The Mordell-Weil Theorem states that the group of rational points on an elliptic curve defined over a number field is finitely generated, meaning it decomposes into a finite torsion subgroup together with a free abelian group of finite rank. Named for Louis Mordell, who proved the case of the rational numbers, and Andre Weil, who extended it to general number fields, it is a foundational result of arithmetic geometry underlying the modern study of elliptic curves.
Facts
StatementThe group of rational points on an abelian variety defined over a number field is finitely generated; for an elliptic curve over the rational numbers this group is called the Mordell-Weil group. 1 Proof YearLouis Mordell proved the special case for an elliptic curve over the rational numbers in 1922, answering a question Henri Poincare had posed around 1901. Andre Weil generalized the result to abelian varieties over arbitrary number fields in his 1928 doctoral dissertation, the form this entity states. Classification
Statement FormCharacterization Theorem 1 Connections
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Proved By
Source Mordell-Weil Theorem (Wikipedia)
Sources
1. Mordell-Weil Theorem (Wikipedia)
Wikimedia FoundationLead paragraph, mid-sentence
is a finitely-generated abelian group, called the Mordell-Weil group.
History section, second paragraph
Some years later Andre Weil took up the subject, producing the generalisation to Jacobians of higher genus curves over arbitrary number fields in his doctoral dissertation published in 1928.
Proved By: Andre Weil, Lead paragraph
In mathematics, the Mordell-Weil theorem states that for an abelian variety
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