SWhat 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.
Scientific scope
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. 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.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. 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.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
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