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I am studying algebraic number theory but one of the major challenges I face is understanding what the maps between ideals, orders, class groups does? The research paper I am referring doesn't explicitly state anything and I guess that's why I am here. Please help me understand what the map does.enter image description here

In the above image, $I(\mathcal{O}_\Delta,f)$ is the set of invertible fractional $\mathcal{O}_\Delta$ - ideals prime to $f$ where $\mathcal{O}_\Delta$ is the order of an imaginary quadratic field $K$.

I am curious as to what does the $\varphi_f$ map does? It doesn't really tell anything as to what the map operation is apart from the fact that the map is an isomorphism and it sends one $\mathcal{O}$ ideal prime to $f$ to its corresponding $\mathcal{O}_K$ ideal prime to $f$. What does the map do? Also, how does one understand the map operations like the one given? Is there any trick or procedure to be able to actually define the map operation? For example, I would like to figure out what the inverse map $\mathcal{\varphi^{-1}}$ does so again, how does one go about it?

Thank you.

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  • $\begingroup$ If $A \subset B \subset C$ are abelian groups, then there is a natural projection map $C/A \to C/B$ sending $cA$ to $cB$ (example: residue classes mod $4$ define residue classes mod $2$). And maps that are not injective do not have an inverse. $\endgroup$
    – user23365
    Commented Jun 7, 2017 at 8:04
  • $\begingroup$ It's true that non-injective maps do not have inverses, but the map in question is an isomorphism, so that shouldn't be a problem. $\endgroup$ Commented Jun 7, 2017 at 12:48
  • $\begingroup$ @Tony: you're right - I was thinking about the ideal classes in the last part of the quotation. $\endgroup$
    – user23365
    Commented Jun 7, 2017 at 13:11

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In your question, you state, "it sends one $\mathcal{O}$ ideal prime to $f$ to its corresponding $\mathcal{O}_K$ ideal prime to $f$". I'm not sure what you mean when you ask, "what does the $\varphi_f$ map do?", as your statement seems to be the answer to your question. The map is a correspondence between $\mathcal{O}$ ideals and $\mathcal{O}_K$ ideals, and the fact that it is one-to-one and onto verifies that the two sets are isomorphic.

Additionally, it "induces a surjection $\overline{\varphi}_f$" between the class groups of the respective rings. In order for this to be true, the map $\varphi_f$ must be well-defined with respect to class groups, i.e., if two ideals are equivalent in the domain, then their images are equivalent in the codomain. Once this is verified, we have a map between class groups, and the author seems interested in the order of its kernel, i.e., how many classes of ideals in $\mathcal{O}_{\Delta_p}$ map to the class of principal ideals in $\mathcal{O}_{\Delta_K}$?

To see the inverse map $\varphi^{-1}_f$, you would need to take an arbitrary fractional ideal of $O_{\Delta_K}$ prime to $f$, and figure out for which $\mathfrak{a}\in I(O_{\Delta_f},f)$ your ideal is equal to $\mathfrak{a}O_{\Delta_K}$. I.e., if $\mathfrak{b}\in I(O_{\Delta_K},f)$, and $\mathfrak{b}=\mathfrak{a}O_{\Delta_K}$ for some $\mathfrak{a}\in I(O_{\Delta_f},f)$, then $\varphi^{-1}_f(\mathfrak{b})=\mathfrak{a}$ Does that make any sense?

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