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.

Thank you for your help!


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