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| In [[mathematics]], the '''elliptic modular lambda''' function λ(τ) is a highly symmetric holomorphic function on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.
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| The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)|nome]], is given by:
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| : <math> \lambda(\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \dots</math>. {{oeis|id=A115977 }}
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| By symmetrizing the lambda function under the canonical action of the symmetric group ''S''<sub>3</sub> on ''X''(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group <math>SL_2(\mathbb{Z})</math>, and it is in fact Klein's modular [[j-invariant]].
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| ==Modular properties==
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| The function <math> \lambda(\tau) </math> is invariant under the group generated by<ref name=C115>Chandrasekharan (1985) p.115</ref>
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| :<math> \tau \mapsto \tau+2 \ ;\ \tau \mapsto \frac{\tau}{1-2\tau} \ . </math>
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| The generators of the modular group act by<ref name=C109>Chandrasekharan (1985) p.109</ref>
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| :<math> \tau \mapsto \tau+1 \ :\ \lambda \mapsto \frac{\lambda}{\lambda-1} \, ;</math>
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| :<math> \tau \mapsto -1/\tau \ :\ \lambda \mapsto 1 - \lambda \ . </math>
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| Consequently, the action of the modular group on <math> \lambda(\tau) </math> is that of the [[anharmonic group]], giving the six values of the [[cross-ratio]]:<ref name=C110>Chandrasekharan (1985) p.110</ref>
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| :<math> \left\lbrace { \lambda, \frac{1}{1-\lambda}, \frac{\lambda-1}{\lambda}, \frac{1}{\lambda}, \frac{\lambda}{\lambda-1}, 1-\lambda } \right\rbrace \ .</math>
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| == Other appearances ==
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| ===Other elliptic functions===
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| It is the [[Square (algebra)|square]] of the [[Jacobi modulus]],<ref name=C108>Chandrasekharan (1985) p.108</ref> that is, λ(τ) = ''k''<sup>2</sup>(τ). In terms of [[theta function]]s,<ref name=C108/>
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| :<math> \lambda(\tau) = \frac{\theta_1^4(0,\tau)}{\theta_3^4(0,\tau)} </math>
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| where<ref name=C63>Chandrasekharan (1985) p.63</ref>
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| :<math>\theta_1(0,\tau) = \sum_{n=-\infty}^\infty q^{\left({n+\frac12}\right)^2} \mathrm{ and } \
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| \theta_3(0,\tau) = \sum_{n=-\infty}^\infty q^{n^2} </math>
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| for the [[Nome (mathematics)|nome]] <math>q = e^{\pi i \tau}</math>.
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| In terms of the half-periods of [[Weierstrass's elliptic functions]], let [ω<sub>1</sub>, ω<sub>2</sub>] be a [[fundamental pair of periods]] with τ = ω<sub>2</sub>/ω<sub>1</sub>.
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| :<math> e_1 = \wp(\omega_1/2), e_2 = \wp(\omega_2/2), e_3 = \wp((\omega_1+\omega_2)/2) </math>
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| we have<ref name=C108/>
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| :<math> \lambda = \frac{e_3-e_2}{e_1-e_2} \, . </math>
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| Since the three half-period values are distinct, this shows that λ does not take the value 0 or 1.<ref name=C108/>
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| The relation to the ''j''-invariant is<ref name=C117>Chandrasekharan (1985) p.117</ref><ref>Rankin (1977) pp.226–228</ref>
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| :<math> j(\tau) = 256 \frac{(1-\lambda+\lambda^2)^3}{\lambda^2 (1-\lambda)^2} \ . </math> | |
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| which is the ''j''-invariant of the elliptic curve of [[Legendre form]] <math>y^2=x(x-1)(x-\lambda)</math>
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| ===Little Picard theorem===
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| The lambda function is used in the original proof of the [[Little Picard theorem]], that an [[entire function|entire]] non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard in 1879.<ref>Chandrasekharan (1985) p.121</ref> Suppose if possible that ''f'' is entire and does not take the values 0 and 1. Since λ is holomorphic, it has a local holomorphic inverse ω defined away from 0,1,∞. Consider the function ''z'' → ω(''f''(''z'')). By the [[Monodromy theorem]] this is holomorphic and maps the complex plane '''C''' to the upper half plane. From this it is easy to construct a holomorphic function from '''C''' to the unit disc, which by [[Liouville's theorem (complex analysis)|Liouville's theorem]] must be constant.<ref>Chandrasekharan (1985) p.118</ref>
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| ===Moonshine===
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| The function <math>\frac{16}{\lambda(2\tau)} - 8</math> is the normalized [[Hauptmodul]] for the group <math>\Gamma_0(4)</math>, and its ''q''-expansion <math>q^{-1} + 20q - 62q^3 + \dots</math> is the graded character of any element in conjugacy class 4C of the [[monster group]] acting on the [[monster vertex algebra]].
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| == References ==
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| {{reflist}}
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| *{{Citation | editor1-last=Abramowitz | editor1-first=Milton | editor1-link=Milton Abramowitz | editor2-last=Stegun | editor2-first=Irene A. | editor2-link=Irene Stegun | title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-61272-0 | year=1972 | zbl=0543.33001 }}
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| * {{citation | last=Chandrasekharan | first=K. | authorlink=K. S. Chandrasekharan | title=Elliptic Functions | series=Grundlehren der mathematischen Wissenschaften | volume=281 | publisher=[[Springer-Verlag]] | year=1985 | isbn=3-540-15295-4 | zbl=0575.33001 | pages=108–121 }}
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| *{{citation|first1=John Horton|last1=Conway|author1-link=John Horton Conway|first2=Simon|last2=Norton|author2-link=Simon P. Norton|title=Monstrous moonshine|journal=Bulletin of the London Mathematical Society|volume=11|issue=3|year=1979|pages=308–339|mr=0554399|zbl=0424.20010 |doi=10.1112/blms/11.3.308}}
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| * {{citation | last=Rankin | first=Robert A. | authorlink=Robert Alexander Rankin | title=Modular Forms and Functions | publisher=[[Cambridge University Press]] | year=1977 | isbn=0-521-21212-X | zbl=0376.10020 }}
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| *{{dlmf|id=23.15.E6|title=Elliptic Modular Function|first= W. P. |last=Reinhardt|first2=P. L.|last2= Walker}}
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| {{DEFAULTSORT:Modular Lambda Function}}
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| [[Category:Modular forms]]
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| [[Category:Elliptic functions]]
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Got nothing to say about myself I think.
Hurrey Im here and a part of wmflabs.org.
I just hope Im useful at all
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