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

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

13 Jul 2022arXiv:2207.06287links table onlyarchive 2025-07-28

Daniel Delbourgo, Heiko Knospe

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Following both Ernvall-Mets\"{a}nkyl\"{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 ≤1000 and for odd primes p of small size.

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