Classical modular curve: Difference between revisions

Revision as of 15:35, 4 July 2016

Revision as of 15:36, 4 July 2016

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A curve {{mvar|C}}, over {{math|'''Q'''}} is called a [[modular curve]] if for some {{mvar|n}} there exists a surjective morphism {{math|''φ'' : ''X''₀(''n'') → ''C''}}, given by a rational map with integer coefficients. The famous [[modularity theorem]] tells us that all [[elliptic curve]]s over {{math|'''Q'''}} are modular.

Mappings also arise in connection with {{math|''X''₀(''n'')}} since points on it correspond to some {{mvar|n}}-isogenous pairs of elliptic curves. An ''isogeny'' between two elliptic curves is a non-trivial morphism of varieties (defined by a rational map) between the curves which also respects the group laws 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. Such a map is always surjective and has a finite kernel, the order of which is the ''degree'' of the isogeny. Points on {{math|''X''₀(''n'')}} correspond to pairs of elliptic curves admitting an isogeny of degree {{mvar|n}} with cyclic kernel.

When {{math|''X''₀(''n'')}} has genus one, it will itself be isomorphic to an elliptic curve, which will have the same [[j-invariant|{{mvar|j}}-invariant]].