Nonvanishing of a certain Bernoulli number and a related topic

Humio Ichimura. "Nonvanishing of a certain Bernoulli number and a related topic." Acta Arithmetica 159.4 (2013): 375-386. <http://eudml.org/doc/279585>.

@article{HumioIchimura2013,
abstract = {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.},
author = {Humio Ichimura},
journal = {Acta Arithmetica},
keywords = {Bernoulli number; class number; -extension},
language = {eng},
number = {4},
pages = {375-386},
title = {Nonvanishing of a certain Bernoulli number and a related topic},
url = {http://eudml.org/doc/279585},
volume = {159},
year = {2013},
}

TY - JOUR
AU - Humio Ichimura
TI - Nonvanishing of a certain Bernoulli number and a related topic
JO - Acta Arithmetica
PY - 2013
VL - 159
IS - 4
SP - 375
EP - 386
AB - 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.
LA - eng
KW - Bernoulli number; class number; -extension
UR - http://eudml.org/doc/279585
ER -