Réseaux idéaux sur des corps CM et des corps totalement réels

This thesis deals with the study of ideal lattices over number fields. Let K be a number field, which is assumed to be CM or totally real. An ideal lattice over K is a pair (I,b), where I is a fractional ideal of K and b : I × I → R is a symmetric positive definite bilinear form such that b(x, y) = Tr(αxy) for a totally positive α ∈ K ⊗ R. The ideal lattice defined previously will be denoted by (I,α). In the first part, we will focus on constructing modular lattices over number fields. In particular the case of cyclotomic fields will be treated more carefully. A modular lattice is a lattice which is similar to its dual lattice. H.-G. Quebbemann introduced this notion in 1995 and in 1997, and he also defined a notion of analytic extremality for these lattices. It seemed promising to look for ideal lattices which were modular. This investigation led to the introduction of Arakelov modular lattices. This notion has turned out to be very interesting. If K = Q(ζn ) is a cvclotomic field, then the set of levels ℓ for which there exists an Arakelov modular lattice of level ℓ over Q(ζn ) is explicitly given in this thesis. Moreover, assume that K is a CM field, and that we are given an Arakelov modular lattice (I,α) of level ℓ over K. Under an assumption on K (which is satisfied for cyclotomic fields), we describe a way to compute all the Arakelov modular lattices of level ℓ over K. In the second part of this thesis, a construction is introduced which enables us to associate a totally real field K' to a given CM-field K. This field K' has the following property : if (I,α) is an ideal lattice over K. where I is an ambiguous ideal, then there exists an ideal lattice over K' isometric (as a lattice) to (I,α). This leads to the construction of several classic lattices over totally real fields. It also enables us to bound the Euclidean minimum of the field K' with respect to that of the field K.