Modular lambda function

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.

Modular lambda function in the complex plane.

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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