# Automorphism groups of indefinite non-unimodular integer lattices

Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they don't have to have determined the full group: aspects such as conjugacy classes or conjugacy classes of finite subgroups or similar would all be of interest to me. I'm mostly trying to learn some general ideas and techniques: structural results exist for unimodular lattices and orthogonal groups over fields but less so for the integral indefinite non-unimodular case.

Thanks!

-

## 1 Answer

Most of the published stuff is for unimodular, but there is a case of general interest, where you take a positive lattice and add a hyperbolic plane. The result, as far as automorphism group, is the celebrated result of Conway and, later, Borcherds, on the automorphism group of the Leech lattice plus a hyperbolic plane. See SPLAG but see, especially, Wolfgang Ebeling, Lattices and Codes, at least the second edition(not in the first edition).

The reason this was interesting for me was the calculation of the genus of the original, positive, integral, lattice, as the classes in the same genus as $L$ are the integral classes in $L+U,$ where $U$ is the said hyperbolic plane, Gram matrix corresponding to $2xy.$ This is a single paragraph in SPLAG, page 378 in the first edition. What this accomplishes is the ability to relate the covering radius of $L,$ in the beginning assumed to be integral "even," and the number of classes in the genus. One of those things.

-