Class Groups and Modular Lattices

We show that in the case of 2-dimensional lattices, Quebbemann's notion of modular and strongly modular lattices has a natural extension to the class group of a given discriminant, in terms of a certain set of translations. In particular, a 2-dimensional lattice has "extra" modularities essentially when it has order 4 in the class group. This allows us to determine the conditions on D under which there exists a strongly modular 2-dimensional lattice of discriminant D, as well as how many such lattices there are. The technique also applies to the question of when a lattice can be similar to its even sublattice.