On the Iwasawa invariants for links and Kida’s formula

Abstract

Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M. Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida’s formula on [Formula: see text]-invariants in a [Formula: see text]-extension of [Formula: see text]-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa’s second proof, with use of [Formula: see text]-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched [Formula: see text]-cover as an inverse system of cyclic branched [Formula: see text]-covers of 3-manifolds, generalize the Iwasawa type formula, and compute the Tate cohomology of 2-cycles explicitly.