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    "title": "Mordell: finite generation of rational points on genus-one curves",
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        "description": "Finite generation is not finiteness of the set of points and is not a computable complete generating algorithm for every input curve. The paper is over the rationals; Weil's later number-field/abelian-variety generalization is not included. A genus-one curve needs a rational base point to be viewed as an elliptic-curve group. The concluding genus>1 finiteness assertion is explicitly a conjecture here, not a 1922 proof of the theorem later established by Faltings.",
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          "citation": "On the Rational Solutions of the Indeterminate Equations of the Third and Fourth Degrees",
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      "statement": "# Mordell: finite generation of rational points on genus-one curves\n\n## Infinitely many points, finitely many starting pieces\n\nA curve can have infinitely many rational points and still have a finite generating description. Rational means that a coordinate is a ratio of integers. Mordell's result concerns the rational solutions of particular genus-one equations and, with a rational base point, the group of points on an elliptic curve over the rationals. Finite generation says that repeated allowed combinations of finitely many starting points can produce the whole group. It does not say that the group itself is finite.\n\n## Try a point and its tangent\n\nUse the invented classroom curve y squared equals x cubed minus two. The point (3, 5) lies on it because 25=27-2. The tangent slope there is 3x squared divided by 2y, which gives 27/10. The elliptic-curve doubling formula produces x=129/100 and y=-383/1000. These are again rational coordinates. The operation combines a point with itself; the geometric construction uses a tangent intersection followed by reflection. This finite calculation illustrates an operation, not the full theorem.\n\n## What a generator means\n\nFor an ordinary analogy, all even integers arise by adding or subtracting copies of two. One generator gives infinitely many elements. Elliptic-curve addition is a different operation with a geometric definition, but finite generation has the same broad structure: finitely many ingredients suffice. The number of independent infinite-order generators is called the rank. The theorem does not automatically tell you the rank or supply a complete practical algorithm finding generators for every input curve.\n\n## Several famous results must stay separate\n\nThe 1922 paper works over the rationals, not the later general setting of arbitrary number fields and abelian varieties. A genus-one curve without a rational base point cannot simply be treated as an elliptic-curve group in this way. The paper's concluding finiteness assertion for higher-genus curves is a conjecture there. The later theorem associated with Faltings is a different contribution, not an extra result already proved by Mordell.\n\n## Leave a check another person can repeat\n\nAsk your agent to verify the doubled point by exact fraction arithmetic and list the formulas used. State explicitly that this does not prove the point generates the group or establish its rank. A useful contribution might compare the paper's finite-generation statement with a modern textbook formulation, recording which hypotheses were added, translated or generalized. That turns a celebrated name into a precise reusable mathematical claim.\n\n## Scientific scope\n\nFinite generation is not finiteness of the set of points and is not a computable complete generating algorithm for every input curve. The paper is over the rationals; Weil's later number-field/abelian-variety generalization is not included. A genus-one curve needs a rational base point to be viewed as an elliptic-curve group. The concluding genus>1 finiteness assertion is explicitly a conjecture here, not a 1922 proof of the theorem later established by Faltings.\n\n## Sources for this explanation\n\n- [On the Rational Solutions of the Indeterminate Equations of the Third and Fourth Degrees](https://www.ellipticcurve.info/images/e/ef/Proceedingscambr21camb-193-206-pp-179-192.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n",
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      "title": "Mordell: finite generation of rational points on genus-one curves"
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    "title": "Mordell: finite generation of rational points on genus-one curves",
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    "what_changed": "Mordell proved that rational solutions of the nonsingular genus-one cubic/quartic equations treated in his 1922 paper can all be generated from finitely many rational points by the classical chord/tangent and corresponding quartic constructions. For an elliptic curve over the rationals this gives finite generation of its rational-point group; infinite rational points can therefore have a finite generating set.",
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        "title": "Infinitely many points, finitely many starting pieces",
        "text": "A curve can have infinitely many rational points and still have a finite generating description. Rational means that a coordinate is a ratio of integers. Mordell's result concerns the rational solutions of particular genus-one equations and, with a rational base point, the group of points on an elliptic curve over the rationals. Finite generation says that repeated allowed combinations of finitely many starting points can produce the whole group. It does not say that the group itself is finite."
      },
      {
        "title": "Try a point and its tangent",
        "text": "Use the invented classroom curve y squared equals x cubed minus two. The point (3, 5) lies on it because 25=27-2. The tangent slope there is 3x squared divided by 2y, which gives 27/10. The elliptic-curve doubling formula produces x=129/100 and y=-383/1000. These are again rational coordinates. The operation combines a point with itself; the geometric construction uses a tangent intersection followed by reflection. This finite calculation illustrates an operation, not the full theorem."
      },
      {
        "title": "What a generator means",
        "text": "For an ordinary analogy, all even integers arise by adding or subtracting copies of two. One generator gives infinitely many elements. Elliptic-curve addition is a different operation with a geometric definition, but finite generation has the same broad structure: finitely many ingredients suffice. The number of independent infinite-order generators is called the rank. The theorem does not automatically tell you the rank or supply a complete practical algorithm finding generators for every input curve."
      },
      {
        "title": "Several famous results must stay separate",
        "text": "The 1922 paper works over the rationals, not the later general setting of arbitrary number fields and abelian varieties. A genus-one curve without a rational base point cannot simply be treated as an elliptic-curve group in this way. The paper's concluding finiteness assertion for higher-genus curves is a conjecture there. The later theorem associated with Faltings is a different contribution, not an extra result already proved by Mordell."
      },
      {
        "title": "Leave a check another person can repeat",
        "text": "Ask your agent to verify the doubled point by exact fraction arithmetic and list the formulas used. State explicitly that this does not prove the point generates the group or establish its rank. A useful contribution might compare the paper's finite-generation statement with a modern textbook formulation, recording which hypotheses were added, translated or generalized. That turns a celebrated name into a precise reusable mathematical claim."
      }
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