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  "id": "enwiki-001365947878",
  "slug": "modular-lambda-function",
  "title": "Modular lambda function",
  "qid": "",
  "lead": "In mathematics, the modular lambda function λ(τ) 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 \\mathbb{C}/\\langle 1, \\tau \\rangle , where the map is defined as the quotient by the [−1] involution. The q-expansion, where q = e^{\\pi i \\tau} is the nome, is given by: : \\lambda(\\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \\dots . By symmetrizing the lambda function under the canonical action of the symmetric group S 3 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(\\mathbb{Z}) , and it is in fact Klein's modular j-invariant.",
  "body": "In mathematics, the modular lambda function λ(τ) 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 \\mathbb{C}/\\langle 1, \\tau \\rangle , where the map is defined as the quotient by the [−1] involution. The q-expansion, where q = e^{\\pi i \\tau} is the nome, is given by: \\lambda(\\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \\dots . By symmetrizing the lambda function under the canonical action of the symmetric group S 3 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(\\mathbb{Z}) , and it is in fact Klein's modular j-invariant. ==Modular properties== The function \\lambda(\\tau) is invariant under the group generated by \\tau \\mapsto \\tau+2 \\ ;\\ \\tau \\mapsto \\frac{\\tau}{1-2\\tau} \\ . The generators of the modular group act by \\tau \\mapsto \\tau+1 \\ :\\ \\lambda \\mapsto \\frac{\\lambda}{\\lambda-1} \\, ; \\tau \\mapsto -\\frac{1}{\\tau} \\ :\\ \\lambda \\mapsto 1 - \\lambda \\ . Consequently, the action of the modular group on \\lambda(\\tau) is that of the anharmonic group, giving the six values of the cross-ratio: \\left\\lbrace { \\lambda, \\frac{1}{1-\\lambda}, \\frac{\\lambda-1}{\\lambda}, \\frac{1}{\\lambda}, \\frac{\\lambda}{\\lambda-1}, 1-\\lambda } \\right\\rbrace \\ . ==Relations to other functions== It is the square of the elliptic modulus, that is, \\lambda(\\tau)=k^2(\\tau) . In terms of the Dedekind eta function \\eta(\\tau) and theta functions, \\lambda(\\tau) = \\Bigg(\\frac{\\sqrt{2}\\,\\eta(\\tfrac{\\tau}{2})\\eta^2(2\\tau)}{\\eta^3(\\tau)}\\Bigg)^8 = \\frac{16}{\\left(\\frac{\\eta(\\tau/2)}{\\eta(2\\tau)}\\right)^8 + 16} =\\frac{\\theta_2^4(\\tau)}{\\theta_3^4(\\tau)} and, \\frac{1}{\\big(\\lambda(\\tau)\\big)^{1/4}}-\\big(\\lambda(\\tau)\\big)^{1/4} = \\frac{1}{2}\\left(\\frac{\\eta(\\tfrac{\\tau}{4})}{\\eta(\\tau)}\\right)^4 = 2\\,\\frac{\\theta_4^2(\\tfrac{\\tau}{2})}{\\theta_2^2(\\tfrac{\\tau}{2})} where \\theta_2(\\tau) =\\sum_{n=-\\infty}^\\infty e^{\\pi i\\tau (n+1/2)^2} \\theta_3(\\tau) = \\sum_{n=-\\infty}^\\infty e^{\\pi i\\tau n^2} \\theta_4(\\tau) = \\sum_{n=-\\infty}^\\infty (-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=\\frac{\\omega_2}{\\omega_1} . e_1 = \\wp\\left(\\frac{\\omega_1}{2}\\right), \\quad e_2 = \\wp\\left(\\frac{\\omega_2}{2}\\right),\\quad e_3 = \\wp\\left(\\frac{\\omega_1+\\omega_2}{2}\\right) we have \\lambda = \\frac{e_3-e_2}{e_1-e_2} \\, . Since the three half-period values are distinct, this shows that \\lambda does not take the value 0 or 1. The relation to the j-invariant is j(\\tau) = \\frac{256(1-\\lambda(1-\\lambda))^3}{(\\lambda(1-\\lambda))^2} = \\frac{256(1-\\lambda+\\lambda^2)^3}{\\lambda^2 (1-\\lambda)^2} \\ . which is the j-invariant of the elliptic curve of Legendre form y^2=x(x-1)(x-\\lambda) Given m\\in\\mathbb{C}\\setminus\\{0,1\\} , let \\tau=i\\frac{K\\{1-m\\}}{K\\{m\\}} where K is the complete elliptic integral of the first kind with parameter m=k^2 . Then \\lambda (\\tau)=m. ==Modular equations== The modular equation of degree p (where p is a prime number) is an algebraic equation in \\lambda (p\\tau) and \\lambda (\\tau) . If \\lambda (p\\tau)=u^8 and \\lambda (\\tau)=v^8 , the modular equations of degrees p=2,3,5,7 are, respectively, (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_{k=1}^\\infty \\tanh\\frac{(k-1/2)\\pi i}{\\tau}=\\sqrt{2}e^{\\pi i\\tau/8}\\frac{\\sum_{k\\in\\mathbb{Z}}e^{(2k^2+k)\\pi i\\tau}}{\\sum_{k\\in\\mathbb{Z}}e^{k^2\\pi i\\tau}}\\\\ &=\\cfrac{\\sqrt{2}e^{\\pi i\\tau/8}}{1+\\cfrac{e^{\\pi i\\tau}}{1+e^{\\pi i\\tau}+\\cfrac{e^{2\\pi i\\tau}}{1+e^{2\\pi i\\tau}+\\cfrac{e^{3\\pi i\\tau}}{1+e^{3\\pi i\\tau}+\\ddots}}}}\\end{align} Since \\lambda(i)=1/2 , the modular equations can be used to give algebraic values of \\lambda(pi) for any prime p . The algebraic values of \\lambda(ni) are also given by \\lambda (ni)=\\prod_{k=1}^{n/2} \\operatorname{sl}^8\\frac{(2k-1)\\varpi}{2n}\\quad (n\\,\\text{even}) \\lambda (ni)=\\frac{1}{2^n}\\prod_{k=1}^{n-1} \\left(1-\\operatorname{sl}^2\\frac{k\\varpi}{n}\\right)^2\\quad (n\\,\\text{odd}) where \\operatorname{sl} is the lemniscate sine and \\varpi is the lemniscate constant. ==Lambda-star== ===Definition and computation of lambda-star=== The function \\lambda^*(x) (where x\\in\\mathbb{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: \\frac{K\\left[\\sqrt{1-\\lambda^*(x)^2}\\right]}{K[\\lambda^*(x)]} = \\sqrt{x} The values of \\lambda^*(x) can be computed as follows: \\lambda^*(x) = \\frac{\\theta^2_2(i\\sqrt{x})}{\\theta^2_3(i\\sqrt{x})} \\lambda^*(x) = \\left[\\sum_{a=-\\infty}^\\infty\\exp[-(a+1/2)^2\\pi\\sqrt{x}]\\right]^2\\left[\\sum_{a=-\\infty}^\\infty\\exp(-a^2\\pi\\sqrt{x})\\right]^{-2} \\lambda^*(x) = \\left[\\sum_{a=-\\infty}^\\infty\\operatorname{sech}[(a+1/2)\\pi\\sqrt{x}]\\right]\\left[\\sum_{a=-\\infty}^\\infty\\operatorname{sech}(a\\pi\\sqrt{x})\\right]^{-1} The functions \\lambda^* and \\lambda are related to each other in this way: \\lambda^*(x) = \\sqrt{\\lambda(i\\sqrt{x})} ===Properties of lambda-star=== Every \\lambda^* value of a positive rational number is a positive algebraic number: \\lambda^*(x) \\in \\overline{\\mathbb{Q}}_+ \\quad \\forall x\\in\\mathbb{Q}^+. K(\\lambda^*(x)) and E(\\lambda^*(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\\mathbb{Q}^+ , as Selberg and Chowla proved in 1949. The following expression is valid for all n \\in \\mathbb{N} : \\sqrt{n} = \\sum_{a = 1}^{n} \\operatorname{dn}\\left[\\frac{2a}{n}K\\left[\\lambda^*\\left(\\frac{1}{n}\\right)\\right];\\lambda^*\\left(\\frac{1}{n}\\right)\\right] where \\operatorname{dn} is the Jacobi elliptic function delta amplitudinis with modulus k . By knowing one \\lambda^* value, this formula can be used to compute related \\lambda^* values: \\lambda^*(n^2x) = \\lambda^*(x)^n\\prod_{a=1}^{n}\\operatorname{sn}\\left\\{\\frac{2a-1}{n}K[\\lambda^*(x)];\\lambda^*(x)\\right\\}^2 where n\\in\\mathbb{N} and \\operatorname{sn} is the Jacobi elliptic function sinus amplitudinis with modulus k . Further relations: \\lambda^*(x)^2 + \\lambda^*(1/x)^2 = 1 [\\lambda^*(x)+1][\\lambda^*(4/x)+1] = 2 \\lambda^*(4x) = \\frac{1-\\sqrt{1-\\lambda^*(x)^2}}{1+\\sqrt{1-\\lambda^*(x)^2}} = \\tan\\left\\{\\frac{1}{2}\\arcsin[\\lambda^*(x)]\\right\\}^2 \\lambda^*(x) - \\lambda^*(9x) = 2[\\lambda^*(x)\\lambda^*(9x)]^{1/4} - 2[\\lambda^*(x)\\lambda^*(9x)]^{3/4} \\begin{align} & a^{6}-f^{6} = 2af +2a^5f^5\\, &\\left(a = \\left[\\frac{2\\lambda^*(x)}{1-\\lambda^*(x)^2}\\right]^{1/12}\\right) &\\left(f = \\left[\\frac{2\\lambda^*(25x)}{1-\\lambda^*(25x)^2}\\right]^{1/12}\\right) \\\\ &a^{8}+b^{8}-7a^4b^4 = 2\\sqrt{2}ab+2\\sqrt{2}a^7b^7\\, &\\left(a = \\left[\\frac{2\\lambda^*(x)}{1-\\lambda^*(x)^2}\\right]^{1/12}\\right) &\\left(b = \\left[\\frac{2\\lambda^*(49x)}{1-\\lambda^*(49x)^2}\\right]^{1/12}\\right) \\\\ & a^{12}-c^{12} = 2\\sqrt{2}(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]^{1/12}\\right) &\\left(c = \\left[\\frac{2\\lambda^*(121x)}{1-\\lambda^*(121x)^2}\\right]^{1/12}\\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^{13}d^{13}\\, &\\left(a = \\left[\\frac{2\\lambda^*(x)}{1-\\lambda^*(x)^2}\\right]^{1/12}\\right) &\\left(d = \\left[\\frac{2\\lambda^*(169x)}{1-\\lambda^*(169x)^2}\\right]^{1/12}\\right) \\end{align} Lambda-star values of integer numbers of 4n-3-type: \\lambda^*(1) = \\frac{1}{\\sqrt{2}} \\lambda^*(5) = \\sin\\left[\\frac{1}{2}\\arcsin\\left(\\sqrt{5}-2\\right)\\right] \\lambda^*(9) = \\frac{1}{2}(\\sqrt{3}-1)(\\sqrt{2}-\\sqrt[4]{3}) \\lambda^*(13) = \\sin\\left[\\frac{1}{2}\\arcsin(5\\sqrt{13}-18)\\right] \\lambda^*(17) = \\sin\\left\\{\\frac{1}{2}\\arcsin\\left[\\frac{1}{64}\\left(5+\\sqrt{17}-\\sqrt{10\\sqrt{17}+26}\\right)^3\\right]\\right\\} \\lambda^*(21) = \\sin\\left\\{\\frac{1}{2}\\arcsin[(8-3\\sqrt{7})(2\\sqrt{7}-3\\sqrt{3})]\\right\\} \\lambda^*(25) = \\frac{1}{\\sqrt{2}}(\\sqrt{5}-2)(3-2\\sqrt[4]{5}) \\lambda^*(33) = \\sin\\left\\{\\frac{1}{2}\\arcsin[(10-3\\sqrt{11})(2-\\sqrt{3})^3]\\right\\} \\lambda^*(37) = \\sin\\left\\{\\frac{1}{2}\\arcsin[(\\sqrt{37}-6)^3]\\right\\} \\lambda^*(45) = \\sin\\left\\{\\frac{1}{2}\\arcsin[(4-\\sqrt{15})^2(\\sqrt{5}-2)^3]\\right\\} \\lambda^*(49) = \\frac{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) \\lambda^*(57) = \\sin\\left\\{\\frac{1}{2}\\arcsin[(170-39\\sqrt{19})(2-\\sqrt{3})^3]\\right\\} \\lambda^*(73) = \\sin\\left\\{\\frac{1}{2}\\arcsin\\left[\\frac{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: \\lambda^*(2) = \\sqrt{2}-1 \\lambda^*(6) = (2-\\sqrt{3})(\\sqrt{3}-\\sqrt{2}) \\lambda^*(10) = (\\sqrt{10}-3)(\\sqrt{2}-1)^2 \\lambda^*(14) = \\tan\\left\\{\\frac{1}{2}\\arctan\\left[\\frac{1}{8}\\left(2\\sqrt{2}+1-\\sqrt{4\\sqrt{2}+5}\\right)^3\\right]\\right\\} \\lambda^*(18) = (\\sqrt{2}-1)^3(2-\\sqrt{3})^2 \\lambda^*(22) = (10-3\\sqrt{11})(3\\sqrt{11}-7\\sqrt{2}) \\lambda^*(30) = \\tan\\left\\{\\frac{1}{2}\\arctan[(\\sqrt{10}-3)^2(\\sqrt{5}-2)^2]\\right\\} \\lambda^*(34) = \\tan\\left\\{\\frac{1}{4}\\arcsin\\left[\\frac{1}{9}(\\sqrt{17}-4)^2\\right]\\right\\} \\lambda^*(42) = \\tan\\left\\{\\frac{1}{2}\\arctan[(2\\sqrt{7}-3\\sqrt{3})^2(2\\sqrt{2}-\\sqrt{7})^2]\\right\\} \\lambda^*(46) = \\tan\\left\\{\\frac{1}{2}\\arctan\\left[\\frac{1}{64}\\left(3+\\sqrt{2}-\\sqrt{6\\sqrt{2}+7}\\right)^6\\right]\\right\\} \\lambda^*(58) = (13\\sqrt{58}-99)(\\sqrt{2}-1)^6 \\lambda^*(70) = \\tan\\left\\{\\frac{1}{2}\\arctan[(\\sqrt{5}-2)^4(\\sqrt{2}-1)^6]\\right\\} \\lambda^*(78) = \\tan\\left\\{\\frac{1}{2}\\arctan[(5\\sqrt{13}-18)^2(\\sqrt{26}-5)^2]\\right\\} \\lambda^*(82) = \\tan\\left\\{\\frac{1}{4}\\arcsin\\left[\\frac{1}{4761}(8\\sqrt{41}-51)^2\\right]\\right\\} Lambda-star values of integer numbers of 4n-1-type: \\lambda^*(3) = \\frac{1}{2\\sqrt{2}}(\\sqrt{3}-1) \\lambda^*(7) = \\frac{1}{4\\sqrt{2}}(3-\\sqrt{7}) \\lambda^*(11) = \\frac{1}{8\\sqrt{2}}(\\sqrt{11}+3)\\left(\\frac{1}{3}\\sqrt[3]{6\\sqrt{3}+2\\sqrt{11}}-\\frac{1}{3}\\sqrt[3]{6\\sqrt{3}-2\\sqrt{11}}+\\frac{1}{3}\\sqrt{11}-1\\right)^4 \\lambda^*(15) = \\frac{1}{8\\sqrt{2}}(3-\\sqrt{5})(\\sqrt{5}-\\sqrt{3})(2-\\sqrt{3}) \\lambda^*(19) = \\frac{1}{8\\sqrt{2}}(3\\sqrt{19}+13)\\left[\\frac{1}{6}(\\sqrt{19}-2+\\sqrt{3})\\sqrt[3]{3\\sqrt{3}-\\sqrt{19}}-\\frac{1}{6}(\\sqrt{19}-2-\\sqrt{3})\\sqrt[3]{3\\sqrt{3}+\\sqrt{19}}-\\frac{1}{3}(5-\\sqrt{19})\\right]^4 \\lambda^*(23) = \\frac{1}{16\\sqrt{2}}(5+\\sqrt{23})\\left[\\frac{1}{6}(\\sqrt{3}+1)\\sqrt[3]{100-12\\sqrt{69}}-\\frac{1}{6}(\\sqrt{3}-1)\\sqrt[3]{100+12\\sqrt{69}}+\\frac{2}{3}\\right]^4 \\lambda^*(27) = \\frac{1}{16\\sqrt{2}}(\\sqrt{3}-1)^3\\left[\\frac{1}{3}\\sqrt{3}(\\sqrt[3]{4}-\\sqrt[3]{2}+1)-\\sqrt[3]{2}+1\\right]^4 \\lambda^*(39) = \\sin\\left\\{\\frac{1}{2}\\arcsin\\left[\\frac{1}{16}\\left(6-\\sqrt{13}-3\\sqrt{6\\sqrt{13}-21}\\right)\\right]\\right\\} \\lambda^*(55) = \\sin\\left\\{\\frac{1}{2}\\arcsin\\left[\\frac{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: \\lambda^*(4) = (\\sqrt{2}-1)^2 \\lambda^*(8) = \\left(\\sqrt{2}+1-\\sqrt{2\\sqrt{2}+2}\\right)^2 \\lambda^*(12) = (\\sqrt{3}-\\sqrt{2})^2(\\sqrt{2}-1)^2 \\lambda^*(16) = (\\sqrt{2}+1)^2(\\sqrt[4]{2}-1)^4 \\lambda^*(20) = \\tan\\left[\\frac{1}{4}\\arcsin(\\sqrt{5}-2)\\right]^2 \\lambda^*(24) = \\tan\\left\\{\\frac{1}{2}\\arcsin[(2-\\sqrt{3})(\\sqrt{3}-\\sqrt{2})]\\right\\}^2 \\lambda^*(28) = (2\\sqrt{2}-\\sqrt{7})^2(\\sqrt{2}-1)^4 \\lambda^*(32) = \\tan\\left\\{\\frac{1}{2}\\arcsin\\left[\\left(\\sqrt{2}+1-\\sqrt{2\\sqrt{2}+2}\\right)^2\\right]\\right\\}^2 Lambda-star values of rational fractions: \\lambda^*\\left(\\frac{1}{2}\\right) = \\sqrt{2\\sqrt{2}-2} \\lambda^*\\left(\\frac{1}{3}\\right) = \\frac{1}{2\\sqrt{2}}(\\sqrt{3}+1) \\lambda^*\\left(\\frac{2}{3}\\right) = (2-\\sqrt{3})(\\sqrt{3}+\\sqrt{2}) \\lambda^*\\left(\\frac{1}{4}\\right) = 2\\sqrt[4]{2}(\\sqrt{2}-1) \\lambda^*\\left(\\frac{3}{4}\\right) = \\sqrt[4]{8}(\\sqrt{3}-\\sqrt{2})(\\sqrt{2}+1)\\sqrt{(\\sqrt{3}-1)^3} \\lambda^*\\left(\\frac{1}{5}\\right) = \\frac{1}{2\\sqrt{2}}\\left(\\sqrt{2\\sqrt{5}-2}+\\sqrt{5}-1\\right) \\lambda^*\\left(\\frac{2}{5}\\right) = (\\sqrt{10}-3)(\\sqrt{2}+1)^2 \\lambda^*\\left(\\frac{3}{5}\\right) = \\frac{1}{8\\sqrt{2}}(3+\\sqrt{5})(\\sqrt{5}-\\sqrt{3})(2+\\sqrt{3}) \\lambda^*\\left(\\frac{4}{5}\\right) = \\tan\\left[\\frac{\\pi}{4}-\\frac{1}{4}\\arcsin(\\sqrt{5}-2)\\right]^2 ===Ramanujan's class invariants=== Ramanujan's class invariants G_n and g_n are defined as G_n=2^{-1/4}e^{\\pi\\sqrt{n}/24}\\prod_{k=0}^\\infty \\left(1+e^{-(2k+1)\\pi\\sqrt{n}}\\right), g_n=2^{-1/4}e^{\\pi\\sqrt{n}/24}\\prod_{k=0}^\\infty \\left(1-e^{-(2k+1)\\pi\\sqrt{n}}\\right), where n\\in\\mathbb{Q}^+ . For such n , the class invariants are algebraic numbers. For example g_{58}=\\sqrt{\\frac{5+\\sqrt{29}}{2}}, \\quad g_{190}=\\sqrt{(\\sqrt{5}+2)(\\sqrt{10}+3)}. Identities with the class invariants include G_n=G_{1/n},\\quad g_{n}=\\frac{1}{g_{4/n}},\\quad g_{4n}=2^{1/4}g_nG_n. The class invariants are very closely related to the Weber modular functions \\mathfrak{f} and \\mathfrak{f}_1 . These are the relations between lambda-star and the class invariants: G_n = \\sin\\{2\\arcsin[\\lambda^*(n)]\\}^{-1/12} = 1\\Big /\\left[\\sqrt[12]{2\\lambda^*(n)}\\sqrt[24]{1-\\lambda^*(n)^2}\\right] g_n = \\tan\\{2\\arctan[\\lambda^*(n)]\\}^{-1/12} = \\sqrt[12]{[1-\\lambda^*(n)^2]/[2\\lambda^*(n)]} \\lambda^*(n) = \\tan\\left\\{ \\frac{1}{2}\\arctan[g_n^{-12}]\\right\\} = \\sqrt{g_n^{24}+1}-g_n^{12} ===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. 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. ===Moonshine=== The function \\tau\\mapsto 16/\\lambda(2\\tau) - 8 is the normalized Hauptmodul for the group \\Gamma_0(4) , and its q-expansion q^{-1} + 20q - 62q^3 + \\dots , where q=e^{2\\pi i\\tau } , 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",
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