \(\varepsilon ^ { \prime } , \varepsilon ^ { \prime \prime } \in \mathbb { Z } ^ { g } / 2 \mathbb { Z } ^ { g } \simeq \mathbb { F } _ { 2 } ^ { g }\) .
Example / comparison
For genus \(1\) there are three even theta characteristics and one odd theta characteristic.
Example / comparison
For genus \(3\) there are \(36\) even and \(28\) odd theta characteristics; on a smooth plane quartic the odd theta characteristics correspond to its \(28\) bitangents.
Definition / construction / notation
16.4.C Weil pairing on torsion points of \(J ( C )\) . Suppose that \(D _ { 1 } , D _ { 2 }\) are two degree 0 divisors on the curve \(C\) with \(\alpha ( D _ { j } )\) , \(j = 1 , 2\) , a \(k\) -torsion point. Then \(k D _ { j } = \operatorname { d i v } ( f _ { j } )\) for some non-zero rational functions \(f _ { 1 } , f _ { 2 }\) . If \(D _ { 1 }\) and \(D _ { 2 }\) have disjoint supports, then so have \(f _ { 1 }\) and \(D _ { 2 }\) as well as \(f _ { 2 }\) and \(D _ { 1 }\) . Hence we can define \[ \begin{array} { r } { e _ { k } ( D _ { 1 } , D _ { 2 } ) : = f _ { 1 } ( D _ { 2 } ) / f _ { 2 } ( D _ { 1 } ) , \quad \mathrm { d i v } ( f _ { j } ) = k D _ { j } . } \end{array} \]
Definition / construction / notation
The \(2\)-torsion \(J_2(C)\) with the Weil pairing is a symplectic \(\mathbf F_2\)-space; theta characteristics determine quadratic refinements of this pairing.
Definition / construction
The \(2\)-torsion of the Jacobian acts on quadratic refinements of the Weil pairing by adding the corresponding linear forms.
Definition / construction / notation
On the other hand, a sum \(q + q ^ { \prime }\) of two quadratic forms in \(Q ( V )\) is a linear form, and hence has the shape \(x \mapsto J ( x , v )\) for a vector \(v = v ( q , q ^ { \prime } )\) uniquely determined by \(q\) and \(q ^ { \prime }\) . So adding two quadratic forms \(q\) and \(q ^ { \prime }\) defines a unique vector in \(V\) . The set \(Q ( V )\) is not a vector space, since it has no zero, but the disjoint union of \(V\) and \(Q ( V )\) is a vector space of dimension \(2 n + 1\) if we take the original vector space structure on \(V\) , we use the \(t\) -action on \(Q ( V )\) for addition of vectors \(v \in V\) and \(q \in Q ( V )\) , and, finally, we define \(\boldsymbol { q } + \boldsymbol { q } ^ { \prime } = \boldsymbol { v } ( \boldsymbol { q } , \boldsymbol { q } ^ { \prime } )\) whenever \(q , q ^ { \prime } \in Q ( V )\) .
Proposition 16.4.3
Proposition 16.4.3. There are \(2 ^ { g - 1 } ( 2 ^ { g } + 1 )\) forms with Arf invariant 0 and \(2 ^ { g - 1 } ( 2 ^ { g } - 1 )\) with Arf invariant 1.
Theorem 16.4.4
Theorem 16.4.4. Let \(c\) be a smooth complex algebraic curve of genus \(g ~ \geq ~ 1\) . Then \(c\) has \(2 ^ { g - 1 } ( 2 ^ { g } + 1 )\) even and \(2 ^ { g - 1 } ( 2 ^ { g } - 1 )\) odd theta characteristics. The even theta characteristics correspond to quadratic forms with Arf invariant 0 and the odd ones to quadratic forms with Arf invariant 1.
Remark: Remark
Remark. In [164] D. Mumford gave a geometric argument proving this result. His argument is valid for curves in all characteristics different from 2.