Editing Modular equation

{{Short description|Type of algebraic equation}}
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In [[mathematics]], a '''modular equation''' is an [[algebraic equation]] satisfied by ''moduli'',<ref>{{MathWorld|title=Modular Equation|urlname=ModularEquation}}</ref> in the sense of [[moduli problem|moduli problems]]. That is, given a number of functions on a [[moduli space]], a modular equation is an equation holding between them, or in other words an [[identity (mathematics)|identity]] for moduli.

The most frequent use of the term ''modular equation'' is in relation to the moduli problem for [[elliptic curve]]s. In that case the moduli space itself is of dimension one. That implies that any two [[rational function]]s ''F'' and ''G'', in the [[function field of an algebraic variety|function field]] of the modular curve, will satisfy a modular equation ''P''(''F'',''G'') = 0 with ''P'' a non-zero [[polynomial]] of two variables over the [[complex number]]s. For suitable non-degenerate choice of ''F'' and ''G'', the equation ''P''(''X'',''Y'') = 0 will actually define the modular curve.

This can be qualified by saying that ''P'', in the worst case, will be of high degree and the plane curve it defines will have [[Mathematical singularity|singular points]]; and the [[coefficient]]s of ''P'' may be very large numbers. Further, the 'cusps' of the moduli problem, which are the points of the modular curve not corresponding to honest elliptic curves but degenerate cases, may be difficult to read off from knowledge of ''P''.

In that sense a modular equation becomes the '''equation of a modular curve'''. Such equations first arose in the theory of multiplication of [[elliptic function]]s (geometrically, the ''n<sup>2</sup>''-fold [[covering map]] from a 2-[[torus]] to itself given by the mapping ''x'' → ''n''·''x'' on the underlying group) expressed in terms of [[complex analysis]].

==See also==
* [[Modular lambda function]]
* [[Ramanujan's lost notebook]]

==References==
{{Reflist}}

[[Category:Modular forms]]


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