Classification of integral lattices with large class number

by Rudolf Scharlau and Boris Hemkemeier

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 of $\ell$-elementary lattices, for small prime level $\ell,$ are obtained. For instance, the class number of $12$-dimensional $7$-elementary even lattices of determinant $7^6$ is $395$; no extremal lattice in the sense of Quebbemann exists. The implementation incorporates as essential parts previous programs of W.~Plesken and B.~Souvignier.

Key words and phrases: lattice, integral quadratic form, class number of genus, neighbour method, $p$-elementary lattice, extremal modular lattice.

1991 Mathematics Subject Classification: Primary 11E41, Secondary 11H55, 11--04.

Published in: Mathematics of Computation, Volume 67, Number 222, April 1998

Note: You can retrieve the latest version of the computer program tn described above here (300kByte).tn is written in C and is compliant to POSIX-1003.1 a superset of ANSI-C. It compiles in every UNIX environment and is extensively tested on HP-UX 9.*, 10.*, Convex SPP-HP-UX 5.2, IBM's AIX 4.1.