Galois coinvariants of the unramified Iwasawa modules of multiple $$\mathbb {Z}_p$$-extensions

On the Cyclicity of the Unramified Iwasawa Modules of the Maximal Multiple $\mathbb{Z}_p$-Extensions Over Imaginary Quadratic Fields

“…We can take generators x 1 and x 2 of X K c ∞ satisfying the following condition (CG) (see Section 4 in [11]):…”

“…In [11], the authors give some conditions that X K is to be Z p [[Gal( K/K)]]-cyclic for imaginary quadratic fields K. In this paper, we give methods for computation and examples about the results. In the rest of this section, we prepare notation and introduce the theorems in [11] (Theorems 1.1, 1.2). In §2, we give a method of computation and examples about Theorem 1.1.…”

“…We define the Iwasawa µ-invariant µ(K ∞ /K) and the Iwasawa λ-invariant λ(K ∞ /K) of a Z p -extension K ∞ by i m i and j n j deg f j for M = X K∞ , respectively. Now we introduce the theorems in [11] which give conditions for X K to be (i)(trivial case) Assume that…”

“…([11, Thoerem 1.1]) Let p be an odd prime number and K an imaginary quadratic field such that p does not split.…”

“…The authors would like to express their sincere gratitude to Professor Masato Kurihara. They started a series of studies progressed in this paper and the previous one [11], drawing their inspiration from his brilliant idea…”