Construction of Arakelov-modular Lattices from Number Fields

TL;DR

This paper extends the theory of Arakelov-modular lattices to totally real number fields and CM fields, providing new existence criteria and characterizations for these lattices over various types of number fields.

Contribution

It introduces new definitions and criteria for Arakelov-modular lattices over totally real and CM fields, expanding previous results limited to cyclotomic fields.

Findings

Characterization of Arakelov-modular lattices over maximal real subfields of cyclotomic fields with prime power degree.

Existence criteria for Arakelov-modular lattices over totally real Galois fields with odd degrees.

Characterizations of trace type Arakelov-modular lattices for quadratic and certain cyclotomic subfields.

Abstract

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.