https://wiki.sarg.dev/index.php?action=history&feed=atom&title=Modular_equation 2026-06-25T22:27:56Z Revision history for this page on the wiki MediaWiki 1.44.2 https://wiki.sarg.dev/index.php?title=Modular_equation&diff=273915&oldid=prev 2024-05-13T05:02:28Z

<p>tag as one source</p> <p><b>New page</b></p><div>{{Short description|Type of algebraic equation}}<br /> &lt;!-- {{more citations needed|date=August 2008}} --&gt;<br /> {{one source |date=May 2024}}<br /> In [[mathematics]], a &#039;&#039;&#039;modular equation&#039;&#039;&#039; is an [[algebraic equation]] satisfied by &#039;&#039;moduli&#039;&#039;,&lt;ref&gt;{{MathWorld|title=Modular Equation|urlname=ModularEquation}}&lt;/ref&gt; 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.<br /> <br /> The most frequent use of the term &#039;&#039;modular equation&#039;&#039; 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 &#039;&#039;F&#039;&#039; and &#039;&#039;G&#039;&#039;, in the [[function field of an algebraic variety|function field]] of the modular curve, will satisfy a modular equation &#039;&#039;P&#039;&#039;(&#039;&#039;F&#039;&#039;,&#039;&#039;G&#039;&#039;) = 0 with &#039;&#039;P&#039;&#039; a non-zero [[polynomial]] of two variables over the [[complex number]]s. For suitable non-degenerate choice of &#039;&#039;F&#039;&#039; and &#039;&#039;G&#039;&#039;, the equation &#039;&#039;P&#039;&#039;(&#039;&#039;X&#039;&#039;,&#039;&#039;Y&#039;&#039;) = 0 will actually define the modular curve.<br /> <br /> This can be qualified by saying that &#039;&#039;P&#039;&#039;, 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 &#039;&#039;P&#039;&#039; may be very large numbers. Further, the &#039;cusps&#039; 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 &#039;&#039;P&#039;&#039;.<br /> <br /> In that sense a modular equation becomes the &#039;&#039;&#039;equation of a modular curve&#039;&#039;&#039;. Such equations first arose in the theory of multiplication of [[elliptic function]]s (geometrically, the &#039;&#039;n&lt;sup&gt;2&lt;/sup&gt;&#039;&#039;-fold [[covering map]] from a 2-[[torus]] to itself given by the mapping &#039;&#039;x&#039;&#039; → &#039;&#039;n&#039;&#039;·&#039;&#039;x&#039;&#039; on the underlying group) expressed in terms of [[complex analysis]].<br /> <br /> ==See also==<br /> * [[Modular lambda function]]<br /> * [[Ramanujan&#039;s lost notebook]]<br /> <br /> ==References==<br /> {{Reflist}}<br /> <br /> [[Category:Modular forms]]<br /> <br /> <br /> {{algebra-stub}}</div>

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