Integral quadratic forms

Definition. A unimodular symmetric bilinear form \{Q=\left( L, \langle \,.\,,\,.\,\rangle \right) \} over the integers consists of

  • a free $\mathbb Z$-module $L$ of finite rank and

  • a symmetric bilinear form $\langle \,.\,,\,.\,\rangle : L\times L\to \mathbb Z$
  • such that the map $L\to L^\vee, v\mapsto \langle v,\,.\,\rangle$ is an isomorphism of $L$ with its dual.

    Equivalently, if $v_1,\ldots,v_r$ is a basis for $L$, then the matrix $C=\left(c_{ij}\right)_{ij}$ with entries $$c_{ij}=\langle v_i,v_j\rangle$$ is invertible. In what follows, we will use the term quadratic form as a shorthand synonym.

    Invariants of $Q$.

    • $r(Q)$ the rank of $L$ as a module over the integers.
    • The real quadratic form $Q\otimes \mathbb R$ can be diagonalized with $b_+$ positive and $b_-$ negative entries in the diagonal. The signature $$\sigma(Q):= b_+-b_-$$ is well defined according to Sylvester's law of inertia.
    • The form $Q$ is called even, if the squares $\langle v,v\rangle$ are even integers for any $v\in L$. Otherwise, $Q$ is called odd.

    Examples

    • The quadratic form $\langle 1\rangle := \left(\mathbb Z, (1)\right)$ is odd of rank and signature 1.
    • The hyperbolic plane $$H:= \left(\mathbb Z^2, \left(\begin{matrix}0&1\\1&0\end{matrix}\right)\right)$$ is an even quadratic form of rank 2 and signature 0.
    • The quadratic form $E_8:=\left(\mathbb Z^8, \left(c_{ij}\right)\right)$ is best described by its Dynkin diagram:
      The vertices (or circles) $e_1,\ldots,e_8$ correspond to the standard generators of $\mathbb Z^8$. The associated matrix $C$ has entries $$c_{ij}=\langle e_i, e_j\rangle = \begin{cases}2 & \mbox{if }i=j\\ -1&\mbox{if }e_i\mbox{ and }e_j\mbox{ are connected by an edge}\\0&\mbox{else.}\end{cases}$$The quadratic form $E_8$ is even with both rank and signature 8.
    • If $Q=\left(L,C\right)$ is a quadratic form, then $-Q$ denotes the quadatic form $\left(L,-C\right)$.
    • The orthogonal sum of quadratic forms $Q_1$ and $Q_2$ is denoted by $$Q_1 \perp Q_2.$$