Nonvanishing of a certain Bernoulli number and a related topic

• Autorzy: Humio Ichimura
• Języki publikacji: EN

Abstrakty

EN

Let $p = 1+2^{e+1}q$ be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order $2^{e+1}$ (resp. order $d_{φ}$ dividing q), and let ψₙ be an even character of conductor $p^{n+1}$ and order pⁿ. We put χ = δφψₙ, whose value is contained in $Kₙ = ℚ(ζ_{(p-1)pⁿ})$. It is well known that the Bernoulli number $B_{1,χ}$ is not zero, which is shown in an analytic way. In the extreme cases $d_{φ} = 1$ and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: $Tr_{n/1}(ξB_{1,χ}) ≠ 0$ for any pⁿth root ξ of unity, where $Tr_{n/1}$ is the trace map from Kₙ to K₁.

Kategorie tematyczne

• 11R18: Cyclotomic extensions
• 11R23: Iwasawa theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 159
• Numer: 4
• Strony: 375-386

Daty

wydano

2013

Twórcy

autor

Humio Ichimura

• Faculty of Science, Ibaraki University, Bunkyo 2-1-1, Mito, 310-8512, Japan

Identyfikatory

DOI

10.4064/aa159-4-6