Topological Iwasawa invariants and arithmetic statistics
Cédric Dion, Anwesh Ray · Documenta Mathematica · 2022
Given a prime number p , we study topological analogues of Iwasawa invariants associated to \mathbb{Z}_p -covers of the 3 -sphere that are branched along a link. We prove explicit criteria to detect these Iwasawa invariants, and apply them to the study of links consisting of 2 component knots. Fixing the prime p , we prove statistical results for the average behaviour of p -primary Iwasawa invariants for 2 -bridge links that are in Schubert normal form. Our main result, which is entirely unconditional, shows that the density of 2 -bridge links for which the \mu -invariant vanishes, and the \lambda -invariant is equal to 1 , is (1-\frac{1}{p}) . We also conjecture that the density of 2 -bridge links for which the \mu -invariant vanishes is 1 , and this is significantly backed by computational evidence. Our results are proven in a topological setting, yet have arithmetic significance, as we set out new directions in arithmetic statistics and arithmetic topology.