In mathematics, the modular lambda function λ(τ) is not a modular function (per the Wikipedia definition), but every modular function is a rational function in . Some authors use a non-equivalent definition of "modular functions". 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 , where the map is defined as the quotient by the [−1] involution.
The q-expansion, where is the nome, is given by:
.
By symmetrizing the lambda function under the canonical action of the symmetric group S3 on X(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group , and it is in fact Klein's modular j-invariant.
Modular properties The function is invariant under the group generated byChandrasekharan (1985) p.115
The generators of the modular group act byChandrasekharan (1985) p.109
Consequently, the action of the modular group on is that of the anharmonic group, giving the six values of the cross-ratio:Chandrasekharan (1985) p.110
Relations to other functions It is the square of the elliptic modulus,Chandrasekharan (1985) p.108 that is, . In terms of the Dedekind eta function and theta functions,
and,
whereChandrasekharan (1985) p.63
In terms of the half-periods of Weierstrass's elliptic functions, let be a fundamental pair of periods with .
we have
Since the three half-period values are distinct, this shows that does not take the value 0 or 1.
The relation to the j-invariant isChandrasekharan (1985) p.117Rankin (1977) pp.226–228
which is the j-invariant of the elliptic curve of Legendre form
Given , let
where is the complete elliptic integral of the first kind with parameter . Then
Modular equations The modular equation of degree (where is a prime number) is an algebraic equation in and . If and , the modular equations of degrees are, respectively, p. 103–109, 134
The quantity (and hence ) can be thought of as a holomorphic function on the upper half-plane :
Since , the modular equations can be used to give algebraic values of for any prime .For any prime power, we can iterate the modular equation of degree . This process can be used to give algebraic values of for any The algebraic values of are also given by p. 42 is algebraic for every
where is the lemniscate sine and is the lemniscate constant.
Lambda-star
Definition and computation of lambda-star
The function p. 152 (where ) gives the value of the elliptic modulus , for which the complete elliptic integral of the first kind and its complementary counterpart are related by following expression:
The values of can be computed as follows:
The functions and are related to each other in this way:
Properties of lambda-star
Every value of a positive rational number is a positive algebraic number:
and (the complete elliptic integral of the second kind) can be expressed in closed form in terms of the gamma function for any , as Selberg and Chowla proved in 1949.
The following expression is valid for all :
where is the Jacobi elliptic function delta amplitudinis with modulus .
By knowing one value, this formula can be used to compute related values:
where and is the Jacobi elliptic function sinus amplitudinis with modulus .
Further relations:
Lambda-star values of integer numbers of 4n-3-type:
Lambda-star values of integer numbers of 4n-2-type:
Lambda-star values of integer numbers of 4n-1-type:
Lambda-star values of integer numbers of 4n-type:
Lambda-star values of rational fractions:
Ramanujan's class invariants
Ramanujan's class invariants and are defined as
where . For such , the class invariants are algebraic numbers. For example
Identities with the class invariants include p. 240
The class invariants are very closely related to the Weber modular functions and . These are the relations between lambda-star and the class invariants:
Other appearances
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.Chandrasekharan (1985) p.121 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.Chandrasekharan (1985) p.118
Moonshine The function is the normalized Hauptmodul for the group , and its q-expansion , where , is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.
Footnotes
References Notes
Other
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.
Conway, J. H. and Norton, S. P. "Monstrous Moonshine." Bull. London Math. Soc. 11, 308-339, 1979.
Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J. reine angew. Math. 227, 86-110, 1967.
External links Modular lambda function at Fungrim