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Let $V$ be an $n$-dimensional vector space over a field $F$ equipped with a symmetric bilinear map $B:V\times V\rightarrow F$ and associated quadratic map $Q(v):=B(v,v).$ The pair $(V,Q)$ is a quadratic space. Fixing a basis, $\{v_1,...,v_n\}$ for $V$, the space $(V,Q)$ has an associated quadratic form, that is, a homogeneous degree 2 polynomial, given by \[ f(x_1,...,x_n)=\sum_{i,j}B(v_i,v_j)x_ix_j, \] where $B(v_i,v_j)=B(v_j,v_i)$ since $B$ is symmetric. It is often helpful to think of a quadratic space in terms of its Gram matrix representation.

Let $R$ denote the ring of integers of $F$. An $R$-lattice is a finitely generated $R$-module which is a discrete subset of $V$ endowed with the same bilinear map $B$. In the special case when $F=\mathbb{R}$, an integral lattice is simply a $\mathbb{Z}$-lattice, $L$, with the added restriction that $B(L,L)\subseteq \mathbb Z$.

For example, letting $B(v,w)=v\cdot w$, the standard inner product, Euclidean $n$-space can be thought of as the quadratic space $(\mathbb{R}^n, \cdot)$. In this case, the space $(\mathbb{R}^n, \cdot)$ contains the integral lattice $\mathbb{Z}^n$ as a discrete subset endowed with the standard inner product.

Knowl status:
  • Review status: reviewed
  • Last edited by Kiran S. Kedlaya on 2018-06-19 03:07:24
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