In
mathematics, the modular lambda function λ(τ)[note 1] 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 curveX(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.
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.
A plot of x→ λ(ix)
Modular properties
The function is invariant under the group generated by[1]
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,[8]
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 .[note 2] The algebraic values of are also given by[9][note 3]
The function [10] (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:
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.[15] 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.[16]
^Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience.
ISBN0-471-83138-7. p. 103–109, 134
^Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience.
ISBN0-471-83138-7. p. 152
^ 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".
^For any
prime power, we can iterate the modular equation of degree . This process can be used to give algebraic values of for any
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.
In
mathematics, the modular lambda function λ(τ)[note 1] 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 curveX(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.
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.
A plot of x→ λ(ix)
Modular properties
The function is invariant under the group generated by[1]
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,[8]
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 .[note 2] The algebraic values of are also given by[9][note 3]
The function [10] (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:
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.[15] 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.[16]
^Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience.
ISBN0-471-83138-7. p. 103–109, 134
^Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience.
ISBN0-471-83138-7. p. 152
^ 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".
^For any
prime power, we can iterate the modular equation of degree . This process can be used to give algebraic values of for any
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.