The local global-principle states that two hermitian spaces over some number field K are isometric if and only if they are isometric over every completion of K.

The genus of a lattice L in an hermitian space consists of those lattices which are isometric to L locally everywhere. Every genus decomposes into finitely many isometry classes. The number of isometry classes in a genus is called its class number. Hence the genera with class number one are precisely those lattices for which the local-global principle holds.

For indefinite lattices, the class number can be expressed a-priori in terms of some local invariants. For definite lattices, such a description is not possible. However, up to similarity, there are only finitely many genera with a given class number.

In my talk, I will present the classification of all definite hermitian lattices of class number one.