Eta Products and Theta Series Identities pp 81-95 | Cite as

# Groups of Coprime Residues in Quadratic Fields

## Abstract

For an explicit specification of a Hecke theta series for a quadratic number field *K* we need an explicit definition of characters on the groups \((\mathcal{O}_{K}/\mathfrak{m})^{\times}\) and \((\mathcal{J}_{K}/(M))^{\times}\) where \(\mathfrak{m}\) is a non-zero ideal in \(\mathcal{O}_{K}\), *M* is an ideal number for \(\mathfrak{m}\), and \(\mathcal{J}_{K}\) is a system of integral ideal numbers for *K*. Since \((\mathcal{O}_{K}/\mathfrak{m})^{\times}\) and \((\mathcal {J}_{K}/(M))^{\times}\) are finite abelian groups, they are isomorphic with direct products of cyclic subgroups. When we know generators of the direct factors then we can define a character by specifying its values on the generators. In almost all of the examples in Part II we will define characters in this way. For this purpose we need to know a decomposition of the groups into direct factors, and we need to know generators of the factors.

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