On Leopoldt's and Gross's defects for Artin representations

Maksoud, Alexandre

doi:10.48550/arxiv.2201.08203

Cited by 3 publications

(2 citation statements)

References 13 publications

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“…Theorem 1.3. [Mak22] Let L/K be an extension of number fields and let p be a prime number. Then δ(L, p) ≤ δ(K, p)…”

Section: Introductionmentioning

confidence: 99%

Applications of representation theory and of explicit units to Leopoldt's conjecture

Ferri¹,

Johnston²

2023

Preprint

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Let L/K be a Galois extension of number fields and let G = Gal(L/K). We show that under certain hypotheses on G, for a fixed prime number p, Leopoldt's conjecture at p for certain proper intermediate fields of L/K implies Leopoldt's conjecture at p for L. We also obtain relations between the Leopoldt defects of intermediate extensions of L/K. By applying a result of Buchmann and Sands together with an explicit description of units and a special case of the above results, we show that given any finite set of prime numbers P, there exists an infinite family F of totally real S 3 -extensions of Q such that Leopoldt's conjecture for F at p holds for every F ∈ F and p ∈ P.Theorem 1.1. [Ax65, Bru67] Let K be a finite abelian extension of Q or of an imaginary quadratic field. Then Leo(K, p) holds for every prime number p.

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“…Theorem 1.3. [Mak22] Let L/K be an extension of number fields and let p be a prime number. Then δ(L, p) ≤ δ(K, p)…”

Section: Introductionmentioning

confidence: 99%

Applications of representation theory and of explicit units to Leopoldt's conjecture

Ferri¹,

Johnston²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…While Theorem C (b) proves the validity of condition (F) for all but finitely many Z p -extensions in the described setting, its method of proof does not imply condition (F) for any concrete extensions. To this end, we remark that, after the first version of this article has appeared online, Maksoud has proved the Gross-Kuz'min conjecture for finite abelian extensions of imaginary quadratic fields in [53,Cor. 1.4].…”

Section: Introductionmentioning

confidence: 99%