Generalised Iwasawa invariants and the growth of class numbers

We study the generalised Iwasawa invariants of {\mathbb{Z}_{p}^{d}}-extensions of a fixed number field K. Based on an inequality between ranks of finitely generated torsion {\mathbb{Z}_{p}[\kern-2.133957pt[T_{1},\dots,T_{d}]\kern-2.133957pt]}-modules and their corresponding elementary modules, we prove that these invariants are locally maximal with respect to a suitable topology on the set of {\mathbb{Z}_{p}^{d}}-extensions of K, i.e., that the generalised Iwasawa invariants of a {\mathbb{Z}_{p}^{d}}-extension {\mathbb{K}} of K bound the invariants of all {\mathbb{Z}_{p}^{d}}-extensions of K in an open neighbourhood of {\mathbb{K}}. Moreover, we prove an asymptotic growth formula for the class numbers of the intermediate fields in certain {\mathbb{Z}_{p}^{2}}-extensions, which improves former results of Cuoco and Monsky. We also briefly discuss the impact of generalised Iwasawa invariants on the global boundedness of Iwasawa λ-invariants.

Bounding the Iwasawa invariants of Selmer groups

We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$ -extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$ -extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$ -extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.