Classical modular curve: Difference between revisions

Revision as of 21:25, 28 August 2006

Revision as of 21:28, 28 August 2006

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== Mappings ==

A curve ''C'' over the rationals '''Q''' such that there exists a surjective morphism from ''X''<sub>0</sup>(''n'') to ''C'' for some ''n'', given by a rational map with integer coefficients

:φ:''X''<sub>0</sup>(''n'') →&rarr; ''C]],

is a [[modular curve]]. The famous [[modularity theorem]] tells us that all [[elliptic curve]]s over '''Q''' are modular.

Mappings also arise in connection with X<sub>0</sup>(n) since points on it correspond to n-isogenous pairs of elliptic curves. Two elliptic curves are ''isogenous'' if there is a morphism of varieties (defined by a rational map) between the curves which is also a group homomorphism, respecting the group law on the elliptic curves, and hence which sends the point at infinity (serving as the identity of the group law) to the point at infinity. The isogenies with cyclic kernel of degree n, the cyclic isogenies, correspond to points on X<sub>0</sup>(n).