Modular lambda function

Modular lambda function in the complex plane.

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 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:

. OEISA115977

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

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The function is invariant under the group generated by[1]

The generators of the modular group act by[2]

Consequently, the action of the modular group on is that of the anharmonic group, giving the six values of the cross-ratio:[3]

Relations to other functions

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It is the square of the elliptic modulus,[4] that is, . In terms of the Dedekind eta function and theta functions,[4]

and,

where[5]

In terms of the half-periods of Weierstrass's elliptic functions, let be a fundamental pair of periods with .

we have[4]

Since the three half-period values are distinct, this shows that does not take the value 0 or 1.[4]

The relation to the j-invariant is[6][7]

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

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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]

where is the lemniscate sine and is the lemniscate constant.

Lambda-star

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Definition and computation of lambda-star

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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:

Properties of lambda-star

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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.[11][12]

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:[9]

where and is the Jacobi elliptic function sinus amplitudinis with modulus .

Further relations:

Special values

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

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Ramanujan's class invariants and are defined as[13]

where . For such , the class invariants are algebraic numbers. For example

Identities with the class invariants include[14]

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

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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 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]

Moonshine

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The function is the normalized Hauptmodul for the group , and its q-expansion , OEISA007248 where , is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.

Footnotes

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  1. ^ Chandrasekharan (1985) p.115
  2. ^ Chandrasekharan (1985) p.109
  3. ^ Chandrasekharan (1985) p.110
  4. ^ a b c d Chandrasekharan (1985) p.108
  5. ^ Chandrasekharan (1985) p.63
  6. ^ Chandrasekharan (1985) p.117
  7. ^ Rankin (1977) pp.226–228
  8. ^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 103–109, 134
  9. ^ a b Jacobi, Carl Gustav Jacob (1829). Fundamenta nova theoriae functionum ellipticarum (in Latin). p. 42
  10. ^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 152
  11. ^ Chowla, S.; Selberg, A. (1949). "On Epstein's Zeta Function (I)". Proceedings of the National Academy of Sciences. 35 (7): 373. doi:10.1073/PNAS.35.7.371. PMC 1063041. S2CID 45071481.
  12. ^ Chowla, S.; Selberg, A. "On Epstein's Zeta-Function". EuDML. pp. 86–110.
  13. ^ Berndt, Bruce C.; Chan, Heng Huat; Zhang, Liang-Cheng (6 June 1997). "Ramanujan's class invariants, Kronecker's limit formula, and modular equations". Transactions of the American Mathematical Society. 349 (6): 2125–2173.
  14. ^ Eymard, Pierre; Lafon, Jean-Pierre (1999). Autour du nombre Pi (in French). HERMANN. ISBN 2705614435. p. 240
  15. ^ Chandrasekharan (1985) p.121
  16. ^ Chandrasekharan (1985) p.118

References

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Notes

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  1. ^ 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".
  2. ^ For any prime power, we can iterate the modular equation of degree . This process can be used to give algebraic values of for any
  3. ^ is algebraic for every

Other

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  • 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.
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