Classical modular curve: Difference between revisions

Revision as of 04:33, 24 May 2006

Revision as of 06:34, 24 May 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, given by a rational map with integer coefficients &phi;:X<sub>0</sup>(n) → C, is a [[modular curve]]. The famous [[modularity theorem]] tells us that all [[elliptic curve]]s are modular; theover ''conductor'Q''' ofare the elliptic curve being the minimal n required for this mapping to existmodular.

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 varities (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).