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Construction of Arakelov-modular Lattices from Number Fields

An Arakelov-modular lattice of level $\ell$, where $\ell$ is a positive integer, is an $\ell-$modular lattice constructed from a fractional ideal of a CM field such that the lattice can be obtained from its dual by multiplication of an element with norm $\ell$. The characterization of existence of Arakelov-modular lattices has been completed for cyclotomic fields [4]. In this paper, we extend the definition to totally real number fields and study the criteria for the existence of Arakelov-modular lattices over totally real number fields and CM fields. We give the characterization of Arakelov-modular lattices over the maximal real subfield of a cyclotomic field with prime power degree and totally real Galois fields with odd degrees. Characterizations of Arakelov-modular lattices of trace type, which are special cases of Arakelov-modular lattices, are given for quadratic fields and maximal real subfields of cyclotomic fields with non-prime power degrees.