Well-rounded algebraic lattices in odd prime dimension

Bases of minimal vectors in tame lattices

“…Recently the authors of [9,10] have used a version of the result above to construct bases of well-rounded lattices defined by certain submodules of the ring of integers of a prime degree Galois number field. Suppose that p is an odd prime and K is a degree p tame Galois number field.…”

“…Suppose that p is an odd prime and K is a degree p tame Galois number field. In [9,10] a set of conditions on positive integers m ≡ 1 (mod p) are given so that the sublattice {x ∈ O K : Tr K/Q (x) ≡ 0 (mod m)} of O K has a minimal basis.…”

“…In fact, the results in [4] show that when K is abelian, not necessarily of prime degree but of prime conductor, a Lagrangian basis exists. Since there are number fields with Lagrangian basis and with neither prime conductor nor prime degree (see for instance Example 4.14), the construction of [9,10] can be generalized to an even bigger class of number fields.…”

Dense generic well-rounded lattices

“…A prominent feature of well-rounded lattices is that the set of shortest vectors span the ambient space. Previously, constructions of well-rounded lattices have been considered in, e.g., [12,1,15], and they also relate to the famous Minkowski and Woods conjectures [22].…”

“…In [8, IV.8] Conner and Perlis showed that if K is tame, i.e., no rational prime has wild ramification, then there exists an integral basis for O K , which they called Lagrangian basis, such that the Gram matrix of the integral trace in such basis is a p × p matrix of the form where d is the conductor of K, i.e., the smallest positive integer such that K is inside of the cyclotomic extension Q(ζ d ). Since K is tame, d is also equal to the product of the primes that ramify in K. Based in this explicit description of the trace form over O K , in [15] the authors have constructed families of well-rounded lattices that are sublattices of O K . In [14], the notion of tame lattice is introduced by axiomatizing the key properties of the integral trace over degree p Galois extensions.…”