# properties of anti-cyclotomic extension

Let $$K$$ be an imaginary quadratic field, and $$p$$ be a primes number, there exists an unique $$\mathbb{Z}_p$$-extension of $$K$$, we denote it by $$K_{p,\infty}$$ such that the action of complex conjugate $$c$$ acts on $$Gal(K_{p,\infty}/K)$$ is given by $$c \cdot \tau = c \tau c^{-1} = \tau^{-1}$$ this $$K_{p,\infty}$$ is called the anti-cyclotomic extension of $$K$$. Notice that it's not the cyclotomic extension of $$K$$, which just put all $$p^{\infty}$$ root of unit into $$K$$, since conjugate on elements of Galois group cyclotomic extension doesn't change it.

Anti-cyclotomic extension maybe important for the study of arithmetic of elliptic curves and Iwasawa theory....

I want to know are there any materials explain this extension? Especially how can we construct such an extension, why it is important, and it has what kind of peoperties.