Iwasawa theory: climbing up towers, from numbers to graphs

Iwasawa theory studies the evolutions of certain invariants in a tower of covers of objects. Curves over finite fields and number fields have historically been the first kinds of objects whose invariants have been studied by Iwasawa-theoretic methods. More surprisingly, one can study in an analogous manner towers of hyperbolic manifolds and towers of graphs. In this talk, we will survey these different developments of Iwasawa theory, ending with a survey of a recent joint work between myself and Daniel Vallières. In this work, we provide a “global” analogue of the Iwasawa theory for graphs, by looking at towers parametrized by the integers, instead of looking at towers parametrized by powers of a given prime, which are those considered in the classical case. This allows us to recover combinatorial results of the Mednykhs, and also to generalize a formula for the p-adic valuation of Fibonacci numbers, due to Lengyel.