Modular lambda function explained

C /\langle 1 , \tau \rangle , where the map is defined as the quotient by the [−1] involution.

The q-expansion, where

q = e ^{\pi} is the nome, is given by:λ (\tau) = 16 q - 128 q ^{2} + 704 q ^{3} - 3072 q ^{4} + 11488 q ^{5} - 38400 q ^{6} + ... .

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

\operatorname{SL} _{2(Z)} , and it is in fact Klein's modular j-invariant.

Modular properties

The function

λ (\tau) is invariant under the group generated by^[2] \tau \mapsto \tau +2 ; \tau \mapsto (\tau)/(1-2 \tau) .

The generators of the modular group act by^[3]

\tau \mapsto \tau +1 : λ \mapsto (λ)/(λ-1) ; \tau \mapsto - (1)/(\tau) : λ \mapsto 1 - λ .

Consequently, the action of the modular group on

λ (\tau) is that of the anharmonic group, giving the six values of the cross-ratio:^[4] \left\lbrace { λ , (1)/(1-λ) , (λ-1)/(λ) , (1)/(λ) , (λ)/(λ-1) , 1-λ } \right\rbrace .

Relations to other functions

It is the square of the elliptic modulus,^[5] that is,

λ (\tau) = k ^{2(\tau)} . In terms of the Dedekind eta function η (\tau) and theta functions,^[5] λ (\tau) = ( (\sqrt{ 2)/(η (\tfrac{\tau}{ 2 }) η) ^{2(2\tau)}{η} ^{3(\tau)})} ^{8} = (16)/(= \left( (η (\tau/ 2 ))/(\right) ^{8} + 16 η ( 2 \tau))) (\theta ^{4 (\tau)} _{2})/(\theta ^{4 (\tau)} _{3})

and,

(1)/(( λ (\tau) ) ^{1/4}) - ( λ (\tau) ) ^{1/4} = (1)/(\left( 2) (η (\tfrac{\tau)/(4) )}{ η (\tau)}\right) ^{4} =2 (\theta ^{2 (\tfrac{\tau} _{4})/(2) )}{\theta ^{2 (\tfrac{\tau}{ 2 })}} _{2}

where^[6]

\theta _{2(\tau)} = \sum ^{inf ty} _{n =-inf ty} e ^{\pi i\tau (n +1 / 2 ) ^{2}} \theta _{3(\tau)} = \sum ^{inf ty} _{n =-inf ty} e ^{\pi i\tau n ^{2}} \theta _{4(\tau)} = \sum ^{inf ty} _{n =-inf ty} ( -1 ) ^{n} e ^{\pi i\tau n ^{2}}

In terms of the half-periods of Weierstrass's elliptic functions, let

[\omega _{1,\omega} _{2]} be a fundamental pair of periods with \tau = (\omega _{2})/(\omega _{1}) .e _{1} = \wp\left( (\omega _{1})/(2) \right), e _{2} = \wp\left( (\omega _{2})/(2) \right), e _{3} = \wp\left( (\omega _{1+\omega} _{2})/(2) \right)

we have^[5]

λ = (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.^[5]

The relation to the j-invariant is^{[7] [8]}

j(\tau) = (256 ( 1-λ ( 1-λ )) ^{3})/(( λ ( 1-λ )) ^{2}) = (256 ( 1-λ+λ ^{2)} ^{3})/(λ ^{2} ( 1-λ ) ^{2}) . y ^{2=x(x-1)(x-λ)}

Given

m\in C \setminus\{ 0 , 1 \} , let\tau = i (K\{ 1- m\)/() }where K is the complete elliptic integral of the first kind with parameter m = k ^{2} .Thenλ (\tau) = m.

Modular equations

The modular equation of degree

p (where p is a prime number) is an algebraic equation in λ (p\tau) and λ (\tau) . If λ (p\tau) = u ^{8} and λ (\tau) = v ^{8} , the modular equations of degrees p =2 , 3 , 5 , 7 are, respectively,^[9] ( 1+ u ^{4)} ^{2v} ^{8-4u} ^{4=0,} u ^{4-v} ^{4+2uv(1-u} ^{2v} ^{2)=0,} u ^{6-v} ^{6+5u} ^{2v} ^{2(u} ^{2-v} ^{2)+4uv(1-u} ^{4v} ^{4)=0,} ( 1- u ^{8)(1-v} ^{8)-(1-uv)} ^{8=0.} The quantity v (and hence u ) can be thought of as a holomorphic function on the upper half-plane \operatorname{Im}\tau> 0 :\begin{align}v& = \prod ^{inf ty} _{k =1} \tanh ((k -1 / 2 )\pi i)/(\tau) = \sqrt{ 2 }e ^{\pi} (\sum _{k\in Z} e ^{( 2 k ^{2+k)\pi} i\tau})/(\sum _{k\in Z} e ^{k ^{2\pi} i\tau}) \\ & = \cfrac{\sqrt{ 2 }e ^{\pi} }\endSince λ (i) =1 / 2 , the modular equations can be used to give algebraic values of λ (pi) for any prime p .^[10] The algebraic values of λ (ni) are also given by^{[11] [12]} λ (ni) = \prod ^{n/ 2} _{k =1} \operatorname{sl} ^{8 (( 2 k -1 )\varpi)/(2 n)} (n even ) λ (ni) = (1)/(2 ^{n}) \prod ^{n -1} _{k =1} \left( 1- \operatorname{sl} ^{2 (k\varpi)/(n)} \right) ^{2} (n odd ) where \operatorname{sl} is the lemniscate sine and \varpi is the lemniscate constant.

Lambda-star

Definition and computation of lambda-star

The function

λ ^{*(x)} ^[13] (where x\in R ^{+} ) gives the value of the elliptic modulus k , for which the complete elliptic integral of the first kind K(k) and its complementary counterpart K(\sqrt{ 1- k ^{2})} are related by following expression:(K\left[\sqrt{ 1-λ ^{*(x)} ^{2})/(\right]}{K[ λ) ^{*(x)]}} = \sqrt{x}

The values of

λ ^{*(x)} can be computed as follows:λ ^{*(x)} = (\theta ^{2} _{2 (i\sqrt{x})/()}{\theta) ^{2} ^{3 (i\sqrt{x})}} λ ^{*(x)} = \left[\sum ^{inf ty\exp[ - (a +1 / 2 )} _{a =-inf ty} ^{2\pi\sqrt{x}]\right]} ^{inf ty\exp( - a} ^{a =-inf ty} ^{2\pi\sqrt{x})\right]} ^{-2} λ ^{*(x)} = \left[\sum ^{inf ty\operatorname{sech}(a\pi\sqrt{x})\right]} _{a =-inf ty} ^{-1}

The functions

λ ^{*} and λ are related to each other in this way:λ ^{*(x)} = \sqrt{ λ (i\sqrt{x})}

Properties of lambda-star

Every

λ ^{*} value of a positive rational number is a positive algebraic number:λ ^{*(x} \in Q ^{+)} \in A ^{+.} K( λ ^{*(x))} and E( λ ^{*(x))} (the complete elliptic integral of the second kind) can be expressed in closed form in terms of the gamma function for any x\in Q ^{+} , as Selberg and Chowla proved in 1949.^{[14] [15]}

The following expression is valid for all

n \in N :\sqrt{n} = \sum ^{n} _{a =1} \operatorname{dn}\left[ (2 a)/(n) K\left[ λ ^{*\left( (1)/(n)} \right)\right]; λ ^{*\left( (1)/(n)} \right)\right]

where

\operatorname{dn} is the Jacobi elliptic function delta amplitudinis with modulus k .

By knowing one

λ ^{*} value, this formula can be used to compute related λ ^{*} values:^[11] λ ^{*(n} ^{2x)} = λ ^{*(x)} \operatorname{sn}\left\{ ^{n} _{a =1} (2 a -1)/(n) K[ λ ^{*(x)];λ} ^{*(x)\right\}} ^{2}

where

n\in N and \operatorname{sn} is the Jacobi elliptic function sinus amplitudinis with modulus k .

Further relations:

λ ^{*(x)} ^{2} + λ ^{*(1/x)} ^{2} = 1 [ λ ^{*(x)+1][λ} ^{*(4/x)+1]} = 2 λ ^{*(4x)} = (1- \sqrt{ 1-λ ^{*(x)} ^{2})/() } = \tan\left\^2 λ ^{*(x)} - λ ^{*(9x)} = 2 [ λ ^{*(x)λ} ^{*(9x)]} ^{1/4} - 2 [ λ ^{*(x)λ} ^{*(9x)]} ^{3/4}

& a^-c^ = 2\sqrt(ac+a^3c^3)(1+3a^2c^2+a^4c^4)(2+3a^2c^2+2a^4c^4)\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^\right) &\left(c = \left[\frac{2\lambda^*(121x)}{1-\lambda^*(121x)^2}\right]^\right) \\

& (a^2-d^2)(a^4+d^4-7a^2d^2)[(a^2-d^2)^4-a^2d^2(a^2+d^2)^2] = 8ad+8a^d^\, &\left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^\right) &\left(d = \left[\frac{2\lambda^*(169x)}{1-\lambda^*(169x)^2}\right]^\right) \end

Lambda-star values of integer numbers of 4n-3-type:

λ ^{*(1)} = (1)/(\sqrt{ 2) }λ ^{*(5)} = \sin\left[ (1)/(2) \arcsin\left(\sqrt{ 5 } -2 \right)\right] λ ^{*(9)} = (1)/(2) (\sqrt{ 3 } -1 )(\sqrt{ 2 } - \sqrt[ 4 ]{ 3 }) λ ^{*(13)} = \sin\left[ (1)/(2) \arcsin( 5 \sqrt{ 13 } -18 )\right] λ ^{*(17)} = \sin\left\{ (1)/(\arcsin\left[ 2) (1)/(64) \left( 5+ \sqrt{ 17 } - \sqrt{ 10 \sqrt{ 17 } +26 }\right) ^{3\right]\right\}} λ ^{*(21)} = \sin\left\{ (1)/(2) \arcsin[( 8-3 \sqrt{ 7 })( 2 \sqrt{ 7 } -3 \sqrt{ 3 })]\right\} λ ^{*(25)} = (1)/(\sqrt{ 2) }(\sqrt-2)(3-2\sqrt[4])λ ^{*(33)} = \sin\left\{ (1)/(2) \arcsin[( 10-3 \sqrt{ 11 })( 2- \sqrt{ 3 }) ^{3]\right\}} λ ^{*(37)} = \sin\left\{ (1)/(2) \arcsin[(\sqrt{ 37 } -6 ) ^{3]\right\}} λ ^{*(45)} = \sin\left\{ (1)/(2) \arcsin[( 4- \sqrt{ 15 }) ^{2(\sqrt{5}-2)} ^{3]\right\}} λ ^{*(49)} = (1)/(4) ( 8+3 \sqrt{ 7 })( 5- \sqrt{ 7 } - \sqrt[ 4 ]{ 28 })\left(\sqrt{ 14 } - \sqrt{ 2 } - \sqrt[ 8 ]{ 28 }\sqrt{ 5- \sqrt{ 7 }}\right) λ ^{*(57)} = \sin\left\{ (1)/(2) \arcsin[( 170-39 \sqrt{ 19 })( 2- \sqrt{ 3 }) ^{3]\right\}} λ ^{*(73)} = \sin\left\{ (1)/(\arcsin\left[ 2) (1)/(64) \left( 45+5 \sqrt{ 73 } -3 \sqrt{ 50 \sqrt{ 73 } +426 }\right) ^{3\right]\right\}}

Lambda-star values of integer numbers of 4n-2-type:

λ ^{*(2)} = \sqrt{ 2 } -1 λ ^{*(6)} = ( 2- \sqrt{ 3 })(\sqrt{ 3 } - \sqrt{ 2 }) λ ^{*(10)} = (\sqrt{ 10 } -3 )(\sqrt{ 2 } -1 ) ^{2} λ ^{*(14)} = \tan\left\{ (1)/(\arctan\left[ 2) (1)/(8) \left( 2 \sqrt{ 2 } +1- \sqrt{ 4 \sqrt{ 2 } +5 }\right) ^{3\right]\right\}} λ ^{*(18)} = (\sqrt{ 2 } -1 ) ^{3(2-\sqrt{3})} ^{2} λ ^{*(22)} = ( 10-3 \sqrt{ 11 })( 3 \sqrt{ 11 } -7 \sqrt{ 2 }) λ ^{*(30)} = \tan\left\{ (1)/(2) \arctan[(\sqrt{ 10 } -3 ) ^{2(\sqrt{5}-2)} ^{2]\right\}} λ ^{*(34)} = \tan\left\{ (1)/(\arcsin\left[ 4) (1)/(9) (\sqrt{ 17 } -4 ) ^{2\right]\right\}} λ ^{*(42)} = \tan\left\{ (1)/(2) \arctan[( 2 \sqrt{ 7 } -3 \sqrt{ 3 }) ^{2(2\sqrt{2}-\sqrt{7})} ^{2]\right\}} λ ^{*(46)} = \tan\left\{ (1)/(\arctan\left[ 2) (1)/(64) \left( 3+ \sqrt{ 2 } - \sqrt{ 6 \sqrt{ 2 } +7 }\right) ^{6\right]\right\}} λ ^{*(58)} = ( 13 \sqrt{ 58 } -99 )(\sqrt{ 2 } -1 ) ^{6} λ ^{*(70)} = \tan\left\{ (1)/(2) \arctan[(\sqrt{ 5 } -2 ) ^{4(\sqrt{2}-1)} ^{6]\right\}} λ ^{*(78)} = \tan\left\{ (1)/(2) \arctan[( 5 \sqrt{ 13 } -18 ) ^{2(\sqrt{26}-5)} ^{2]\right\}} λ ^{*(82)} = \tan\left\{ (1)/(\arcsin\left[ 4) (1)/(4761) ( 8 \sqrt{ 41 } -51 ) ^{2\right]\right\}}

Lambda-star values of integer numbers of 4n-1-type:

λ ^{*(3)} = (1)/(2 \sqrt{ 2) }(\sqrt-1)λ ^{*(7)} = (1)/(4 \sqrt{ 2) }(3-\sqrt)λ ^{*(11)} = (1)/(8 \sqrt{ 2) }(\sqrt+3)\left(\frac\sqrt[3]-\frac\sqrt[3]+\frac\sqrt-1\right)^4λ ^{*(15)} = (1)/(8 \sqrt{ 2) }(3-\sqrt)(\sqrt-\sqrt)(2-\sqrt)λ ^{*(19)} = (1)/(8 \sqrt{ 2) }(3\sqrt+13)\left[\frac{1}{6}(\sqrt{19}-2+\sqrt{3})\sqrt[3]-\frac(\sqrt-2-\sqrt)\sqrt[3]-\frac(5-\sqrt)\right]^4λ ^{*(23)} = (1)/(16 \sqrt{ 2) }(5+\sqrt)\left[\frac{1}{6}(\sqrt{3}+1)\sqrt[3]-\frac(\sqrt-1)\sqrt[3]+\frac\right]^4λ ^{*(27)} = (1)/(16 \sqrt{ 2) }(\sqrt-1)^3\left[\frac{1}{3}\sqrt{3}(\sqrt[3]-\sqrt[3]+1)-\sqrt[3]+1\right]^4λ ^{*(39)} = \sin\left\{ (1)/(\arcsin\left[ 2) (1)/(16) \left( 6- \sqrt{ 13 } -3 \sqrt{ 6 \sqrt{ 13 } -21 }\right)\right]\right\} λ ^{*(55)} = \sin\left\{ (1)/(\arcsin\left[ 2) (1)/(512) \left( 3 \sqrt{ 5 } -3- \sqrt{ 6 \sqrt{ 5 } -2 }\right) ^{3\right]\right\}}

Lambda-star values of integer numbers of 4n-type:

λ ^{*(4)} = (\sqrt{ 2 } -1 ) ^{2} λ ^{*(8)} = \left(\sqrt{ 2 } +1- \sqrt{ 2 \sqrt{ 2 } +2 }\right) ^{2} λ ^{*(12)} = (\sqrt{ 3 } - \sqrt{ 2 }) ^{2(\sqrt{2}-1)} ^{2} λ ^{*(16)} = (\sqrt{ 2 } +1 ) ^{2(\sqrt[4]{2}-1)} ^{4} λ ^{*(20)} = \tan\left[ (1)/(4) \arcsin(\sqrt{ 5 } -2 )\right] ^{2} λ ^{*(24)} = \tan\left\{ (1)/(2) \arcsin[( 2- \sqrt{ 3 })(\sqrt{ 3 } - \sqrt{ 2 })]\right\} ^{2} λ ^{*(28)} = ( 2 \sqrt{ 2 } - \sqrt{ 7 }) ^{2(\sqrt{2}-1)} ^{4} λ ^{*(32)} = \tan\left\{ (1)/(2) \arcsin\left[\left(\sqrt{ 2 } +1- \sqrt{ 2 \sqrt{ 2 } +2 }\right) ^{2\right]\right\}} ^{2}

Lambda-star values of rational fractions:

λ ^{*\left( (1)/(2)} \right) = \sqrt{ 2 \sqrt{ 2 } -2 } λ ^{*\left( (1)/(3)} \right) = (1)/(2 \sqrt{ 2) }(\sqrt+1)λ ^{*\left( (2)/(3)} \right) = ( 2- \sqrt{ 3 })(\sqrt{ 3 } + \sqrt{ 2 }) λ ^{*\left( (1)/(4)} \right) = 2 \sqrt[ 4 ]{ 2 }(\sqrt{ 2 } -1 ) λ ^{*\left( (3)/(4)} \right) = \sqrt[ 4 ]{ 8 }(\sqrt{ 3 } - \sqrt{ 2 })(\sqrt{ 2 } +1 )\sqrt{(\sqrt{ 3 } -1 ) ^{3}} λ ^{*\left( (1)/(5)} \right) = (1)/(2 \sqrt{ 2) }\left(\sqrt+\sqrt-1\right)λ ^{*\left( (2)/(5)} \right) = (\sqrt{ 10 } -3 )(\sqrt{ 2 } +1 ) ^{2} λ ^{*\left( (3)/(5)} \right) = (1)/(8 \sqrt{ 2) }(3+\sqrt)(\sqrt-\sqrt)(2+\sqrt)λ ^{*\left( (4)/(5)} \right) = \tan\left[ (\pi)/(- 4) (1)/(4) \arcsin(\sqrt{ 5 } -2 )\right] ^{2}

Ramanujan's class invariants

Ramanujan's class invariants

G _{n} and g _{n} are defined as^[16] G ^{-1 / 4} _{n =2} e ^{\pi\sqrt{n} / 24 }\prod ^{inf ty} _{k =0} \left( 1+ e ^{-(2k+1)\pi\sqrt{n} }\right), g ^{-1 / 4} _{n =2} e ^{\pi\sqrt{n} / 24 }\prod ^{inf ty} _{k =0} \left( 1- e ^{-(2k+1)\pi\sqrt{n} }\right),where n\in Q ^{+} . For such n , the class invariants are algebraic numbers. For exampleg _{58} = \sqrt{ (5+ \sqrt{ 29)/() }}, \quad g_=\sqrt.

Identities with the class invariants include^[17]

G _{n=G} _{1/n} , g _{n} = (1)/(g _{4/n}) , g _{4n} =2 ^{1/4} g _{nG} _{n.} ak{f} and ak{f} _{1} . These are the relations between lambda-star and the class invariants:G _{n} = \sin\{ 2 \arcsin[ λ ^{*(n)]\}} ^{-1/12} = 1 /\left[\sqrt[ 12 ]{ 2λ ^{*(n)}\sqrt[24]{1-λ} ^{*(n)} ^{2}\right]} g _{n} = \tan\{ 2 \arctan[ λ ^{*(n)]\}} ^{-1/12} = \sqrt[ 12 ]{[ 1-λ ^{*(n)} ^{2]/[2λ} ^{*(n)]}} λ ^{*(n)} = \tan\left\{ (1)/(2) \arctan[g ^{-12} _{n} ]\right\} = \sqrt{g ^{24} _{n} +1 } - g ^{12} _{n}

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.^[18] 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.^[19]

Moonshine

The function

\tau\mapsto 16 / λ ( 2 \tau) - 8 is the normalized Hauptmodul for the group \Gamma _{0(4)} , and its q-expansion q ^{-1} + 20 q - 62 q ^{3} + ... , where q = e ^{2\pi} , is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.

References

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

Notes and References

1. λ (\tau) is not a modular function (per the Wikipedia definition), but every modular function is a rational function in λ (\tau) . Some authors use a non-equivalent definition of "modular functions".
2. Chandrasekharan (1985) p.115
3. Chandrasekharan (1985) p.109
4. Chandrasekharan (1985) p.110
5. Chandrasekharan (1985) p.108
6. Chandrasekharan (1985) p.63
7. Chandrasekharan (1985) p.117
8. Rankin (1977) pp.226–228
9. Book: Borwein . Jonathan M. . Borwein. Peter B. . Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity . Wiley-Interscience . 1987 . First . 0-471-83138-7. p. 103–109, 134
10. For any prime power, we can iterate the modular equation of degree p . This process can be used to give algebraic values of λ (ni) for any n\in N .
11. Book: Jacobi . Carl Gustav Jacob . Carl Gustav Jacob Jacobi. Fundamenta nova theoriae functionum ellipticarum. Latin. 1829. p. 42
12. \operatorname{sl}a\varpi is algebraic for every a\in Q .
13. Book: Borwein . Jonathan M. . Borwein. Peter B. . Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity . Wiley-Interscience . 1987 . First . 0-471-83138-7. p. 152
14. On Epstein's Zeta Function (I).. Chowla. S.. Selberg. A.. Proceedings of the National Academy of Sciences . 1949 . 35 . 7 . 373. 10.1073/PNAS.35.7.371 . 45071481 . free. 1063041.
15. Web site: On Epstein's Zeta-Function. Chowla. S.. Selberg. A.. EuDML. 86–110.
16. 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.
17. Book: Eymard . Pierre . Lafon. Jean-Pierre . Autour du nombre Pi . French. HERMANN . 1999 . 2705614435. p. 240
18. Chandrasekharan (1985) p.121
19. Chandrasekharan (1985) p.118