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 Φ_n(x, y)=0, where for the [[j-invariant]] j(τ), x=j(n τ), y=j(τ) is a point on the curve. The curve is sometimes called X₀(n), though often that is used for the abstract curve for which there exist various models. A related object is the ''classical modular polynomial'', a polynomial in one variable defined as Φ_n(x, x).

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

:{{math|Φ_''n''(''x'', ''y'') {{=}} 0}},

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]].

In [[number theory]], the '''classical modular curve''' is an irreducible [[algebraic curve|plane algebraic curve]] given by an equation Φ_n(x, y)=0, where for the [[j-invariant]] j(τ), x=j(n τ), y=j(τ) is a point on the curve. 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 Φ{{math|Φ_''n''(''x'', ''x'')}}.

The 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)}}]]

ΦThe 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, butthe thesecoefficients are sufficiently large, makingthat computational work with the curve computationallycan be difficult. As a polynomial in {{mvar|x}} with coefficients in {{math|'''Z'''[''y'']}}, it has degree ψ{{math|''ψ''(''n'')}}, where ψ{{mvar|ψ}} is the [[Dedekind psi function]]. Since Φ{{math|Φ_''n''(''x'', ''y'') {{=}}   ΦΦ_''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.

The classical modular curve, which we will call X₀(n), is of degree greater than or equal to 2n when n>1, with equality if and only if n is a prime.

Φ_n has integer coefficients, and hence is defined over every field, but these are large, making the curve computationally difficult. As a polynomial in x with coefficients in '''Z'''[y], it has degree ψ(n), where ψ is the [[Dedekind psi function]]. Since Φ_n(x, y) =   Φ_n(y, x), X₀(n) is symmetrical around the line 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 n>2, there are two singularites at infinity, where x=0, y=∞ and 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 ==

WhenFor {{math|''n'' {{=}} 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18}}, or {{math|25}}, {{math|''X''₀(''n'')}} has [[geometric genus|genus]] zero, and hence can be parametrized [http://www.math.fsu.edu/~hoeij/files/X0N/Parametrization] by rational functions. The simplest nontrivial example is {{math|''X''₀(2)}}, where if:

:<math>j_2(q) = q^{-1} - 24 + 276q -2048q^2 + 11202q^3 + \cdots =\left (\frac{\eta(q)}{\eta(q^2)} \right)^{24}</math>

When n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, or 25, X₀(n) has [[geometric genus|genus]] zero, and hence can be parametrized by rational functions. The simplest nontrivial example is X₀(2), where if

is (up to the constant term) the [[Monstrous moonshine|McKay-ThompsonMcKay–Thompson series]] for the class 2B of the [[Monster group|Monster]], and {{mvar|η}} is the [[Dedekind eta function]], then

:<math>j_2(q) = q^{-1} - 24 + 276q -2048q^2 + 11202q^3 + \cdots</math>

:<math>x = \frac{(j_2+256)^3}{j_2^2}, y = \frac{(j_2+16)^3}{j_2}</math>

is (up to the constant term) the [[Monstrous moonshine|McKay-Thompson series]] for the class 2B of the [[Monster group|Monster]], then

:<math>xy \mapsto= \frac{x^5-2x^4(j_2+3x16)^3-2x+1}{x^2(x-1)^2j_2}</math>

parametrizes {{math|''X''₀(2)}} in terms of rational functions of {{math|''j''₂}}. It is not necessary to actually compute {{math|''j''₂}} to use this parametrization; it can be taken as an arbitrary parameter.

:<math>x = \frac{(j_2+256)^3}{j_2^2}, y = \frac{(j_2+16)^3}{j_2}</math>

parametrizes X₀(2) in terms of rational functions of j₂. It is not necessary to actually compute j₂ to use this parametrization; it can be taken as an arbitrary parameter.

== 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''₀(''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''<sub>0</supsub>(''n'')}} since points on it correspond to some {{mvar|n}}-isogenous pairs of elliptic curves. Two elliptic curves areAn ''isogenousisogeny'' ifbetween theretwo elliptic curves is a non-trivial morphism of varitiesvarieties (defined by a rational map) between the curves which is also a group homomorphism, respectingrespects the group law on the elliptic curveslaws, and hence which sends the point at infinity (serving as the identity of the group law) to the point at infinity. TheSuch isogeniesa withmap cyclicis always surjective and has a finite kernel, the order of degreewhich n,is the cyclic''degree'' isogenies,of correspondthe toisogeny. pointsPoints on {{math|''X''<sub>0</supsub>(''n'').}} correspond to pairs of elliptic curves admitting an isogeny of degree {{mvar|n}} with cyclic kernel.

A curve C such that there exists a surjective morphism from X<sub>0</sup>(n) to C, given by a rational map &phi;:X<sub>0</sup>(n) → C, is a [[modular curve]]. The famous [[modularity theorem]] tells us that all [[elliptic curve]]s are modular; the ''conductor'' of the elliptic curve being the minimal n required for this mapping to exist.

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]].

Mappings also arise in connection with X<sub>0</sup>(n) since points on it correspond to n-isogenous pairs of elliptic curves. Two elliptic curves are ''isogenous'' if there is a morphism of varities (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 n, the cyclic isogenies, correspond to points on X<sub>0</sup>(n).

When X<sub>0</sup>(n) has genus one, it will itself be isomorphic to an elliptic curve, which will have the same [[j-invariant]]. For instance, {{math|''X''<sub>0</supsub>(11)}} has {{mvar|j}}-invariant -122023936/161051 = - 2{{math|−2¹²11^-5−531³}}, and is isomorphic to the curve {{math|''y''² + ''y'' {{=}} ''x''³- − ''x''²-10x- − 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''₀(5)}}, corresponding to the six isogenies between these three curves.

Specifically, we get the six rational points x=-122023936/161051, y=-4096/11, x=-122023936/161051, and y=-52893159101157376/11, x=-4096/11, y=y=-52893159101157376/11, plus the three points exchanging x and y, all on X<sub>0</sup>(5), and these correspond to isogenies. If in the curve {{math|''y''² + ''y'' {{=}} ''x''³- − ''x''²-10x- − 10''x'' − 20}}, isomorphic to {{math|''X''<sub>0</supsub>(11)}} we substitute

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

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

and

:<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 -4096{{math|−2¹²11⁻¹}}. Hence both curves are modular of level {{math|11}}, having mappings from {{math|''X''<sub>0</supsub>(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 &Phi;{{math|Φ₀(''n'')}} is a polynomial of degree &psi;{{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|PGL₂(2, ''p'')}}, the [[projective linear group|projective general linear group]] of [[Möbius transformationstransformation|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 reciprocity#Gauss.27s version in Legendre symbols|Gauss]] then:

The Galois theory of the modular curve was investigated by [[Erich Hecke]]. Considered as a polynomial in x with coefficients in '''Z'''[y], the modular equation &Phi;₀(n) is a polynomial of degree &psi;(n) in x, whose roots generate a [[Galois extension]] of '''Q'''(y). In the case of X₀(p) with p prime, where the [[Characteristic (algebra)|characteristic]] of the field is not p, the [[Galois group]] of '''Q'''(x, y)/'''Q'''(y) is PGL₂(p), the [[projective linear group|projective general linear group]] of [[Möbius transformations|linear fractional transformations]] of the [[projective line]] of the field of p elements, which has p+1 points, the degree of X₀(p).

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

This extension contains an algebraic extension <math>F = \mathbf {Q}(\sqrt{(-1)^\frac{p-1}{2}p})</math> of '''Q'''. If we extend the field of constants to be ''F'', we now have an extention with Galois group PSL₂(p), the [[projective linear group|projective special linear group]] of the field with p elements, which is a finite simple group. By specializing y to a specific field element, we can, outside of a thin set, obtain an infinity of examples of groups with Galois group PSL₂(p) over ''F'', and PGL₂(p) over '''Q'''.

This extension contains an algebraic extension <math>F = \mathbf {Q}(\sqrt{(-1)^\frac{p-1}{2}p})</math> of '''Q'''. If we extend the field of constants to be ''{{mvar|F''}}, we now have an extentionextension with Galois group {{math|PSL₂(2, ''p'')}}, the [[projective linear group|projective special linear group]] of the field with {{mvar|p}} elements, which is a finite simple group. By specializing {{mvar|y}} to a specific field element, we can, outside of a thin set, obtain an infinity of examples of groupsfields with Galois group {{math|PSL₂(2, ''p'')}} over ''{{mvar|F''}}, and {{math|PGL₂(2, ''p'')}} over {{math|'''Q'''}}.

When n is not a prime, the Galois groups can be analyzed in terms of the factors of n as a wreath product.

When {{mvar|n}} is not a prime, the Galois groups can be analyzed in terms of the factors of {{mvar|n}} as a [[wreath product]].

== See also ==

== See also ==

*[[Algebraic curves]]

*[[J-invariant]]

*[[Modular curve]]

*[[Modular function]]

== 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]

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

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

*Goro Shimura, ''Introduction to the Arithmetic Theory of Automorphic Functions'', Princeton, 1972

== External links ==

*{{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'')}}

[[Category:Algebraic curves]]

*[http://www.research.att.com/~njas/sequences/A001617]Genus of X₀(n)

[[Category:Modular forms]]

*[http://www.math.uwaterloo.ca/~mrubinst/modularpolynomials/phi_l.html]Coefficients of X₀(n)

[[Category:Analytic number theory]]

== References ==

*Erich Hecke, ''Die eindeutige Bestimmung der Modulfunktionen q-ter Stufe durch algebraische Eigenschaften'', Math. Ann. '''111''' (1935), 293-301, 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]

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

*Goro Shimura, ''Introduction to the Arithmetic Theory of Automorphic Functions'', Princeton, 1972