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Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell , [ 1 ] and known as the Mordell conjecture until its 1983 proof by Gerd Faltings . [ 2 ]
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Upload file; Special pages ... Cite this page; Get shortened URL; Download QR code; Print/export Download as PDF; Printable version ... Rational curves are subdivided ...
The case with an elliptic curve and the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties .
In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems.
The modularity of an elliptic curve E of conductor N can be expressed also by saying that there is a non-constant rational map defined over ℚ, from the modular curve X 0 (N) to E. In particular, the points of E can be parametrized by modular functions. For example, a modular parametrization of the curve y 2 − y = x 3 − x is given by [18]
The conductor of an elliptic curve over the rationals was introduced and named by Weil (1967) as a constant appearing in the functional equation of its L-series, analogous to the way the conductor of a global field appears in the functional equation of its zeta function. He showed that it could be written as a product over primes with exponents ...
Mazur's conjecture B is a weaker variant of the uniform boundedness conjecture that asserts that there should be a number (,,) such that for any algebraic curve defined over having genus and whose Jacobian variety has Mordell–Weil rank over equal to , the number of -rational points of is at most (,,).