Why this work is in the frame
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Bibliographic record
Abstract
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 an elliptic curve defined over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and with complex multiplication. For a prime <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> of good reduction, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E overbar"> <mml:semantics> <mml:mover> <mml:mi>E</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the reduction of <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> modulo <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p period"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We find the density of the primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p less-than-or-equal-to x"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p \leq x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper E With bar left-parenthesis double-struck upper F Subscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mi>E</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\overline {E}(\mathbb {F}_p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.-P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula and also to provide explicit error terms in the formula.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| 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) | 0.001 | 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