Let L/F be a finite Galois extension of number fields with an arbitrary Galois group G. We give an explicit description of the kernel of the natural map on motivic cohomology of the rings of integers <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2024_233_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="229" /> </InlineMediaObject> <EquationSource Format="TEX">$H^2_\mathcal {M}(o_L, {\textbf{Z}}(i))_{G} {\longrightarrow } H^2_\mathcal {M}(o_F, {\textbf{Z}}(i))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mi mathvariant="script">M</mi> <mn>2</mn> </msubsup> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>o</mi> <mi>L</mi> </msub> <mo>,</mo> <mi mathvariant="bold">Z</mi> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>G</mi> </msub> <mo stretchy="false">⟶</mo> <msubsup> <mi>H</mi> <mi mathvariant="script">M</mi> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>o</mi> <mi>F</mi> </msub> <mo>,</mo> <mi mathvariant="bold">Z</mi> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Using the link between motivic cohomology and K-theory, we deduce genus formulae for all even K-groups <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2024_233_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$K_{2i-2}(o_F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mrow> <mn>2</mn> <mi>i</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>o</mi> <mi>F</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the ring of integers. As a by-product, we answer a question raised by B. Kahn about a signature map.

Galois codescent for motivic tame kernels

期刊论文
发表日期：

2025年1月27日

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J. Assim,
A. Movahhedi

摘要

Let L/F be a finite Galois extension of number fields with an arbitrary Galois group G. We give an explicit description of the kernel of the natural map on motivic cohomology of the rings of integers $H^2_\mathcal {M}(o_L, {\textbf{Z}}(i))_{G} {\longrightarrow } H^2_\mathcal {M}(o_F, {\textbf{Z}}(i))$ $H_{M}^{2} {(o_{L}, Z (i))}_{G} ⟶ H_{M}^{2} (o_{F}, Z (i))$ . Using the link between motivic cohomology and K-theory, we deduce genus formulae for all even K-groups $K_{2i-2}(o_F)$ $K_{2 i - 2} (o_{F})$ of the ring of integers. As a by-product, we answer a question raised by B. Kahn about a signature map.