Classification of integral lattices with large class number

Abstract

A detailed exposition of Kneser’s neighbour method for quadratic lattices over totally real number fields, and of the sub-procedures needed for its implementation, is given. Using an actual computer program which automatically generates representatives for all isomorphism classes in one genus of rational lattices, various results about genera ofℓ\ell-elementary lattices, for small prime levelℓ,\ell ,are obtained. For instance, the class number of1212-dimensional77-elementary even lattices of determinant767^6is395395; no extremal lattice in the sense of Quebbemann exists. The implementation incorporates as essential parts previous programs of W. Plesken and B. Souvignier.