An Analogue of Greenberg's Conjecture for CM Fields

2024/10/08 by Peikai, Qi, Stokes, Matt
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.05706

Abstract

Let K be a CM field and K⁺ be the maximal totally real subfield of K. Assume that the primes above p in K⁺ split in K. Let S be a set containing exactly half of the prime ideals in K above p. We show, assuming Leopoldt's conjecture is true for K and p, that there is a unique ℤ_p-extension of K unramified outside of S (the S-ramified ℤ_p-extension of K). Such ℤ_p-extensions for CM fields have similar properties to the cyclotomic ℤ_p-extensions of a totally real field. For example, Greenberg proved some criterion for the Iwasawa invariants μ=λ=0 of the cyclotomic ℤ_p-extension of a totally real field, and we will prove analogous results for the S-ramified ℤ_p-extension of a CM field. We also give a numerical criterion for the Iwasawa invariants μ=λ=0 for an imaginary biquadratic field, which is analogous to the one given by Fukuda and Komatsu for real quadratic fields.

Related

• Weak Greenberg's generalized conjecture for imaginary quadratic fields
• The Leopoldt and Iwasawa Conjectures for CM fields
• On the triviality of the unramified Iwasawa modules of the maximal multiple ℤ_p-extensions
• On Greenberg's generalized conjecture for families of number fields
• A weak form of Greenberg's conjecture for the cyclotomic ℤ₂-extension of real quadratic fields