Torsion subgroups of elliptic curves in short Weierstrass form
Bibliographic record
Abstract
In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E slash double-struck upper Q"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">E/\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in short Weierstrass form, subject to certain inequalities for their coefficients. We provide a series of counterexamples to this claim and explore a number of related results. In particular, we show that, for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> ε </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\varepsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , all but finitely many curves <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript upper A comma upper B Baseline colon y squared equals x cubed plus upper A x plus upper B comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> <mml:mspace width="thickmathspace"/> <mml:mo>:</mml:mo> <mml:mspace width="thickmathspace"/> <mml:msup> <mml:mi>y</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E_{A,B} \; : \; y^2 = x^3 + A x + B,</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are integers satisfying <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A greater-than StartAbsoluteValue upper B EndAbsoluteValue Superscript 1 plus epsilon Baseline greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>B</mml:mi> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi> ε </mml:mi> </mml:mrow> </mml:msup> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">A>|B|^{1+\varepsilon }>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , have rational torsion subgroups of order either one or three. If we modify our demands upon the coefficients to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper A EndAbsoluteValue greater-than StartAbsoluteValue upper B EndAbsoluteValue Superscript 2 plus epsilon Baseline greater-than 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>B</mml:mi> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi> ε </mml:mi> </mml:mrow> </mml:msup> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">|A|>|B|^{2+\varepsilon }>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript upper A comma upper B"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">E_{A,B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> now have trivial rational torsion, with at most finitely many exceptions, at least under the assumption of the abc-conjecture of Masser and Oesterlé.
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How this classification was reachedexpand
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) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".