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.
2001 Academic Press
An n-dimensional integral lattice 4 is modular [5] if there exists a similarity _ such that _(4*)=4, where 4* is the dual of 4. More generally, if 6 is a set of primes, define the 6-dual of 4 to consist of the vectors v # 4 Q such that v } 4/Z p for p # 6 and v } 4*/Z p for p Γ 6. Then a similarity _ will be said to be a 6-modularity of 4 if _(4 V6 )=4; if _ multiplies norms by N, it will be said to be a modularity of level N. A lattice is strongly modular if it has 6-modularities for all subsets of the primes dividing its level. (This concept was introduced in [6], but we are using the notation of [8]).
In the present note, we consider the case n=2. It turns out that, properly defined, 6-duality acts as a translation on the class group. This allows us to answer the following question: Given a discriminant D and a set of primes 6, when does there exist a 2-dimensional lattice of that discriminant possessing a 6-modularity, and if one exists, how many are there? In particular, we answer this question for strongly modular lattices. The connection between class groups and modularities was alluded to in the case of square-free discriminant in [6], which also stated the appropriate special case of Corollary 10 below.
It will turn out to be convenient in the sequel to use nonstandard conventions when discussing quadratic forms. The basic difference is that the