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 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
C
/
⟨
1
,
τ
⟩
{\displaystyle \mathbb {C} /\langle 1,\tau \rangle }
, where the map is defined as the quotient by the [−1] involution.
The q-expansion, where
q
=
e
π
i
τ
{\displaystyle q=e^{\pi i\tau }}
is the nome , is given by:
λ
(
τ
)
=
16
q
−
128
q
2
+
704
q
3
−
3072
q
4
+
11488
q
5
−
38400
q
6
+
…
{\displaystyle \lambda (\tau )=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+\dots }
. OEIS : A115977
By symmetrizing the lambda function under the canonical action of the symmetric group S 3 on X (2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group
S
L
2
(
Z
)
{\displaystyle SL_{2}(\mathbb {Z} )}
, and it is in fact Klein's modular j-invariant .
Modular properties
The function
λ
(
τ
)
{\displaystyle \lambda (\tau )}
is invariant under the group generated by[ 1]
τ
↦
τ
+
2
;
τ
↦
τ
1
−
2
τ
.
{\displaystyle \tau \mapsto \tau +2\ ;\ \tau \mapsto {\frac {\tau }{1-2\tau }}\ .}
The generators of the modular group act by[ 2]
τ
↦
τ
+
1
:
λ
↦
λ
λ
−
1
;
{\displaystyle \tau \mapsto \tau +1\ :\ \lambda \mapsto {\frac {\lambda }{\lambda -1}}\,;}
τ
↦
−
1
τ
:
λ
↦
1
−
λ
.
{\displaystyle \tau \mapsto -{\frac {1}{\tau }}\ :\ \lambda \mapsto 1-\lambda \ .}
Consequently, the action of the modular group on
λ
(
τ
)
{\displaystyle \lambda (\tau )}
is that of the anharmonic group , giving the six values of the cross-ratio :[ 3]
{
λ
,
1
1
−
λ
,
λ
−
1
λ
,
1
λ
,
λ
λ
−
1
,
1
−
λ
}
.
{\displaystyle \left\lbrace {\lambda ,{\frac {1}{1-\lambda }},{\frac {\lambda -1}{\lambda }},{\frac {1}{\lambda }},{\frac {\lambda }{\lambda -1}},1-\lambda }\right\rbrace \ .}
Other appearances
Other elliptic functions
It is the square of the Jacobi modulus ,[ 4] that is,
λ
(
τ
)
=
k
2
(
τ
)
{\displaystyle \lambda (\tau )=k^{2}(\tau )}
. In terms of the Dedekind eta function
η
(
τ
)
{\displaystyle \eta (\tau )}
and theta functions ,[ 4]
λ
(
τ
)
=
(
2
η
(
τ
2
)
η
2
(
2
τ
)
η
3
(
τ
)
)
8
=
16
(
η
(
τ
/
2
)
η
(
2
τ
)
)
8
+
16
=
θ
2
4
(
0
,
τ
)
θ
3
4
(
0
,
τ
)
{\displaystyle \lambda (\tau )={\Bigg (}{\frac {{\sqrt {2}}\,\eta ({\tfrac {\tau }{2}})\eta ^{2}(2\tau )}{\eta ^{3}(\tau )}}{\Bigg )}^{8}={\frac {16}{\left({\frac {\eta (\tau /2)}{\eta (2\tau )}}\right)^{8}+16}}={\frac {\theta _{2}^{4}(0,\tau )}{\theta _{3}^{4}(0,\tau )}}}
and,
1
(
λ
(
τ
)
)
1
/
4
−
(
λ
(
τ
)
)
1
/
4
=
1
2
(
η
(
τ
4
)
η
(
τ
)
)
4
=
2
θ
4
2
(
0
,
τ
2
)
θ
2
2
(
0
,
τ
2
)
{\displaystyle {\frac {1}{{\big (}\lambda (\tau ){\big )}^{1/4}}}-{\big (}\lambda (\tau ){\big )}^{1/4}={\frac {1}{2}}\left({\frac {\eta ({\tfrac {\tau }{4}})}{\eta (\tau )}}\right)^{4}=2\,{\frac {\theta _{4}^{2}(0,{\tfrac {\tau }{2}})}{\theta _{2}^{2}(0,{\tfrac {\tau }{2}})}}}
where[ 5] for the nome
q
=
e
π
i
τ
{\displaystyle q=e^{\pi i\tau }}
,
θ
2
(
0
,
τ
)
=
∑
n
=
−
∞
∞
q
(
n
+
1
2
)
2
{\displaystyle \theta _{2}(0,\tau )=\sum _{n=-\infty }^{\infty }q^{\left({n+{\frac {1}{2}}}\right)^{2}}}
θ
3
(
0
,
τ
)
=
∑
n
=
−
∞
∞
q
n
2
{\displaystyle \theta _{3}(0,\tau )=\sum _{n=-\infty }^{\infty }q^{n^{2}}}
θ
4
(
0
,
τ
)
=
∑
n
=
−
∞
∞
(
−
1
)
n
q
n
2
{\displaystyle \theta _{4}(0,\tau )=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}}
In terms of the half-periods of Weierstrass's elliptic functions , let
[
ω
1
,
ω
2
]
{\displaystyle [\omega _{1},\omega _{2}]}
be a fundamental pair of periods with
τ
=
ω
2
ω
1
{\displaystyle \tau ={\frac {\omega _{2}}{\omega _{1}}}}
.
e
1
=
℘
(
ω
1
2
)
,
e
2
=
℘
(
ω
2
2
)
,
e
3
=
℘
(
ω
1
+
ω
2
2
)
{\displaystyle e_{1}=\wp \left({\frac {\omega _{1}}{2}}\right),e_{2}=\wp \left({\frac {\omega _{2}}{2}}\right),e_{3}=\wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)}
we have[ 4]
λ
=
e
3
−
e
2
e
1
−
e
2
.
{\displaystyle \lambda ={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}}\,.}
Since the three half-period values are distinct, this shows that λ does not take the value 0 or 1.[ 4]
The relation to the j-invariant is[ 6] [ 7]
j
(
τ
)
=
256
(
1
−
λ
(
1
−
λ
)
)
3
(
λ
(
1
−
λ
)
)
2
=
256
(
1
−
λ
+
λ
2
)
3
λ
2
(
1
−
λ
)
2
.
{\displaystyle j(\tau )={\frac {256(1-\lambda (1-\lambda ))^{3}}{(\lambda (1-\lambda ))^{2}}}={\frac {256(1-\lambda +\lambda ^{2})^{3}}{\lambda ^{2}(1-\lambda )^{2}}}\ .}
which is the j -invariant of the elliptic curve of Legendre form
y
2
=
x
(
x
−
1
)
(
x
−
λ
)
{\displaystyle y^{2}=x(x-1)(x-\lambda )}
Little Picard theorem
The lambda function is used in the original proof of the Little Picard theorem , that an entire non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard in 1879.[ 8] 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 must be constant.[ 9]
Moonshine
The function
16
λ
(
2
τ
)
−
8
{\displaystyle {\frac {16}{\lambda (2\tau )}}-8}
is the normalized Hauptmodul for the group
Γ
0
(
4
)
{\displaystyle \Gamma _{0}(4)}
, and its q -expansion
q
−
1
+
20
q
−
62
q
3
+
…
{\displaystyle q^{-1}+20q-62q^{3}+\dots }
, OEIS : A007248 where
q
=
e
2
π
i
τ
{\displaystyle q=e^{2\pi i\tau }}
, is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra .
^ Chandrasekharan (1985) p.115
^ Chandrasekharan (1985) p.109
^ Chandrasekharan (1985) p.110
^ a b c d Chandrasekharan (1985) p.108
^ Chandrasekharan (1985) p.63
^ Chandrasekharan (1985) p.117
^ Rankin (1977) pp.226–228
^ Chandrasekharan (1985) p.121
^ Chandrasekharan (1985) p.118
References
Abramowitz, Milton ; Stegun, Irene A. , eds. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , New York: Dover Publications , ISBN 978-0-486-61272-0 , Zbl 0543.33001
Chandrasekharan, K. (1985), Elliptic Functions , Grundlehren der mathematischen Wissenschaften, vol. 281, Springer-Verlag , pp. 108–121, ISBN 3-540-15295-4 , Zbl 0575.33001
Conway, John Horton ; Norton, Simon (1979), "Monstrous moonshine", Bulletin of the London Mathematical Society , 11 (3): 308–339, doi :10.1112/blms/11.3.308 , MR 0554399 , Zbl 0424.20010
Rankin, Robert A. (1977), Modular Forms and Functions , Cambridge University Press , ISBN 0-521-21212-X , Zbl 0376.10020
Reinhardt, W. P.; Walker, P. L. (2010), "Elliptic Modular Function" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN 978-0-521-19225-5 , MR 2723248 .