TITLE: Explicit map for Scholz reflection principle
QUESTION [7 upvotes]: The question is about the specific case of reflection theorems (copied straight from Franz Lemmermeyer's "Class Groups of Dihedral Extensions"):

Let $k^+ = \mathbb{Q}(\sqrt{m})$ with $m\in \mathbb{N}$, and put $k^- = \mathbb{Q}(\sqrt{-3m})$; then the 3-ranks $r_3^+$ and $r_3^-$ of $Cl(k^+)$ and $Cl(k^-)$ satisfy the inequalities $r_3^+ \le r_3^- \le r_3^+ + 1$.

The proofs I have seen either use p-adic arguments or galois actions.
Question
Is there an explicit surjective map from $Cl(k^-)[3]$ to $Cl(k^+)[3]$ that might, as the theorem suggests, have kernel of size 3?
At the least, an algorithm for such a map?

REPLY [2 votes]: Leopoldt's reflection theorem, of which Scholz's result (discovered independently by Reichardt) is a special case, bounds the sizes of certain eigenspaces of the class group 
of abelian extensions. I see the main reason for their existence in the fact that abelian 
extensions over fields containing the appropriate roots of unity are Kummer extensions.
Anyway what you are looking for is not so much a map between class groups of different fields as a map between eigenspaces of the class group of one single field, which can be pulled down to subfields in some cases.
If you're interested in an explicit map in Scholz's case, you should have a look at Bhargava's
excellent Higher composition laws. I: A new view on Gauss composition, and quadratic generalizations. There he mentions a map defined in terms of binary quadratic forms studied already by Eisenstein, which I think might have something to do with the question 
you're asking. I always wanted to study this part in detail, but haven't yet found time to do so.  If you come to understand Eisenstein's result before I do let me know -)