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I just read Iwasawa's review of Meyer's "Die Berechnung der Klassenzahl abelscher Körper über quadratischen Zahlkörpern" and wonder how the problems Iwasawa mentions at the end of it developed, esp. that on the real case and Hecke's notice?

(Quote: "In case F is imaginary, the class number of K is expressed in terms of certain singular values of the functions which are familiar in the theory of elliptic modular functions, and this naturally suggests that there exists a deep relation between the class number formula and the theory of complex multiplication which is yet unknown to us. On the other hand, if F is real, the class number of K is expressed by means of logarithmic integrals of the kind of functions used in the imaginary case and the analogy with the imaginary case suggests that functions related with these integrals might be used in constructing abelian extensions over F. Finally, as Hecke noticed, it may be also quite interesting if we can find from these class number formulae some special kind of units in K which correspond to the circular units in a cyclotomic field.") https://www.ams.org/journals/bull/1958-64-04/S0002-9904-1958-10216-5/S0002-9904-1958-10216-5.pdf

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    $\begingroup$ Constructions of abelian extension of real quadratic fields is an active area. See for instance this article by Darmon and Vonk. I fear as stated this question is a bit too broad. What exactly are you looking for? $\endgroup$
    – Chris Wuthrich
    Commented Jul 2 at 13:01
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    $\begingroup$ If there are two particular questions in which you are interested, could you reproduce them here to make your post self-contained? $\endgroup$
    – LSpice
    Commented Jul 2 at 14:23

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