Galois co-descent for étale wild kernels and capitulation
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Bibliographic record
Abstract
Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> be a number field with ring of integers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>o</mml:mi> <mml:mi>F</mml:mi> </mml:msub> </mml:math> . For a fixed prime number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> the étale wild kernels <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mover accent="true"> <mml:mi mathvariant="normal">e</mml:mi> <mml:mo>´</mml:mo> </mml:mover> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> are defined as kernels of certain localization maps on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>i</mml:mi> </mml:math> -fold twist of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -adic étale cohomology groups of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>spec</mml:mi> <mml:mspace width="0.166667em"/> <mml:msub> <mml:mi>o</mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:mrow> <mml:mo>[</mml:mo> </mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>p</mml:mi> </mml:mfrac> <mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . These groups are finite and coincide for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -part of the classical wild kernel <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . They play a role similar to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -part of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -class group of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> . For class groups, Galois co-descent in a cyclic extension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> </mml:math> is described by the ambiguous class formula given by genus theory. In this formula, the only factor which is not well mastered is the norm index <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:msubsup> <mml:mi>U</mml:mi> <mml:mi>F</mml:mi> <mml:mo>′</mml:mo> </mml:msubsup> <mml:mo>:</mml:mo> <mml:msubsup> <mml:mi>U</mml:mi> <mml:mi>F</mml:mi> <mml:mo>′</mml:mo> </mml:msubsup> <mml:mo>∩</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mo>*</mml:mo> </mml:msup> <mml:mo>)</mml:mo> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -units <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>U</mml:mi> <mml:mi>F</mml:mi> <mml:mo>′</mml:mo> </mml:msubsup> </mml:math> . The aim of this paper is the study of the Galois co-descent for wild kernels: Given a cyclic extension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> </mml:math> of degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> with Galois group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , we show that the transfer map <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mover accent="true"> <mml:mi mathvariant="normal">e</mml:mi> <mml:mo>´</mml:mo> </mml:mover> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> <mml:msub> <mml:mo>)</mml:mo> <mml:mi>G</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:mi>W</mml:mi> <mml:msubsup> <mml:mi>K</mml:mi>
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it