On Iwasawa $λ$-invariants for abelian number fields and random matrix heuristics

Abstract

Following both Ernvall-Metsänkylä 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) $\text{ord}(χ)$ 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 $\mathbb{Q}(χ)$. Finally in an extensive Appendix, we tabulate the values of the $λ$-invariant for every character $χ$ of conductor $\leq 1000$ and for odd primes $p$ of small size.