The structure of higher-dimensional Iwasawa modules under a far-fetched assumption
Let \(K\) be a number field; let \(p\) be an odd prime. For \(\mathbb{Z}_p\)-extensions \(M_\infty/K\) of \(K\) \((M_\infty= \bigcup_n M_n)\) the growth of the \(p\)-class groups \(A(M_n)= \text{Cl}(M_n)\{p\}\) is well understood by a classical result of Iwasawa; in particular the growth of \(\log_p(A(M_n))\) is up to an additive constant given by \(\mu p^n+\lambda n\). Such a result can be proved by studying the so-called Iwasawa module \(X=\varprojlim A(M_n)\) over the Iwasawa algebra \(\mathbb{Z}_pT\) and its relation to the modules \(A(M_n)\) at the finite levels of the tower. In the case of multiple \(\mathbb{Z}_p\)-extensions, the situation is much more complicated. The authors look at \(\mathbb{Z}_p^2\)-extensions \(L_\infty/K\), and prove the following results (Theorems 1.1 and 1.2): ``From the assumption that Leopoldt's conjecture fails and some mild extra assumptions, we deduce the existence of multiple \(\mathbb{Z}_p\)-extensions whose Iwasawa modules are `large' in a precise sense''.
scientific article; zbMATH DE number 7135250