# Lattice Theory :::{.remark} We refer to: - \cite{vinberg1985hyperbolic-groups} - \cite{Vin75} ::: ## Basic Theory :::{.remark} The study of semitoroidal compactifications of moduli spaces of Coble surfaces largely reduces to lattice theory, of which we will now recall the essential notions. ::: :::{.remark title="Basic invariants"} By a **lattice**, we mean a free $\ZZ$-module $L$ of finite rank equipped with a nondegenerate[^1] symmetric integral bilinear form $\beta_L: L \otimes_\ZZ L \to \ZZ$. We abbreviate $vw \da \beta_L(v, w)$ and $v^2 \da \beta_L(v, v)$ and refer to the latter as the \textbf{norm} of $v$. We write $L_R \da L\otimes_{\ZZ} R$ and $\beta_{L_R}$ for $R = \QQ, \RR, \CC$ for the $\ZZ$-linear extensions of $(L, \beta_L)$ to the rational, real, and complex numbers respectively. A submodule $M\subseteq L$ is a **sublattice** if the restricted bilinear form $\ro{\beta_L}{M}$ endows $M$ with the structure of a lattice. A vector $v\in L$ is **isotropic** if $v^2 = 0$, i.e. it is norm zero, and more generally a sublattice $M \subseteq L$ is isotropic if $\ro{\beta_L}{M} \equiv 0$. A lattice is said to be **even** if $x^2\in 2\ZZ$ for all $x\in L$, and \textbf{odd} otherwise. A nondegenerate symmetric bilinear form can be linearly extended to $L_\RR$ and by Sylvester's theorem, diagonalized with only $1$ or $-1$ on the diagonal. We write $n_+$ and $n_-$ respectively for the number of $\pm 1$ entries on the diagonal. The **signature** of $L$ is the pair $(n_+, n_-)$, and the **index** is $n_+ - n_-$. We say $L$ is **definite** if either $n_+$ or $n_-$ is zero, and **indefinite** otherwise. More precisely, if $n_- = 0$ we say $L$ is **positive definite**, and similarly if $n_+ = 0$, we say $L$ is **negative definite**. A negative definite lattice generated by elements of norm $-2$ is said to be a **root lattice**, and such elements are referred to as **roots**. The **rank** $r$ of a lattice is its rank as a free $\ZZ$-module and is given by $r=n_+ + n_- = \dim_\QQ L_\QQ$. An indefinite lattice of signature $(1, r-1)$ is said to be **hyperbolic**. Fixing a generating set $e_i$ of $L$, we define the **Gram matrix** of $L$ as the matrix $G_L \da (\beta_L(e_i, e_j))_{ij}$, and the **discriminant** as $\operatorname{disc} L \da \det G_L$. The discriminant is independent of the choice of generating set. ::: :::{.remark title="Discriminant forms"} The **dual lattice** to $L$ is denoted $L\dual \da \Hom_\ZZ(L, \ZZ)$, and there is an morphism $$\begin{aligned} \iota: L &\injects L\dual \\ x &\mapsto \beta_L(x, \cdot) \end{aligned}$$ which, if $L$ is nondegenerate, is an injection with finite index image. The **discriminant group** is $A_L \da \coker \iota \cong L\dual/L$; this is a finite order group of order $\abs{\operatorname{disc} L}$. We say $L$ is **unimodular** if any of the following equivalent conditions hold: 1. $A_L$ is the trivial group, 2. $\iota$ is an isomorphism and $L\cong L\dual$, 3. $\abs{ \operatorname{disc} L} = 1$. If $A_L \cong (\ZZ/p\ZZ)^a$ for some $a$, we say $L$ is **$p$-elementary**; in our applications we will often have $p=2$. For even lattices, the form $\beta_L$ descends to a well-defined quadratic form $$\begin{aligned} q_L: A_L &\to \QQ/2\ZZ \\ x + L &\mapsto \beta_{L_\QQ}(x, x) \mod 2\ZZ \end{aligned}$$ We call the pair $(A_L, q_L)$ the **discriminant quadratic form** of $L$. A **morphism** between two lattices is a morphism of $\ZZ$-modules $\eta: L\to L'$ respecting the bilinear forms in the sense that $\beta_L(x, y) = \beta_{L'}(\eta(x), \eta(y))$, and is a **primitive embedding** if $\eta$ is injective and $\coker \eta$ is torsionfree. An **isometry** of lattices is an isomorphism, defined in the obvious way. We write $\Orth(L)$ for the group of lattice automorphisms of $L$, denoted the **orthogonal group** of $L$, and similarly $\Orth(q_L)$ for the $\ZZ$-module automorphisms of of the discriminant form $A_L$ which preserve the quadratic form. There is a natural group homomorphism $\Orth(L)\to\Orth(q_L)$, the kernel is denoted $\tilde \Orth(L)$. ::: :::{.remark title="Orthogonal complements"} Given two lattices $L_1, L_2$ we write $L_1\oplus L_2$ for the **orthogonal direct sum**, which is the direct sum of the underlying modules with bilinear form defined by $$\beta_{L_1\oplus L_2}(v_1 + v_2, w_1 + w_2)\da \beta_{L_1}(v_1, w_1) + \beta_{L_2}(v_2, w_2) .$$ We write $L^{\oplus n}$ for the direct sum of $n$ copies of $L$. Let $\eta: M \injects L$ be a primitive embedding of lattices, for example the inclusion of a sublattice. We write $$M^{\perp L} \da \ts{x\in L \mid \beta_L(x, M) = 0} .$$ If the ambient lattice $L$ is understood, we often abuse notation and simply write $M^{\perp}$ without reference to $L$. Note that $M^{\perp L}\oplus M \subseteq L$ may not be saturated, and is generally a finite index sublattice of $L$. We note that $M^{\perp L}\intersect M \neq \ts{0}$ in general.[^2] We also note that for any lattices $L_i$, \[ A_{L_1 \oplus \cdots \oplus L_n} = A_{L_1} \oplus \cdots \oplus A_{L_n} \] ::: :::{.remark title="2-elementary lattices"} Let $L$ be a 2-elementary lattice. The **divisibility** of a vector $v\in L$, denoted $\operatorname{div}_L(v)$, is defined by $\beta_L(v, L) = \operatorname{div}_L(v)\ZZ$, i.e. the positive integral generator of the image of the map $\beta_L(v, \cdot): L\to \ZZ$. For 2-elementary lattices, one always has $\operatorname{div}_L(v) \in \ts{1, 2}$. We set $v^* \da v/\operatorname{div}_L(v)\in A_L$. Letting $q_L:A_L \to {1\over 2}\ZZ/\ZZ$ be the induced quadratic form on $A_L$, we say $v^*$ is **characteristic** if $q_L(x) = \beta_L(v^*, x)\mod \ZZ$ for all $x\in A_L$, and is **ordinary** otherwise. We say that a primitive isotropic vector $e\in L$ is 1. **odd** if $\operatorname{div}_L(e) = 1$, 2. **even ordinary** if $\operatorname{div}_L(e) = 2$ and $e^*$ is ordinary, or 3. **even characteristic** if $\operatorname{div}_L(e) = 2$ and $e^*$ is characteristic. The 2-elementary hyperbolic lattices admitting a primitive embedding into $\lkt$ were classified by Nikulin in \cite[\S 3.6.2]{nikulin1979integer-symmetric}. An indefinite 2-elementary lattice is determined up to isometry by a triple of invariants $(r,a,\delta)$. Here, $r\da \operatorname{rank}_\ZZ(L)$ is the rank, $a = \operatorname{rank}_{\bF_2}A_L$ is the exponent appearing in $A_L = (\ZZ/2\ZZ)^a$, and $\delta \in \ts{0, 1}$ is the **coparity**: we set $\delta = 0$ if $q_L(A_L) \subseteq \ZZ$, so $q_L(x) \equiv 0 \mod \ZZ$ for all $x\in A_L$, and $\delta=1$ otherwise. We accordingly specify such lattices using the notation $(r,a,\delta)_{n_+}$. ::: :::{.remark title="Twists of a lattice"} If $L$ is a lattice with bilinear for $\beta_L$, define $L(n)$ to be the twist of $L$ by $n$, which has the same underlying $\ZZ$-module but is equipped with the scaled bilinear form \[\beta_{L(n)}(v,w) \da n\cdot \beta_L(v, w).\] ::: :::{.remark title="The lattice $\gens{n}$"} The lattice $\gens{n}$ is defined as the rank 1 lattice $\ZZ$ with one generator $v$ satisfying $\beta_{\gens{n}}(v,v) = n$. The Gram matrix is the $1\times 1$ matrix $G_{\gens n} = [n]$, and the associated quadratic form is $q_{\gens{n}}(x) = nx^2$. ::: :::{.remark title="The hyperbolic lattice"} In rank 2, there are two unimodular hyperbolic lattices: the odd $\rm{I}_{1, 1} \da \gens{1} \oplus \gens{-1}$, and the even $U\da \rm{II}_{1, 1}$. We refer to the latter as the \textbf{hyperbolic lattice}, which can be realized as $U \da \ZZ e \oplus \ZZ f$ with $e^2=f^2 = 0$ and $ef = 1$, and thus the following Gram matrix: \[ G_U = \begin{bmatrix}0&1\\1&0\end{bmatrix} .\] ::: :::{.remark title="ADE lattices"} Any Dynkin diagram of type $A_n, D_n, E_6, E_7, E_8$ corresponds to a root lattice of the respective type. By convention, we take the negative definite twists of these lattices. Of particular importance to us is the $E_8$ lattice associated to the following Dynkin diagram: \begin{figure}[H] \centering \input{tikz/e8_coxeter_diagram.tikz} \caption{The Dynkin diagram $E_{8}$} \label{fig:e8-coxeter-diagram} \end{figure} ::: :::{.remark title="The lattice ${{\rm{I} }_{p, q}}$."} For any pair of non-negative integers $(p, q)$, there exists an odd indefinite unimodular lattice determined up to isomorphism by its rank and signature: \[ \mathrm{I}_{p, q} \da \gens{1}^{\oplus p}\oplus \gens{-1}^{\oplus q} .\] ::: :::{.remark title="The lattice ${\rm{II}}_{p, q}$"} Let $L$ be an even indefinite unimodular lattice of signature $(p, q)$. Then $p-q\equiv 0 \pmod 8$, and $L$ is uniquely determined up to isomorphism by its rank and signature: \[ {\rm{II}}_{p, q} \da \begin{cases} U ^{\oplus p}\oplus E_8^{\oplus {q-p\over 8}}, & p < q \\ U^{\oplus q} \oplus E_8(-1)^{\oplus {p-q\over 8}}, & p > q .\end{cases} \] ::: [^1]: A bilinear form is *nondegenerate* if for any $x$, $\beta_L(x, L) = 0$ implies $x=0$. [^2]: For example, let $L = U = \gens{e,f}$ and $M = \gens{e}$. Then $M^{\perp L} = M$.