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

159 pages, 2 figures

References in corpus (1)

• Dirichlet Series Expansions of p-adic L-Functions

Cited by in corpus (2)

• Special values of $p$-adic $L$-functions and Iwasawa $\lambda$-invariants of Dirichlet characters
• On large Iwasawa $\lambda$-invariants of imaginary quadratic function fields