5
$\begingroup$

In the book Elliptic Functions, Lang uses a definition for the ray class group of a number field that I'm not completely comfortable with. I'm trying to understand why his definition is justified.

For context, I normally think of the ray class group of a number field $k$ in the following way. Let $\mathfrak f$ be a fractional ideal of $k$. Let $F(\mathfrak f)$ be the group of fractional ideals prime to $\mathfrak f$ and let $S(\mathfrak f)$ be the subgroup of principal ideals generated by elements in the ray modulo $\mathfrak f$. Then define the ray class group modulo $\mathfrak f$ as the quotient group $F(\mathfrak f)/S(\mathfrak f)$.

Here is Lang's definition when $k$ is quadratic imaginary. This occurs in Chapter 22 at the beginning of Section 2.

Again let $k$ be an imaginary quadratic field with $\mathfrak o_k = \mathfrak o$ and let $\mathfrak f$ be an [integral] ideal of $\mathfrak o, \mathfrak f \neq \mathfrak o$. Let $G_\mathfrak f = I(\mathfrak f)/P_1(\mathfrak f)$ be the ray class group where $I(\mathfrak f)$ denotes the monoid of [integral] ideals prime to $\mathfrak f$, and $P_1(\mathfrak f)$ denotes the subset consisting of those principal ideals $(\alpha)$ such that $\alpha \equiv 1 \mod \mathfrak f$.

The key difference between this and how I think about the ray class group, is that Lang is working with integral ideals, and the set of integral ideals prime to $\mathfrak f$ forms a monoid, rather than a group.

From the notation, I would guess that the set $I(\mathfrak f)/P(\mathfrak f)$ is the collection of cosets of the form $\mathfrak aP_1(\mathfrak f)$ for $\mathfrak a \in I(\mathfrak f)$. But since $I(\mathfrak f)$ is not a group and $P_1(\mathfrak f)$ is not a subgroup, it is not even clear to me that the cosets $\mathfrak aP_1(\mathfrak f)$ actually form a partition of $I(\mathfrak f)$ in the first place. This, I think, is the heart of my issue with the definition.

I think that I have shown that one can define an equivalence relation on $I(k)$ by saying $\mathfrak a \sim \mathfrak b$ if and only if there exist $(\alpha), (\beta) \in P_1(\mathfrak f)$ such that $\alpha\mathfrak a = \beta \mathfrak b$, and in this case the set of equivalence classes do form a group which is isomorphic to $F(\mathfrak f)/S(\mathfrak f)$, the ray class group as I normally think of it. This seems very related, but I'm not convinced on why it is the same as the definition given by Lang in terms of cosets.

My Question: How is the set $I(\mathfrak f)/P_1(\mathfrak f)$ defined? Is it a collection of cosets, or something else? If it is just a collection of cosets, how exactly are the cosets defined, and how do we show that these cosets partition $I(\mathfrak f)$?


Additional comments which may be relevant.

From what I understand, if you have a monoid $M$ and a congruence $\sim$ on $M$ then the equivalence classes $M/\sim$ form a monoid. If $M$ is in fact a group then the set of congruences on $M$ are in one to one correspondence with normal subgroups of $M$ and so it makes sense to use the standard notation $M/N$, where $N$ is some normal subgroup, for the equivalence classes. In general though if $M$ is not a group you do not have such a correspondence, and it doesn't make sense to define quotients in terms of submonoids like we do with subgroups. You must stick with using equivalence relations.

However, here Lang seems to be using notation analogous to the notation that we use with groups, which really makes me think there is a natural equivalence relation that can be associated to the submonoid $P_1(\mathfrak f)$. If this is true, it seems like the equivalence classes of this relation should just be the cosets $\mathfrak a P_1(\mathfrak f)$.

$\endgroup$
1
$\begingroup$

The equivalence relation is the same as in the group of fractional ideals generated by $I(\mathfrak f),P_1(\mathfrak f)$, it is $$I \sim J \qquad \iff \exists (a),(b)\in P_1(\mathfrak f),\quad (a)I=(b)J$$

Given $I\in I(\mathfrak f)$, then $I \, P_1(\mathfrak f)$ is a subset of $I(\mathfrak f)$, the relation $I \sim J $ if $J\in I P_1(\mathfrak f)$ is not an equivalence relation, what you need is to extend it transitively: if $I\sim J$ and $J\sim K$ then $I\sim K$, doing so many times gives the equivalence relation and the cosets $$I(\mathfrak f)/P_1(\mathfrak f)=\bigcup_{n=1}^N \{ J\in I(\mathfrak f), J\sim I_n\}$$ it does have a natural monoid law, and since it is a finite monoid with cancellation, $I(\mathfrak f)/P_1(\mathfrak f)$ is a finite group, the same as the quotient of the groups of fractional ideals generated by $I(\mathfrak f),P_1(\mathfrak f)$.

If $\mathfrak f$ is a modulus instead of an ideal then the quotient isn't finite and it is only a monoid, the group it generates is the Galois group of an infinite abelian extension.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.