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Record W2037512858 · doi:10.1515/forum.2007.026

Dirichlet series and hyperelliptic curves

2007· article· en· W2037512858 on OpenAlex

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueForum Mathematicum · 2007
Typearticle
Languageen
FieldMathematics
TopicAlgebraic Geometry and Number Theory
Canadian institutionsQueen's University
Fundersnot available
KeywordsMathematicsDirichlet seriesHyperelliptic curveSeries (stratigraphy)ConjecturePure mathematicsZero (linguistics)Degree (music)Quadratic equationMathematical analysisDirichlet distributionCombinatoricsAlgebra over a fieldGeometry

Abstract

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Abstract. For a fixed hyperelliptic curve C given by the equation y2 f ðxÞ with f A ZxŠ having distinct roots and degree at least 5, we study the variation of rational points on the quadratic twists Cm whose equation is given by my2 f ðxÞ. More precisely, we study the Dirichlet series Df ðsÞ P0 m00aCmðQÞjmj s where the summation is over all non-zero squarefree integers. We show that Df ðsÞ converges for <ðsÞ> 1. We extend its range of convergence assuming the ABC conjecture. This leads us to study related Dirichlet series attached to binary forms. We are then led to investigate the variation of rational points on twists of superelliptic curves. We apply this study to certain classical problems of analytic number theory such as the number of powerfree values of a fixed polynomial in ZxŠ. 2000 Mathematics Subject Classification: 11G30; 11M41. 1

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Full frame distilled prediction

Teacher imitation

Not 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.

metaresearch head score (Codex)0.001
metaresearch head score (Gemma)0.001
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesnone
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: Empirical
Teacher disagreement score0.033
Threshold uncertainty score0.724

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0010.001
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0000.000

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.

Opus teacher head0.022
GPT teacher head0.282
Teacher spread0.260 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it