Index-$p$ abelianization data of $p$-class tower groups

Given a fixed prime number $p$, the multiplet of abelian type invariants of the $p$-class groups of all unramified cyclic degree $p$ extensions of a number field $K$ is called its IPAD (index-$p$ abelianization data). These invariants have proved to be a valuable information for determining the Galois group $G_p^2$ of the second Hilbert $p$-class field and the $p$-capitulation type $\varkappa$ of $K$. For $p=3$ and a number field $K$ with elementary $p$-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated IPADs of second order are used to identify the group $G_p^\infty$ of the maximal unramified pro-$p$ extension of $K$.

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