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If $K$ is an imaginary quadratic field, then the $\mathbf{Z}_p$-rank of $K$ is $2$, meaning that the Galois group of the compositum of all the $\mathbf{Z}_p$-extensions of $K$ in an algebraic closure $\overline{K}$ is isomorphic to $\mathbf{Z}_p^2$. It is known that the compositum is generated by two special $\mathbf{Z}_p$-extensions: the cyclotomic and the anticyclotomic $\mathbf{Z}_p$-extensions. Both are Galois over $\mathbf{Q}$ (I believe they are the only $\mathbf{Z}_p$-extensions of $K$ which are Galois over $\mathbf{Q}$). The cyclotomic extension is abelian over $\mathbf{Q}$ and is equal to the compositum of $K$ and the cyclotomic (the only!) $\mathbf{Z}_p$-extension of $\mathbf{Q}$. The anticyclotomic extension is "generalized dihedral" over $\mathbf{Q}$, which means that the unique non-trivial element of $\mathrm{Gal}(K/\mathbf{Q})$ acts on $\mathrm{Gal}(K_\infty^{anti}/K)$ by inversion.

I learned these facts from various sources, after being told by my advisor what the anticyclotomic $\mathbf{Z}_p$-extension was (she gave me the generalized dihedral definition). My question is: when did $K_\infty^{anti}$ first appear in the Iwasawa theory literature? Does it appear in the work of Iwasawa (I'm not all that familiar with his work)? Maybe the work of Greenberg or Washington? Was the primary motivation for studying it the connection with CM elliptic curves?

My motivation for asking this is because I'd really like to read a comprehensive exposition of its basic properties (although perhaps this doesn't exist). I've picked up bits and pieces from books and papers. For example, I know that $K_\infty^{anti}$ is the unique $\mathbf{Z}_p$-extension contained in the union of the $p$-power conductor ring class fields of $K$, and the behavior of primes in these class fields can be determined by the ideal-theoretic formulation of global class field theory, but I feel like the basics are not quite as straightforward as for cyclotomic $\mathbf{Z}_p$-extensions (which somehow seem more natural to me).

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I do not really know here it first appears, but I think you should have a look at the 1973 paper by Iwasawa "On the $\mu$ invariants in $\mathbb{Z}_\ell$-extensions". Note that for imaginary quadratic fields, Leopoldt conjecture is almost obvious, as it is for $\mathbb{Q}$, because there are no global units beside roots of units, so it was a standard fact that the compositum of all $\mathbb{Z}_p$-extensions of an imaginary quadratic field K has Galois group $\mathbb{Z}_p^2$, on which $\mathrm{Gal}(K/\mathbb{Q})$ acts, and identifies the anticyclotomic extension as −1-eigenspace... –  Filippo Alberto Edoardo Nov 3 '12 at 5:46
...I would say it is the most obvious example beside cyclotomic one, so I guess Iwasawa himself must have thought about it especially when looking for extensions with non-trivial invariants. –  Filippo Alberto Edoardo Nov 3 '12 at 5:47
I think the idea of $\mathbb Z_{p}$-extension is the kind of idea that have been around at least implicitly for a long time. Certainly Kronecker and Weber knew explicit descriptions of abelian extensions of CM fields, and from that knowledge, introducing the $\mathbb Z_{p}$-extension is just singling out some particularly interesting extensions. I think that Iwasawa is probably the person who first explicitly introduced them. Already in his 1959 article, the general set-up is in place and a general construction of $\mathbb Z_{p}$-extension is given using class field theory. –  Olivier Nov 3 '12 at 8:34
No examples are given, but from his description of the general construction and say the content of Weber Algebra, you get the anticyclotomic extension for free. –  Olivier Nov 3 '12 at 8:37

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up vote 5 down vote accepted

First time I heard about antycyclotomic $\Gamma$-extensions in 1972 from Pavel Kurchanov, in connection with his paper


Actually, his goal was to construct elliptic curves of infinite rank over $\Gamma$-extensions. (According to Mazur's conjecture, one cannot do it over the cyclotomic extensions.)

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