Bibliographic record
Abstract
The primary purpose of this paper is to clarify the relation between previous results in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145], [Amer. J. Math. 141 (2019), pp. 661–703], and [Camb. J. Math. 8 (2020), p. 775–951] via the construction of some interesting locally analytic representations. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a sufficiently large finite extension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q Subscript p"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbf {Q}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript p"> <mml:semantics> <mml:msub> <mml:mi> ρ </mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\rho _p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic semi-stable representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G normal a normal l left-parenthesis bold upper Q Subscript p Baseline overbar slash bold upper Q Subscript p Baseline right-parenthesis right-arrow normal upper G normal upper L Subscript 3 Baseline left-parenthesis upper E right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">a</mml:mi> <mml:mi mathvariant="normal">l</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mover> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> → </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {Gal}(\overline {\mathbf {Q}_p}/\mathbf {Q}_p)\rightarrow \mathrm {GL}_3(E)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the associated Weil–Deligne representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper W normal upper D left-parenthesis rho Subscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">W</mml:mi> <mml:mi mathvariant="normal">D</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi> ρ </mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {WD}(\rho _p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has rank two monodromy and the associated Hodge filtration is non-critical. A computation of extensions of rank one <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis phi comma normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> φ </mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\varphi , \Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -modules shows that the Hodge filtration of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript p"> <mml:semantics> <mml:msub> <mml:mi> ρ </mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\rho _p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depends on three invariants in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We construct a family of locally analytic representations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Superscript normal m normal i normal n Baseline left-parenthesis lamda comma script upper L 1 comma script upper L 2 comma script upper L 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msu
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".