Where in the literature does the anticyclotomic $\mathbf{Z}_p$-extension of an imaginary quadratic field first appear? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:25:54Z http://mathoverflow.net/feeds/question/111324 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111324/where-in-the-literature-does-the-anticyclotomic-mathbfz-p-extension-of-an-im Where in the literature does the anticyclotomic $\mathbf{Z}_p$-extension of an imaginary quadratic field first appear? Keenan Kidwell 2012-11-03T00:28:13Z 2012-11-03T03:51:26Z <p>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 <code>$\mathrm{Gal}(K_\infty^{anti}/K)$</code> by inversion.</p> <p>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 <code>$K_\infty^{anti}$</code> 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?</p> <p>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 <code>$K_\infty^{anti}$</code> 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).</p> http://mathoverflow.net/questions/111324/where-in-the-literature-does-the-anticyclotomic-mathbfz-p-extension-of-an-im/111340#111340 Answer by Yuri Zarhin for Where in the literature does the anticyclotomic $\mathbf{Z}_p$-extension of an imaginary quadratic field first appear? Yuri Zarhin 2012-11-03T03:46:14Z 2012-11-03T03:51:26Z <p>First time I heard about antycyclotomic $\Gamma$-extensions in 1972 from Pavel Kurchanov, in connection with his paper </p> <p><a href="http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;jrnid=sm&amp;paperid=3020&amp;option_lang=eng" rel="nofollow">http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;jrnid=sm&amp;paperid=3020&amp;option_lang=eng</a></p> <p>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.)</p>