Integral aspects of Fourier duality for abelian varieties
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Bibliographic record
Abstract
Abstract We prove several results about integral versions of Fourier duality for abelian schemes, making use of Pappas’s results on an integral version of Grothendieck–Riemann–Roch. If S is smooth quasi-projective of dimension d over a field and $$\pi :X\rightarrow S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>π</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> is a g -dimensional abelian scheme, we prove, under very mild assumptions on X / S , that all classical results about Fourier duality, including the existence of a Beauville decomposition, are valid for the Chow ring $$\textrm{CH}(X;\Lambda )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>CH</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>;</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with coefficients in the ring $$\Lambda = \mathbb {Z}[1/(2g+d+1)!]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>g</mml:mi> <mml:mo>+</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>!</mml:mo> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> . If X admits a polarization $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> of degree $$\nu (\theta )^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> we further construct an $$\mathfrak {sl}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sl</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -action on $$\textrm{CH}(X;\Lambda _\theta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>CH</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>;</mml:mo> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>θ</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with $$\Lambda _\theta = \Lambda [1/\nu (\theta )]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>θ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>Λ</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>ν</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , and we show that $$\textrm{CH}(X;\Lambda _\theta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>CH</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>;</mml:mo> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>θ</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a sum of copies of the symmetric powers $$\textrm{Sym}^n(\textrm{St})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mtext>Sym</mml:mtext> <mml:mi>n</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mtext>St</mml:mtext> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of the 2-dimensional standard representation, for $$n=0,\ldots ,g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi>
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 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) | 0.000 | 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 it