Iwasawa theory and the representations of finite groups
Anwesh Ray
TL;DR
The paper develops a representation-theoretic refinement of Iwasawa theory for finite Cayley graphs by constructing $\mathbb{Z}_\ell$-towers and proving a canonical factorization of the associated Iwasawa polynomial $f_{X,\alpha}(T)$ into factors indexed by irreducible group representations. It defines representation-theoretic Iwasawa polynomials $P_\chi(T)$ attached to each irreducible $\chi$ and introduces invariants $\mu_\chi$ and $\lambda_\chi$ via a Weierstrass factorization, from which global invariants $\mu_\ell$ and $\lambda_\ell$ are aggregations over irreps. The work extends the abelian graph Iwasawa theory to nonabelian Cayley graphs and clarifies how congruences of representations influence the invariants, illustrated by concrete examples (including a GL_2(F_q) case). Overall, the approach yields a representation-theoretic framework for analyzing asymptotic behavior of graph Jacobians in $\mathbb{Z}_\ell$-towers, with potential implications for spectral graph theory and combinatorial number theory.
Abstract
In this note, I develop a representation-theoretic refinement of the Iwasawa theory of finite Cayley graphs. Building on analogies between graph zeta functions and number-theoretic L-functions, I study $\mathbb{Z}_\ell$-towers of Cayley graphs and the asymptotic growth of their Jacobians. My main result establishes that the Iwasawa polynomial associated to such a tower admits a canonical factorization indexed by the irreducible representations of the underlying group. This leads to the definition of representation-theoretic Iwasawa polynomials, whose properties are studied.