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I read about contiguous forms in Achill Sc├╝rmann's thesis on positive quadratic forms. I am wondering about one aspect of the Voronoi algorithm presented in there, that enumerates all arithmetically inequivalent perfect forms. The algorithm stops when all the contiguous forms of some perfect quadratic form $Q$, are arithmetically equivalent to some other already enumerated perfect form. I wonder why this works as a stopping criterion? Is it so that $Q$ together with its contiguous neighbours tile the whole Ryshkov polyhedron with regards to $GL(Z)$, so that all other perfect forms are arithmetically equivalent to $Q$ and its neighbours? Grateful for a clarification of this.

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Do you have the Martinet book yet? – Will Jagy Jan 12 '12 at 19:42
No I don't, so I'm just trying to find things online. Do you know any good paper online describing this? As far as I see it, the contiguous forms of a perfect form (vertex) $Q$ are vertices of the Ryshkov polyhedron that are neighbours to $Q$. Is that correct? – Kap Jan 12 '12 at 19:57
Sorry to hear that. You really need to buy some books. Here is a review with several other useful items in its bibliography,… and here is the page for the Martinet book itself – Will Jagy Jan 12 '12 at 22:06

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