Torsion phenomena for zero-cycles on a product of curves over a number field
Why this work is in the frame
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Bibliographic record
Abstract
Abstract For a smooth projective variety X over an algebraic number field k a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map of X is a torsion group. In this article we consider a product $$X=C_1\times \cdots \times C_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:mrow> </mml:math> of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true for X . For a product $$X=C_1\times C_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> of two curves over $$\mathbb {Q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map $$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊗</mml:mo> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mover> <mml:mo>→</mml:mo> <mml:mi>ε</mml:mi> </mml:mover> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>CH</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is finite, where $$J_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> is the Jacobian variety of $$C_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> . Our constructions include many new examples of non-isogenous pairs of elliptic curves $$E_1, E_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products $$X=C_1\times \cdots \times C_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:mrow> </mml:math> for which the analogous map $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> has finite image.
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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 | 0.008 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 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