SOME MODULAR EQUATIONS IN THE FORM OF SCHLÄFLI 1

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1 italian journal of pure and applied mathematics n ) SOME MODULAR EQUATIONS IN THE FORM OF SCHLÄFLI 1 M.S. Mahadeva Naika Department of Mathematics Bangalore University Central College Campus Bengaluru India msmnaika@rediffmail.com K. Sushan Bairy G Department of Mathematics Vijaya College R.V. Road Basavanagudi Bengaluru India ksbairy@rediffmail.com Abstract. On page 90 of his first notebook S. Ramanujan records Schläfli-type modular equations for degrees and 19. In this paper we establish Schläfli-type modular equations for degrees and 19 which are recorded by Ramanujan in his first notebook. We also establish several new Schläfli-type modular equations of degrees and 71. As an application we deduce some explicit evaluations of Ramanujan-Weber class invariants. Keywords: Schläfli-type modular equation Class invariant. 010 Mathematics Subject Classification: 11B65 D15 11F7. 1. Introduction In 1] Schläfli established modular equations for degrees and 19 which were also recorded by S. Ramanujan on page 90 of his first notebook 1]. In 11] Ramanathan gave a proof of the equation of degree 11. In 14] G.N. Watson gave a proof of Schläfli-type modular equation of degree thirteen and also examined Schläfli-type modular equations in 15]. Using the theory of modular forms Berndt pp ] pp ] has verified these modular equations. Recently William Hart 6] has proved several Schläfli-type modular equations of degrees and 17 by using modular forms in different levels. 1 Research supported by DST grant SR/S4/MS:509/07 Govt. of India.

2 4 m.s. mahadeva naika k. sushan bairy We define 1.1) Kk) := π 0 dϕ 1 k sin ϕ = π n=0 1 ) n n!) kn = π 1 F 1 1 ) ; 1; k where 0 < k < 1 and F 1 is the ordinary or Gaussian hypergeometric function defined by a) F 1 a b; c; z) := n b) n c) n n! zn 0 z < 1 where n=0 a) 0 = 1 a) n = aa + 1) a + n 1) for n a positive integer and a b c are complex numbers such that c The number k is called the modulus of K and k := 1 k is called the complementary modulus. Let K K L and L denote the complete elliptic integrals of the first kind associated with the moduli k k l and l respectively. Suppose that the equality 1.) n K K = L L holds for some positive integer n. Then a modular equation of degree n is a relation between the moduli k and l which is induced by 1.). Following Ramanujan set α = k and β = l. Then we say β is of degree n over α. The multiplier m is defined by 1.) m = K L. Ramanujan s class invariant G n is defined by 1.4) G n := 1/4 q 1/4 χq) = {4α1 α)} 1/4 where χq) = q; q ) q = exp π n) and a; q) := 1 aq n ) q < 1. n=0 M.S. Mahadeva Naika 8] and Mahadeva Naika and K. Sushan Bairy 9] have obtained several new explicit evaluations of the Ramanujan-Weber class invariants using modular equations. In this paper we establish several modular equations in the form of Schläfli for degrees and 71. As an application we obtain several explicit evaluations of Ramanujan-Weber class invariants.

3 some modular equations in the form of schläfli 5. reliminary results In this section we collect several results which are useful in proving our main Schläfli-type modular equations. Lemma.1. Eq. 4.1) p. 15] If β is of degree over α then.1) β = ) 1 1 α α Lemma.. Eq. 4.) p. 15] If β is of degree 4 over α then.) β = ) α α Lemma.. Entry Ch.6 pp ] Let.).4).5) u = 1 αβ 1 α)1 β) ] v = 64 αβ + 1 α)1 β) αβ1 α)1 β) w = αβ1 α)1 β). 1. If β is of degree 9 over α then.6) u 6 w 14u + uv ) w = 0.. If β is of degree 1 over α then.7) u u + 8w ) w 11u + v ) = 0.. If β is of degree 17 over α then.8) u w 1/ 10u + v ) + 1w / u + 1w = If β is of degree 9 over α then.9) u u + 17uw 1/ 9w /) w 1/6 9u + v 1uw 1/ + 15w /) = 0. Lemma.4. Entry Ch.6 p. 85] Let.10).11).1) U = 1 ± αβ) 1/8 ± 1 α)1 β)] 1/8 ] V = 4 αβ) 1/8 + {1 α)1 β)} 1/8 ± {αβ1 α)1 β)} 1/8 W = 4{αβ1 α)1 β)} 1/8.

4 6 m.s. mahadeva naika k. sushan bairy 1. Let U V and W be given by.10).1) with the plus signs taken. If β is of degree 15 over α then.1) U U V ) + W = 0.. Let U V and W be given by.10).1) with the plus signs taken. If β is of degree 1 over α then.14) U V = UW.. Let U V and W be given by.10).1) with the plus signs taken. If β is of degree 47 over α then.15) U V UW 1/ W / = Let U V and W be given by.10).1) with the minus signs taken. If β is of degree 71 over α then.16) U W 1/ 4U + V ) + UW / W = 0. Lemma.5. Entry 7 Ch. 0 p. 6] If β is of degree 11 over α then.17) αβ) 1/4 + {1 α)1 β)} 1/4 + 16αβ1 α)1 β)] 1/1 = 1. Lemma.6. Entry 58 Ch. 6 p. 86] Let.18).19).0) A = 1 αβ) 1/4 1 α)1 β)] 1/4 ] B = 16 αβ) 1/4 + {1 α)1 β)} 1/4 {αβ1 α)1 β)} 1/4 C = 16{αβ1 α)1 β)} 1/4. If β is of degree 19 over α then.1) A 5 7A C BC = 0. Lemma.7. Entry 15 Ch. 0 p. 411] If β is of degree over α then.) αβ) 1/8 + {1 α)1 β)} 1/8 + / {αβ1 α)1 β)} 1/4 = 1. Lemma.8. Entry 15 i) ii) Ch. 19 p. 91] If β is of degree 5 over α then.) ) 1/8 β + α ) 1/8 1 β 1 α ) 1/8 β1 β) α1 α) ) 1/1 β1 β) = mm ) 1/ α1 α).4) ) 1/8 α + β ) 1/8 1 α 1 β ) 1/8 α1 α) β1 β) ) 1/1 α1 α) = β1 β) 5 mm ) 1/.

5 some modular equations in the form of schläfli 7 Lemma.9 Identity Theorem). Suppose fz) is analytic in a domain D and that {z n } is a sequence of distinct points converging to a point z 0 in D. If fz n ) = 0 for each n then fz) 0 throughout D.. Main results In this section we prove Schläfli-type modular equations of degrees and 19 recorded by Ramanujan in his notebooks. We also establish several new Schläfli-type modular equations of degrees and 71. First we set.1) := {16αβ1 α)1 β)} 1/4 and.) Q := { } 1/4 β1 β) α1 α) where β is of degree n over α. Using.1) and.) we obtain the following lemma. Lemma.1. We have.) α = 1 + r and.4) β = 1 + s where r = ± 1 1 Q and s = ± 1 1 Q 1. 1 Theorem.1. If β is of degree over α then 6.5) Q ] ] ] = 0. 8 Q where and Q are defined as in.1) and.) respectively. roof. Equation.1) can be written as.6) 1 a b = ab where a = 1 α and b = β. Squaring both sides of equation.6) we find that.7) 1 a b + a + ab + b a b = 0.

6 8 m.s. mahadeva naika k. sushan bairy Isolating the terms containing a on one side of the equation.7) squaring both sides and then using.) and.4) we deduce that.8) r rs + rs + + s 48b + 0s 16sb = 0. Again isolating the terms containing b on one side of the equation.8) squaring both sides and then using.) and.4) we find that.9) 56rs 1 Q Q Q 1 s 16rQ s + 16rsQ Q 4 4 Q 6 + 4s 4 Q 1 48r 1 Q 4 7s 1 Q 4 r 4 Q Q Q 4 = 0. Eliminating r and s in the same manner we deduce that.10) 56Q Q Q 1 8 Q Q 4 4 ) Q Q 4 + Q Q 4 64Q Q Q Q Q Q Q Q Q Q Q Q 4 ) = 0. By examining the factors near q = 0 it can be seen that there is a neighbourhood about the origin where the first factor vanish but the second factor does not. By the Identity Theorem.9 the first factor vanishes identically. Hence we obtain.5). Theorem.. If β is of degree 4 over α then Q 8 + 1Q ) ) Q 4 + 1Q ) ) 4.11) Q Q ) 1 Q8 Q Q ) = 0. Q 16 where and Q are defined as in.1) and.) respectively Q )] Q 1 roof of the identity.11) is similar to the proof of the identity.5) given above except that in place of result.1) result.) is used. Theorem.. If β is of degree 9 over α then ) 1 + Q + 1Q ] 1 = 94 + Q ) + 8 Q ) Q 18 Q.1) Q ) Q 9 + 1Q ) Q Q 6 + 1Q ) Q + 1Q ) 6

7 some modular equations in the form of schläfli 9 where and Q are defined as in.1) and.) respectively. roof. Using.1) in.).5) we find that.1).14).15) u = 1 a 1 1 4a v = 64a a 1 w = 8 1 where a 1 = {αβ} 1/. Using.1).15) in.6) we deduce that.16) 480a 4 a aa a a a a a a a a a a a a 4 a 1 180a a a a a a a a a 5 a a a a a a a a a a a a 6 = 0 where a = αβ. Isolating the terms involving a 1 on one side of the equation squaring both sides and eliminating r and s we find that.17) 64 4 Q Q 1 64Q Q Q + 1 Q Q Q Q Q Q Q Q Q Q Q Q Q 4 ) Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 54 ) 64 4 Q Q 1 64Q 18 64Q Q + 1 Q Q Q 64 4 Q 15 64Q Q Q Q Q Q Q Q Q 4 ) = 0.

8 40 m.s. mahadeva naika k. sushan bairy utting n = 1/9 in.1) and.) we find that.18) = 1 G 9 and Q = 1. Using.18) in.17) we find that the last factor vanishes for the specific value of q = e π 1/9 then the last factor vanishes in a neighbourhood of q = e π 1/9. This proves the theorem. Theorem.4. 1 p. 90] Entry 41 p. 78] If β is of degree 11 over α then.19) Q Q = ) where and Q are defined as in.1) and.) respectively. roof. Using.1) in.17) we find that.0) {αβ} 1/4 + Squaring both sides of.0) we find that 6 {16αβ} 1/4 = ) {αβ} 1/ + {16αβ} = 1 ) 6. 1/ Again squaring both sides of.1) we find that.) {αβ} + 4 {16αβ} = 1 ) 6 Using.) and.4) in.) we find that ) 1..) 0 Q Q 1 10 Q 4 5 Q Q Q Q Q Q Q 1 + Q Q 1 = 0. Simplifying the above equation we obtain the required result.19). Theorem.5. 1 p. 90] Entry 41 p. 78] If β is of degree 1 over α then Q 7 + 1Q ) + 1 Q 5 + 1Q ) + 5 Q + 1Q ).4) 7 ) Q + 1 Q = ) where and Q are defined as in.1) and.) respectively.

9 some modular equations in the form of schläfli 41 roof. Using.1).15) in.7) we deduce that a a a a a a a a a 4 a a 5 a a a a a a 1 4.5) a a a 4 a 1 4 8a a a a a a a a a a a 4 a a 5 a a 6 a a a a a a a a a a a a 1 + 8a 7 + 6a 60 = 0. Isolating the terms involving a 1 on one side of the above equation squaring on both sides and then eliminating r and s we find that.6) 8Q Q 7 + 5Q Q Q Q Q Q Q 8 ) 8Q Q 7 + 5Q Q Q Q Q Q Q 1 ) Q Q Q Q Q 14 6Q Q Q Q Q Q Q Q Q Q Q 4 1 )178 1 Q Q Q Q Q Q Q Q Q Q Q 4 1 1Q Q Q Q Q Q Q Q 1 080Q Q Q Q Q Q Q Q Q Q Q Q Q 17 6 )178 1 Q Q Q Q Q Q Q 14 64Q Q Q Q 4 1 1Q 6 1 8Q Q Q Q Q Q Q 1 080Q Q Q Q Q Q Q Q Q Q Q Q 7 6 8Q 17 6 )4096Q Q Q 8 + 6Q Q Q Q Q 6 8Q Q Q Q Q Q Q Q Q Q Q Q Q 8 4

10 4 m.s. mahadeva naika k. sushan bairy 64Q Q Q Q Q Q Q Q Q 4 096Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 6 1 8Q Q Q Q Q Q 56 4 ) = 0. By examining the factors near q = 0 it can be seen that there is a neighbourhood about the origin where the first factor vanish but the other factors does not. By the Identity Theorem the first factor vanishes identically. This proves the theorem. Theorem.6. If β is of degree 15 over α then Q Q + Q ) ) 6 Q ) ) ] Q ) { ) ) Q ) ) ) ) 9.7) )} ] { = ) ) ) ) ) ) ) )} where and Q are defined as in.1) and.) respectively.

11 some modular equations in the form of schläfli 4 roof of the identity.7) is similar to the proof of the identity.4) given above except that in place of result.7) result.1) is used. Theorem.7. 1 p. 90] Entry 41 p. 78] If β is of degree 17 over α then Q 9 + 1Q ) Q + 1Q ) 4 Q 6 + 1Q ) ) 9 6.8) Q + 1Q ) ) ) + 40 = where and Q are defined as in.1) and.) respectively. The proof of the identity.8) is similar to the proof of the identity.4) except that in place of result.7) result.8) is used. Theorem.8. 1 p. 90] Entry 41 pp ] If β is of degree 19 over α then Q Q 6 + 1Q ) Q 4 + 1Q ) 1 ) Q ) + 19 Q + 1 ) ) 8 Q ) = 0 where and Q are defined as in.1) and.) respectively. roof. Using.1) in.18).0) we find that.0).1).) A = 1 c c 1 B = 16c C = 8 6. c where c 1 = {αβ} 1/4. Using.0).) in.1) we find that 1040a 1 6 a 640a 4 a a 1 a a 1 0 a a 1 a a 1 4 a + 104a a 6 a a 0 a a 6 a a 1 a 1.) a 18 a a 1 a a 1 4 a 104a 1 a a a + 180a a a a a a a 1 a + 915a a a a a 1 a a a a a a 4 1 0a = 0. Isolating the terms involving a 1 to one side of the above equation and squaring both sides substituting a = 1+r)1+s)/4 and eliminating r and s we find that

12 44 m.s. mahadeva naika k. sushan bairy.4) Q Q Q Q Q Q Q Q Q Q Q Q Q Q 1 740Q Q 6 51Q Q Q Q Q 984Q Q Q Q Q Q Q 51Q Q 740Q Q 8 18 ) Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 60 04Q Q Q Q 7 ) +9178Q Q Q Q Q Q Q Q Q Q Q Q Q 5 1 ) 1040Q Q Q Q Q 4 8Q Q Q 6 ) Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q + Q Q Q 1196Q Q Q Q 1 ) 1479Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 44 ) = 0.

13 some modular equations in the form of schläfli 45 By examining the factors near q = 0 it can be seen that there is a neighbourhood about the origin where the first factor vanish but the other factors does not. By the Identity Theorem the first factor vanishes identically. This proves the theorem. Theorem.9. If β is of degree over α then Q = ) ) Q ) ) ) 9 8.5) ) ) ) ) ) )] where and Q are defined as in.1) and.) respectively. The proof of the identity.5) is similar to the proof of the identity.19) except that in place of result.17) result.) is used. Theorem.10. If β is of degree 5 over α then ) 1 + Q + 1 ) + Q + 1Q )] = Q ) 1 Q Q Q ) + 01 Q ) Q ) Q 14 Q 1 Q Q ) Q ) Q 9 + 1Q ) Q 11 Q.6) Q 8 + 1Q ) Q 7 + 1Q ) Q 6 + 1Q ) Q 5 + 1Q ) Q 4 + 1Q ) Q + 1Q ) Q + 1Q ) Q + 1 ) Q where and Q are defined as in.1) and.) respectively. roof. Using.) in.) and.4) we find that.7) d 1Q + d 4 1 d 1 d 1Q + Q 6 Q d 1 Q 4 d 1 Q 6 d 1 Q d 1 + Q 4 d 1 Q 5 d 1 Q d 1 + Q d 1 = 0 ) 1/8 β where d 1 =. Substituting d 1 = d and isolating the terms involving d 1 α on one side of equation squaring both sides and again substituting d = d and d = d 4 we find that

14 46 m.s. mahadeva naika k. sushan bairy.8) 4d Q d 4d d Q 4d d Q 5 4d Q 4 d d d Q 6 d Q 9 6d Q d 4Q 8 d 4Q 4 d d Q d d +d 4 6d Q 9 6d Q 6 4Q 10 d 4Q 8 d d Q 6 4d Q 7 d Q 1 4d Q 11 + Q 1 8d Q 7 8d Q 5 = 0. Now isolating the terms involving d on one side of the above equation squaring both sides we find that d 4 d Q 1 d d Q 4 48d Q 1 104d Q d Q 18 + d 4 Q 18.9) 80Q 0 d + Q 4 80Q 16 d + d 4 Q 6 48d Q 15 + d 4 Q 1 104Q 19 d 4Q 14 d 8Q 1 d 4d Q 8d Q 8d 4 d Q d Q 5 d 4 80Q 4 d d 4 48d Q 9 d 4 80Q 8 d d 4 104d 4 d Q 7 48d Q d 4 114d Q 6 d 4 4Q d 4 d 8d 4 d Q d 4 d Q 1 4d 4 d Q 10 + d 4 = 0 where d 4 = β α. Isolating the terms involving d on one side of the above equation and squaring on both sides and eliminating r and s we find that.40) 64Q 15 64Q 17 64Q 16 64Q 14 64Q Q Q Q 1 + 6Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 15 64Q 17 4 ) Q + Q + 1) 4 Q + 1) 4 Q 8 Q 4 + 1) Q 4 + 1) Q + 1) = 0. By examining the factors near q = 0 it can be seen that there is a neighbourhood about the origin where the first factor vanish but the other factors does not. By the Identity Theorem the first factor vanishes identically. This proves the theorem. Theorem.11. If β is of degree 9 over α then Q ) 58 1 ) Q ) Q15 14 Q ) +185 ]Q 9 + 1Q ) )Q 6 + 1Q ) ) ) ) ] Q + 1Q ) 8 4 = ) ) ) 6 10 where and Q are defined as in.1) and.) respectively.

15 some modular equations in the form of schläfli 47 Theorem.1. If β is of degree 1 over α then.4) Q Q Q Q 4 + 1Q ) ) ) ) Q 4 + 1Q ) 4 Q 1 ) Q 8 + 1Q ) ) ) Q 4 + 1Q ) 1744 ] Q 4 + 1Q ) ] ) Q 8 + 1Q ) ] = Q 8 + 1Q ) 8 Q Q where and Q are defined as in.1) and.) respectively. Theorem.1. If β is of degree 47 over α then.4) Q Q 1 1 ) ) Q4 Q ) ) ) ) ) ) ) ) ) ) ] = ) ) ] ] ] ] ] ] ] ] 14 1

16 48 m.s. mahadeva naika k. sushan bairy ] ] ] ] ] ] ] ] ] ] ] ] where and Q are defined as in.1) and.) respectively. Theorem.14. If β is of degree 71 over α then Q Q ) ] Q6 Q ] ] ] ] ] ] ] ] ] ]).44) Q ) ] ] Q ] ] ] ] ] ] ] ] 15 14

17 some modular equations in the form of schläfli ] ] ] ] ] ] ] ] ] ] ] ) = ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] 9 8 ] ]

18 50 m.s. mahadeva naika k. sushan bairy ] ] ] ] ] ] ] where and Q are defined as in.1) and.) respectively. roofs of the identities.41).44) are similar to the proof of.4) except that in place of result.7) result.9) is used to prove.41); result.14) is used to prove.4); result.15) is used to prove.4) and the result.16) is used to prove.44). Remark.1. The identities.19).4).8) and.9) have been verified using modular forms in Entry 41 pp ] and other identities.5).11).1).7).5).6).41).44) are appears to be new. 4. Explicit evaluations of class invariants In this section we establish several explicit evaluations of Ramanujan-Weber class invariants G n using modular equations obtained in Section as an application. Theorem 4.1. We have ) G 8 = G 8 4 = 1 + ) 4.) ) + 1 G 8 16 = ) 4.) + ) ) G 9 = 4.5) 4.6) 4.7) 4.8) G 11 = 1 11) 1/ + + ] 11) 1/ 1/6 1/ G 17 = G 4 1 = + 1 G 6 15 = 1 + ) /

19 some modular equations in the form of schläfli ) 4.10) 4.11) 4.1) G 19 = 1 ) 1/ ) 1/ ) 1/ ) 1/ ) 1/ G = G 5 = G 4 9 = 1 { / 16 87) 1/ ] 87) 1/ / ) 1/ } ]) 87) 1/ 1/ 4.1) G 1 = ) 4.15) G 81 = { ) 1/ ) 1/ } 47 ) / ) ) 1/ ) G 11 = / / ) 1/ ) ) 1/ /6 11 1/ 7 ) 1/ 7 + ] )] ) 1/. roof of 4.1). utting n = 1 using the fact that G n = G 1/n and by the definition of G n in.5) we find that 4.16) 4G 16 4G 8 1 ) G ) G 4 1 ) G ) = 0 where G := G. But 4.17) 4G 16 4G 8 1 = 0. Solving the above equation and G > 1 we obtain the required result 4.1). roofs of the identities 4.) 4.15) are similar to the proof of 4.1) except that in place of result.5) result.11) is used to evaluate 4.) and 4.); result.1) is used to prove 4.4) and 4.14); result.19) is used to prove 4.5) and 4.15); result.4) is used to prove 4.6); result.7) is used to prove 4.7); result.8) is used to prove 4.8); result.9) is used to prove 4.9); result.5) is used to prove 4.10); result.6) is used to prove 4.11); result.41) is used to prove 4.1) and result.4) is used to prove 4.1). Acknowledgement. The authors are thankful to referee and rof. Bruce C. Berndt for their useful comments.

20 5 m.s. mahadeva naika k. sushan bairy References 1] Adiga C. Mahadeva Naika M.S. and Shivashankara K. On some -Q eta-function identities of Ramanujan Indian J. Math. 44 ) 00) ] Berndt B.C. Ramanujan s Notebooks art III Springer-Verlag New York ] Berndt B.C. Ramanujan s Notebooks art V Springer-Verlag New York ] Bhargava S. Adiga C. and Mahadeva Naika M.S. A new class of modular equations akin to Ramanujan s -Q eta-function identities and some evaluations there from Adv. Stud. Contemp. Math. 5 1) 00) ] Bhargava S. Adiga C. and Mahadeva Naika M.S. A new class of modular equations in Ramanujan s alternative theory of elliptic function of signature 4 and some new -Q eta-function identities Indian J. Math. 45 1) 00) -9. 6] Hart W. Schläfli modular equations for generalised Weber functions Ramanujan J ) ] Mahadeva Naika M.S. -Q eta-function identities and computation of Ramanujan-Weber class invariants J. Indian Math. Soc ) 00) ] Mahadeva Naika M.S. Some new explicit values for Ramanujan class invariants Adv. Stud. Contemp. Math. 0 4) 010) ] Mahadeva Naika M.S. and Sushan Bairy K. On some new explicit evaluations of class invariants Vietnam J. Math. 6 1) 008) ] Mahadeva Naika M.S. and Sushan Bairy K. On some new Schläfli type mixed modular equations Adv. Stud. Contemp. Math. 1 ) 011) ] Ramanathan K.G. Ramanujan s Modular Equations Acta. Arith ) ] Ramanujan S. Notebooks volumes) Tata Institute of Fundamental Research Bombay ] Schläfli L. Beweis der Hermiteschen Verwandlungstafeln für elliptischen Modularfunctionen J. Reine Angrew. Math ) ] Watson G.N. Some Singular moduli. II Quart.J.Math. 19) ] Watson G.N. Über die Schläflischen Modulargleichungen J.Reine Angew. Math. 1919) Accepted: