On automorphisms of some semidirect product groups and ranks of Iwasawa modules

2025/04/19 by Fujii, Satoshi
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.14284

Abstract

Let p be an odd prime number and k an imaginary quadratic field in which p does not split. Based on their heuristic, Kundu and Washington posed a question which asks whether λ- and μ-invariant of the anti-cyclotomic \Bbb Z_p-extension k_∞^a of k are always trivial. Also, if k_∞^a/k is totally ramified, for n≥ 1, they showed that the p-part of the ideal class group of the nth layer of the anti-cyclotomic \Bbb Z_p-extension of k is not cyclic. In this article, inspired by their paper, we study anti-cyclotomic like \Bbb Z_p-extensions, extending both the above question and Kundu-Washington's result. We show that the values of λ of certain anti-cyclotomic like \Bbb Z_p-extensions are always even. We also show the p-part of the ideal class groups of certain anti-cyclotomic like \Bbb Z_p-extensions of CM-fields are always not cyclic.

Related

• The first level of ℤ_p-extensions and compatibility of heuristics
• Galois coinvariants of the unramified Iwasawa modules of multiple ℤ_p-extensions
• Weak Greenberg's generalized conjecture for imaginary quadratic fields
• Growth of p-parts of ideal class groups and fine Selmer groups in ℤ_q-extensions with p≠ q
• On a bound of p-ranks of Iwasawa modules of ℤ_p-extensions over a quartic CM-field