On Iwasawa $λ$-invariants for Abelian Number Fields and Random Matrix Heuristics

Following both Ernvall-Mets¨ankyl¨a and Ellenberg-Jain-Venkatesh, we study the density of the number of zeroes (i.e. the cyclotomic λ-invariant) for the p-adic zeta-function twisted by a Dirichlet character χ of any order. We are interested in two cases: (i) the character χ is fixed and the prime p varies, and (ii) ord(χ) and the prime p are both fixed but χ is allowed to vary. We predict distributions for these λ-invariants using p-adic random matrix theory and provide numerical evidence for these predictions. We also study the proportion of χ-regular primes, which depends on how p splits inside Q(χ). Finally in an extensive Appendix, we tabulate the values of the λ-invariant for every character χ of conductor 6 1000 and for odd primes p of small size.