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.

**Authors:**

**Knowl status:**

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

**Referred to by:**

- dq.lattice.completeness
- lattice.24.1.1.24.1.top
- lattice.8.1.1.1.1.bottom
- lattice.automorphism_group
- lattice.class_number
- lattice.density
- lattice.determinant
- lattice.dimension
- lattice.dual
- lattice.genus
- lattice.group_order
- lattice.hermite_number
- lattice.history
- lattice.isometry
- lattice.level
- lattice.minimal_vector
- lattice.name
- lattice.normalized_minimal_vector
- lattice.postive_definite
- lattice.primitive
- lattice.search_input
- lattice.theta
- lattice.unimodular
- mf.gl2.history.theta
- lmfdb/lattice/lattice_stats.py (line 16)
- lmfdb/lattice/templates/lattice-single.html (line 101)

**History:**(expand/hide all)

- 2018-06-19 03:07:24 by Kiran S. Kedlaya (Reviewed)