An irrational dual-extreme form. (Une forme dual-extrême irrationnelle.) (French) Zbl 0997.11050

To approach the sphere packing problem from the point of view of duality, the author and J. Martinet have introduced an invariant \(\gamma'\) which, to a pair \((L,L^*)\) of dual lattices, attaches the geometric mean of the Hermite invariants \(\gamma(L)\) and \(\gamma(L^*)\). Its upper limit in dimension \(n\), the Bergé-Martinet constant, has been determined only for \(n=1,2,3,4,8\), where it is attained by the lattices extreme for Hermite’s constant. The conjecture that the same holds for \(n=5\) is supported here by the proof of such a result for 5-dimensional lattices having an automorphism of order 5. Moreover, in this equivariant situation the author finds all pairs of dual lattices on which \(\gamma'\) attains a local maximum; remarkably one of them is irrational (a phenomenon impossible in the classical theory).


11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11H56 Automorphism groups of lattices
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