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This paper partially addresses the problem of characterizing the lengths ofvectors in a family of Euclidean lattices that arise from any CM number field.We define a modified quadratic form on these lattices, the weighted norm, thatcontains the standard field trace as a special case. Using this modifiedquadratic form, we obtain a bound on the field norm of any vector that has aminimal length in any of these lattices, in terms of a basis for the group ofunits of the ring of integers of the field. For any CM number field F, we provethat there exists a finite set of elements of F which allows one to find theset of minimal vectors in every principal ideal of the ring of integers of F.We interpret our result in terms of the asymptotic behavior of a Hilbertmodular form, and consider some of the computational implications of ourtheorem. Additionally, we show how our result can be applied to the specificCraig's Difference Lattice problem, which asks us to find the minimal vectorsin lattices arising from cyclotomic number fields.

A Bound on the Norm of Shortest Vectors in Lattices Arising from CM Number Fields