From the Birch and Swinnerton-Dyer conjecture to Nagao’s conjecture
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.
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 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> with discriminant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript upper E"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>E</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Delta _E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For primes <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 N Subscript p"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">N_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the number of points modulo <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> and write <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Subscript p Baseline equals p plus 1 minus a Subscript p"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">N_p=p+1-a_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript x right-arrow normal infinity Endscripts StartFraction 1 Over log x EndFraction sigma-summation Underscript StartLayout 1st Row p less-than-or-equal-to x 2nd Row p does-not-divide normal upper Delta Subscript upper E Baseline EndLayout Endscripts StartFraction a Subscript p Baseline log p Over p EndFraction equals negative r plus one half comma"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:munder> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:mfrac> <mml:munder> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mstyle scriptlevel="1"> <mml:mtable rowspacing="0.1em" columnspacing="0em" displaystyle="false"> <mml:mtr> <mml:mtd> <mml:mi>p</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>x</mml:mi> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>p</mml:mi> <mml:mo> ∤ </mml:mo> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>E</mml:mi> </mml:mrow> </mml:msub> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mstyle> </mml:mrow> </mml:munder> <mml:mfrac> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mo> − </mml:mo> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \lim _{x\to \infty }\frac {1}{\log x}\sum _{\substack {p\leq x\\ p\nmid \Delta _{E
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.000 | 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