Classical modular curve: Difference between revisions

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== Mappings ==

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''<sub>0</supsub>(''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 {{mvar|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 {{mvar|n}}, the cyclic isogenies, correspond to points on {{math|''X''₀(''n'')}}.

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]].

For instance, {{math|''X''₀(11)}} has {{mvar|j}}-invariant {{math|−2¹²11⁻⁵31³}}, and is isomorphic to the curve {{math|''y''² + ''y'' {{=}} ''x''³ − ''x''² − 10''x'' − 20}}. If we substitute this value of {{mvar|j}} for {{mvar|y}} in {{math|''X''<sub>0</supsub>(5)}}, we obtain two rational roots and a factor of degree four. The two rational roots correspond to isomorphism classes of curves with rational coefficients which are 5-isogenous to the above curve, but not isomorphic, having a different function field. Specifically, we have the six rational points: x=-122023936/161051, y=-4096/11, x=-122023936/161051, y=-52893159101157376/11, and x=-4096/11, y=-52893159101157376/11, plus the three points exchanging {{mvar|x}} and {{mvar|y}}, all on {{math|''X''<sub>0</supsub>(5)}}, corresponding to the six isogenies between these three curves.

If in the curve {{math|''y''² + ''y'' {{=}} ''x''³ − ''x''² − 10''x'' − 20}}, isomorphic to {{math|''X''₀(11)}} we substitute

:<math>x \mapsto \frac{x^5-2x^4+3x^3-2x+1}{x^2(x-1)^2}</math>

:<math>y \mapsto y-\frac{(2y+1)(x^4+x^3-3x^2+3x-1)}{x^3(x-1)^3}</math>

and factor, we get an extraneous factor of a rational function of {{mvar|x}}, and the curve {{math|''y''² + ''y'' {{=}} ''x''³ − ''x''²}}, with {{mvar|j}}-invariant {{math|−2¹²11⁻¹}}. Hence both curves are modular of level {{math|11}}, having mappings from {{math|''X''₀(11)}}.