Rational lattice
Positive definite lattice
A rational lattice is said to be positive definite iff for any nonzero . geo
Equivalently, the quadratic space is (“Euclidean space”).[^1988]
Properties
- There exist finitely many lattice points of a given norm, i.e. for any .
- Assume is ^integral and . Then and
\begin{align*}
\langle \alpha,\beta \rangle = -2 &\iff \alpha+\beta = 0 \
\langle \alpha,\beta \rangle = -1 &\iff \alpha + \beta \in L_{2}
\end{align*}