The (S,{2})-Iwasawa theory

Iwasawa made the fundamental discovery that there is a close connection between the ideal class groups of $\mathbb{Z}_{p}$-extensions of cyclotomic fields and the $p$-adic analogue of Riemann's zeta functions $ζ(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.$ In this paper, we show that there may also exist a parallel Iwasawa's theory corresponding to the $p$-adic analogue of Euler's deformation of zeta functions $ϕ(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}.$

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