1. Introduction. Let p be a prime number and ${\mathbb{Z}}_p$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, $k_{\infty }$ a ${\mathbb{Z}}_p$-extension of k, $k_n$ the nth layer of $k_{\infty }/k$, and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$. Iwasawa proved the following well-known theorem about the order $A_n$ of $A_n$: Theorem A (Iwasawa). Let $k_{\infty }/k$ be a ${\mathbb{Z}}_p$-extension and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$, where $k_n$ is the $n$th layer of $k_{\infty }/k$. Then there exist integers $\lambda =\lambda (k_{\infty }/k)\ge 0$, $\mu =\mu (k_{\infty }/k)\ge 0$, $\nu =\nu (k_{\infty }/k)$, and n₀ ≥ 0 such that $A_n=p^{\lambda n+\mu p^n+\nu }$ for...

The purpose of this work is to carry out the first step in our four-step program towards the main conjecture for ${\text{GL}}_2\times {𝒦}^{\times }$ by the method of Eisenstein congruence on $GU(3,1)$, where $𝒦$ is an imaginary quadratic field. We construct a $p$-adic family of ordinary Eisenstein series on the group of unitary similitudes $GU(3,1)$ with the optimal constant term which is basically the product of the Kubota-Leopodlt $p$-adic $L$-function and a $p$-adic $L$-function for ${\text{GL}}_2\times {𝒦}^{\times }$. This construction also provides a different point of view of $p$-adic...