\(
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The Coble lattice as a twist
Proposition 41.2 ($T_\Co$ is a twisted odd unimodular lattice) Set \[
B \definedas \generators{1}\oplus E_{10} = \generators{1}\oplus U\oplus E_8
.
\] Then \(B\) is odd and unimodular of signature \((2,9)\), so \(B\cong\latI_{2,9}\), and \[
T_\Co = \generators{2}\oplus E_{10}(2) = B(2) \cong \latI_{2,9}(2)
.
\] Consequently \(\Orth(T_\Co) = \Orth(B)\) as subgroups of \(\GL(B)\), and a vector of \(T_\Co\) is isotropic, respectively primitive, exactly when it is isotropic, respectively primitive, as a vector of \(B\).
Proof. The lattice \(E_{10} = U\oplus E_8\) is even and unimodular of signature \((1,9)\) (the Enriques-lattice definition (Definition 39.6)), so \(B = \generators{1}\oplus E_{10}\) is unimodular of signature \((2,9)\) and is odd because the generator of \(\generators{1}\) has norm \(1\). By the classification of indefinite unimodular lattices (the indefinite unimodular classification (Theorem 38.20)) an odd unimodular lattice of signature \((2,9)\) is isometric to \(\latI_{2,9}\). Twisting distributes over orthogonal direct sums and \(\generators{1}(2) = \generators{2}\), whence \(B(2) = \generators{2}\oplus E_{10}(2) = T_\Co\).
A twist leaves the underlying \(\ZZ\)-module unchanged and multiplies the form by a nonzero scalar, so an automorphism of the module preserves \(\beta_B\) if and only if it preserves \(2\beta_B = \beta_{B(2)}\); this gives \(\Orth(T_\Co) = \Orth(B)\). The same scaling shows \(\beta_{T_\Co}(v,v) = 2\beta_B(v,v)\) vanishes exactly when \(\beta_B(v,v)\) does, and primitivity is a property of the module alone.
Proposition 41.4 ($T_\Co$ is the twist of a split maximal lattice) The lattice \(B\cong\latI_{2,9}\) of the twist proposition (Proposition 41.2) is maximal in the sense of the maximal-lattice definition (Definition 38.15) and split in the sense of the split-maximal definition (Definition 38.17): \[
B \cong U\oplus U\oplus\latI_{0,7}
.
\] In a decomposition \(T_\Co = B(2)\) with \(B = U\oplus U\oplus\latI_{0,7}\), an isotropic generator \(e\) of one hyperbolic summand and the plane \(J\) spanned by isotropic generators of both have \[
e^{\perp}/e \cong \latI_{1,8}(2) = (9,9,1)_1,
\qquad
J^{\perp}/J \cong \latI_{0,7}(2) \cong A_1^{\oplus 7} = (7,7,1)_0
.
\]
Proof. \(B\) is unimodular, hence maximal by the maximal-overlattice lemma (Lemma 38.16). The lattice \(U\oplus U\oplus\latI_{0,7}\) is unimodular of signature \((2,9)\) and odd, because \(\latI_{0,7}\) is, so it is isometric to \(\latI_{2,9}\cong B\) by the indefinite unimodular classification (Theorem 38.20); this exhibits \(B\) in the shape required by the split-maximal definition (Definition 38.17), as the maximal-splitting theorem (Theorem 38.18) also predicts for \(n = 9\geq 5\). For the quotients, take \(e\) to be an isotropic generator of the first hyperbolic summand of \(B\cong U\oplus U\oplus\latI_{0,7}\); then \(e^{\perp B} = \ZZ e\oplus U\oplus\latI_{0,7}\) and \(e^{\perp}/e\cong U\oplus\latI_{0,7}\cong\latI_{1,8}\), which is odd unimodular of signature \((1,8)\). Taking \(J\) to be spanned by isotropic generators of the two hyperbolic summands gives \(J^{\perp B} = J\oplus\latI_{0,7}\) and \(J^{\perp}/J\cong\latI_{0,7}\). Twisting by \(2\) and reading off invariants gives \(\latI_{1,8}(2) = (9,9,1)\) of signature \((1,8)\) and \(\latI_{0,7}(2) = \generators{-2}^{\oplus 7} = (7,7,1)\) of signature \((0,7)\).
From discriminant classes to lattice vectors
Proposition 41.12 (Primitive isotropic vectors and their discriminant classes) Every primitive isotropic \(v\in T_\Co\) has \(\div_{T_\Co}(v) = 2\), and its associated class \[
v^* \definedas \tfrac{1}{2}v + T_\Co \in A_{T_\Co}
\] is a nonzero isotropic class. All such classes lie in the single nonzero orbit of the isotropic-class orbit theorem (Theorem 41.11).
Proof. The divisibility statement is the divisibility lemma (Lemma 63.2), and \(v^*\) has order \(\div_{T_\Co}(v) = 2\) in \(A_{T_\Co}\) (the discriminant-form definition (Definition 37.8) and the divisibility conventions of the Lattice Theory section), so \(v^*\neq 0\). Writing \(v\in B\) and using \(\beta_{T_\Co} = 2\beta_B\), isotropy of \(v\) gives \(\beta_B(v,v) = 0\), hence \(Q(v) = 0\) and \(q_{T_\Co}(v^*) = 0\) by the twisted-unimodular discriminant proposition (Proposition 41.6). The final assertion is the isotropic-class orbit theorem (Theorem 41.11), which has a single orbit of nonzero isotropic classes.