Classical modular curve: Difference between revisions

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{{Short description|Plane algebraic curve}}

In [[number theory]], the '''classical modular curve''' is an irreducible [[algebraic curve|plane algebraic curve]] given by an equation

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The curve is sometimes called {{math|''X''₀(''n'')}}, though often that notation is used for the abstract [[algebraic curve]] for which there exist various models. A related object is the '''classical modular polynomial''', a polynomial in one variable defined as {{math|Φ_''n''(''x'', ''x'')}}.

It is important to note that theThe classical modular curves are part of the larger theory of [[modular curve]]s. In particular it has another expression as a compactified quotient of the complex [[upper half-plane]] {{math|'''H'''}}.

== Geometry of the modular curve ==

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== Galois theory of the modular curve ==

The [[Galois theory]] of the modular curve was investigated by [[Erich Hecke]]. Considered as a polynomial in x with coefficients in {{math|'''Z'''[''y'']}}, the modular equation {{math|Φ₀(''n'')}} is a polynomial of degree {{math|''ψ''(''n'')}} in {{mvar|x}}, whose roots generate a [[Galois extension]] of {{math|'''Q'''(''y'')}}. In the case of {{math|''X''₀(''p'')}} with {{mvar|p}} prime, where the [[Characteristic (algebra)|characteristic]] of the field is not {{mvar|p}}, the [[Galois group]] of {{math|'''Q'''(''x'', ''y'')/'''Q'''(''y'')}} is {{math|PGL(2, ''p'')}}, the [[projective linear group|projective general linear group]] of [[Möbius transformation|linear fractional transformations]] of the [[projective line]] of the field of {{mvar|p}} elements, which has {{math|''p'' + 1}} points, the degree of {{math|''X''₀(''p'')}}.

This extension contains an algebraic extension {{math|''F''/'''Q'''}} where if <math>p^* = (-1)^{(p-1)/2}p</math> in the notation of [[Quadratic_reciprocityQuadratic reciprocity#Gauss.27s_version_in_Legendre_symbols27s version in Legendre symbols|Gauss]] then:

:<math>F = \mathbf{Q}\left(\sqrt{p^*}\right).</math>

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== References ==

*{{citation|first=Erich |last=Hecke, ''|title=Die eindeutige Bestimmung der Modulfunktionen q-ter Stufe durch algebraische Eigenschaften'',|journal=[[Mathematische Math. Ann. '''Annalen]]|volume=111''' (|year=1935),|pages= 293-301293–301|url=https://eudml.org/doc/159776|doi=10.1007/BF01472221}}, reprinted in ''Mathematische Werke'', third edition, Vandenhoeck & Ruprecht, Göttingen, 1983, 568-576 [http://dz-srv1.sub.uni-goettingen.de/sub/digbib/pdftermsconditions?did=D37958&p=297]{{dead link|date=August 2017 |bot=InternetArchiveBot |fix-attempted=yes }}

*Anthony Knapp, ''Elliptic Curves'', Princeton, 1992

*[[Serge Lang]], ''Elliptic Functions'', Addison-Wesley, 1973