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The shadow theory of modular and unimodular lattices. (English) Zbl 0917.11026
The first main result of the paper is a bound that had previously been proved only in weaker versions: the minimal nonzero “norm” of an odd self-dual lattice \(L\) in \(n\)-dimensional euclidean space, \(n\not= 23\), can be at most \(2[n/24] + 2\) (for even self-dual lattices this bound has been known for thirty years). The proof uses the “shadow” or characteristic coset of \(2L\) in \(L\) and a subtle analysis of its theta transformation behaviour.
Secondly, the authors extend previous work by the reviewer on lattices of level \(N>1\) whose classes are invariant under a group of involutions corresponding to the Atkin-Lehner involutions on modular forms. Certain known “extremal” lattices of minimal norm 4 for \(N = 2,\ldots,8,11,14,15,23\) are described in a uniform way as sublattices of the Leech lattice, new examples including an odd Coxeter-Todd lattice (\(n=12\)) and an odd Barnes-Wall lattice (\(n=16\)) are provided, and again upper bounds are extended to odd lattices. One question that remains open for general \(n\) and \(N\) is which involution-invariant genera do contain an involution-invariant class, i.e. when does a “strongly modular” lattice exist. For a special type of cases the Appendix of this paper gives a non-existence result, based on an interesting study of reduction modulo 2 for Atkin-Lehner eigenforms.

MSC:
11H06 Lattices and convex bodies (number-theoretic aspects)
Software:
Magma
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