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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such that {{math|(''x'', ''y'') {{=}} (''j''(''nτ''), ''j''(''τ''))}} is a point on the curve. Here {{math|''j''(''τ'')}} denotes the [[j-invariant|{{mvar|j}}-invariant]].

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 ==

[[Image:Modknot11.png|thumb|Knot at infinity of {{math|''X''₀(11)}}]]

The classical modular curve, which we will call {{math|''X''₀(''n'')}}, is of degree greater than or equal to {{math|2''n''}} when {{math|''n'' > 1}}, with equality if and only if {{mvar|n}} is a prime. The polynomial {{math|Φ_''n''}} has integer coefficients, and hence is defined over every field. However, the coefficients are sufficiently large that computational work with the curve can be difficult. As a polynomial in {{mvar|x}} with coefficients in {{math|'''Z'''[''y'']}}, it has degree {{math|''ψ''(''n'')}}, where {{mvar|ψ}} is the [[Dedekind psi function]]. Since {{math|Φ_''n''(''x'', ''y'') {{=}} Φ_''n''(''y'', ''x'')}}, {{math|''X''₀(''n'')}} is symmetrical around the line {{math|''y'' {{=}} ''x''}}, and has singular points at the repeated roots of the classical modular polynomial, where it crosses itself in the complex plane. These are not the only singularities, and in particular when {{math|''n'' > 2}}, there are two singularitessingularities at infinity, where {{math|''x'' {{=}} 0, ''y'' {{=}} ∞}} and {{math|''x'' {{=}} ∞, ''y'' {{=}} 0}}, which have only one branch and hence have a knot invariant which is a true knot, and not just a link.

== Parametrization of the modular curve ==

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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)}}.

By a theorem of [[Henri Carayol]], if an elliptic curve {{mvar|E}} is modular then its [[conductor of an elliptic curve|conductor]], an isogeny invariant described originally in terms of [[cohomology]], is the smallest integer {{mvar|n}} such that there exists a rational mapping {{math|''φ'' : ''X''₀(''n'') → ''E''}}. Since we now know all elliptic curves over {{math|'''Q'''}} are modular, we also know that the conductor is simply the level {{mvar|n}} of its minimal modular parametrization.

== 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

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== External links ==

*{{Cite OEIS el|1=A001617|2=Genus of modular group Gamma_0(n). Or, genus of modular curve X_0(n)}}

*[http://www.math.uwaterloo.ca/~mrubinst/modularpolynomials/phi_l.html] Coefficients of {{math|''X''₀(''n'')}}