Classical modular curve: Difference between revisions

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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. Two elliptic curves areAn ''isogenousisogeny'' ifbetween theretwo elliptic curves is a non-trivial morphism of varieties (defined by a rational map) between the curves which is also a group homomorphism, respectingrespects the group lawlaws 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. TheSuch isogeniesa withmap cyclicis always surjective and has a finite kernel, the order of degreewhich {{mvar|n}},is the cyclic''degree'' isogenies,of correspondthe toisogeny. pointsPoints 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]].