Let F be a totally real number field and let p be an odd prime. For an odd integer <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1228_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$i\le -1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>≤</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we give an analytical upper bound for the twisted Kummer-Leopoldt constant <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1228_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\kappa _i(F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> introduced in Assim et al. (N. Y. J. Math. 30:369–397, 2024). As a by-product of the proof, we provide a characterization of the vanishing of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1228_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\kappa _i(F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of the value of the zeta function at the integer i. We also investigate the triviality of the twisted Kummer-Leopoldt constant in the cyclotomic tower. In particular, we characterize this triviality as an equality of lambda-invariants of some Iwasawa modules.

On the vanishing of the twisted Kummer-Leopoldt constant of totally real number fields

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发表日期：

2025年11月5日

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J. Assim,
Z. Boughadi,
A. Driwach

摘要

Let F be a totally real number field and let p be an odd prime. For an odd integer $i\le -1$ $i \leq - 1$ , we give an analytical upper bound for the twisted Kummer-Leopoldt constant $\kappa _i(F)$ $κ_{i} (F)$ introduced in Assim et al. (N. Y. J. Math. 30:369–397, 2024). As a by-product of the proof, we provide a characterization of the vanishing of $\kappa _i(F)$ $κ_{i} (F)$ in terms of the value of the zeta function at the integer i. We also investigate the triviality of the twisted Kummer-Leopoldt constant in the cyclotomic tower. In particular, we characterize this triviality as an equality of lambda-invariants of some Iwasawa modules.