On the involutions fixing the class of a lattice

With any integral lattice Λ in n-dimensional Euclidean space we associate an elementary abelian 2-group I(Λ) whose elements represent parts of the dual lattice that are similar to Λ. There are corresponding involutions on modular forms for which the theta series of Λ is an eigenform; previous work has focused on this connection. In the present paper I(Λ) is considered as a quotient of some finite 2-subgroup of . We establish upper bounds, depending only on n, for the order of I(Λ), and we study the occurrence of similarities of specific types.