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 André 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
Has Statement Form
In Branch
Source Wolfram MathWorld: Mordell-Weil Theorem
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
Some years later André 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: André Weil, Lead paragraph
In mathematics, the Mordell-Weil theorem states that for an abelian variety
View the Source Wolfram MathWorld: Mordell-Weil Theorem
Wolfram MathWorldIn Branch: Field Theory, Subject classificationsQuote, In Branch: Field Theory, Subject classifications
Algebra Field Theory
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