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 \lambda -invariant) for the p p -adic zeta-function twisted by a Dirichlet character χ \chi of any order. We are interested in two cases: (i) the character χ \chi is fixed and the prime p p varies, and (ii) o r d ( χ ) ord(\chi ) and the prime p p are both fixed but χ \chi is allowed to vary. We predict distributions for these λ \lambda -invariants using p p -adic random matrix theory and provide numerical evidence for these predictions. We also study the proportion of χ \chi -regular primes, which depends on how p p splits inside Q ( χ ) \mathbb {Q}(\chi ) . Finally, we tabulate the values of the λ \lambda -invariant for every character χ \chi of conductor ⩽ 1000 \leqslant 1000 and for odd primes p p of small size.