Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers and $S$ a finite subset of the real places. Let $\mathfrak{m} \subset \mathcal{O}_K$ an ideal. The ideal theoretic ray class group of $\mathfrak{m} $ and $S$ is the quotient group

$$ I^{\mathfrak{m}}/P^{\mathfrak{m}} $$

where $ I^{\mathfrak{m}} $ is the group of fractional ideals co-prime to $ \mathfrak{m} $, and the "ray" $ P^{\mathfrak{m}}$ is the group consisting of all principal ideals in the ring of integers of $K$ having the form $( \alpha)$ where $ \alpha \in K^*$ is multiplicatively congruent to $ 1 \operatorname{ mod } \mathfrak{m}$ and that $\alpha $ are positive at all real places in $S$.

In this question I asked about the origin of the motivation to introduce ray class group in that way, especially the ray ideal $P^{\mathfrak{m}} $ with it's at first glance strange from heaven fallen congruence relation $1 \operatorname{ mod } \mathfrak{m}$.

Viewed retrospectively this approach provided the "right" generalization of the ideal class group: the "ray class group" $ I^{\mathfrak{m}}/P^{\mathfrak{m}} $ provides the analoga of the "classical" ideal class group in the sense that it carries roughly speaking the same amount of information about finite abelian extensions of $K$ where the only primes of $\mathcal{O}_K$ can ramify, which divide $\mathfrak{m}$ like the usual class group about all abelian finite unramified extensions of $K$. But my qeustion deals with the origin only.

Thanks to excellent answers by Franz Lemmermeyer and SomeCallMeTim of the linked question one can say in summary that Weber, who introduced this notation, was presumingly inspired by methods from Gauss's theory of binary quadratic forms and Dedekind's explanation of these groups in terms of ideal classes in orders, and the "presidential case" $\mathbb{Q}(\zeta_m)/\mathbb{Q}$ where the Galois group $({\mathbb Z}/m{\mathbb Z})^\times$ is exactly $ I^{\mathfrak{m}}/P^{\mathfrak{m}} $ in terms of this new terminology and where the the primes $p$ which split completely are exactly those satisfing the relation $p \equiv 1 \operatorname{ mod } \mathfrak{m}$. So $\mathbb{Q}(\zeta_m)/\mathbb{Q}$ seems to be the "perfectly fitted" inspiration for the definition of ray class group.

My question is if $\mathbb{Q}(\zeta_m)/\mathbb{Q}$ is really the only one concrete example paved to Weber the way for the definition of the ray class group $ I^{\mathfrak{m}}/P^{\mathfrak{m}} $ he did or were at that time to Weber also known some concrete examples of CM-fields ( ... say naturally next to the cyclotomic field on the difficulty scale), eg of form $K= \mathbb{Q}(\sqrt{-d}), d >0$ with known Galois groups isomorphic to $ \cong I^m_K/P^m_K $ and well understood splitting behavior of primes based on the congruence relation from above, which also inspired him to define the ray class group in the way he did it?

The reason why I asking is that I'm pretty sure that at that time ($\sim 1896$ ) the CM-fields ware known and provided a widely active research field - keyword Kronecker's Jugendtraum for imaginary quadratic field - so I'm wondering if these might be also influenced Weber in his definition of ray class group besides the cyclotomic field only.

  • $\begingroup$ Vol III of Weber's Algebra deals with elliptic functions, number fields, and the theory of complex multiplication as far as it was known back then. $\endgroup$ Nov 11, 2022 at 19:50


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