Integral quadratic form

Summary

An integral quadratic form is a quadratic form $\sum_{i \le j} a_{ij} x_i x_j$ whose coefficients $a_{ij}$ are integers. The main questions are arithmetic: which integers the form represents, in how many ways, and when two forms are equivalent under integral changes of variables. Classical examples are sums of squares, such as $x_1^2 + x_2^2 + x_3^2 + x_4^2$. Integral quadratic forms correspond to lattices with an integer-valued quadratic function on them.