Search results
Results From The WOW.Com Content Network
The proof of Lüroth's theorem can be derived easily from the theory of rational curves, using the geometric genus. [2] This method is non-elementary, but several short proofs using only the basics of field theory have long been known, mainly using the concept of transcendence degree. [3]
This definition differs from that of path connectedness only by the nature of the path, but is very different, as the only algebraic curves which are rationally connected are the rational ones. Every rational variety, including the projective spaces , is rationally connected, but the converse is false.
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves E p obtained from an elliptic curve E over the rational numbers by reduction modulo almost all prime numbers p. Mikio Sato and John Tate independently posed the conjecture around 1960.
The naive height or classical height of a vector of rational numbers is the maximum absolute value of the vector of coprime integers obtained by multiplying through by a lowest common denominator. This may be used to define height on a point in projective space over Q , or of a polynomial, regarded as a vector of coefficients, or of an ...
The tangent-chord process (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century.The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group / which forms a major step in the proof.
The 1-dimensional factors are elliptic curves (there can also be higher-dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve (but not, in general, isomorphic to it).
A birational map from X to Y is a rational map f : X ⇢ Y such that there is a rational map Y ⇢ X inverse to f.A birational map induces an isomorphism from a nonempty open subset of X to a nonempty open subset of Y, and vice versa: an isomorphism between nonempty open subsets of X, Y by definition gives a birational map f : X ⇢ Y.
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 ]