Ramanujan and the Modular j-invariant

Transcription

1 Canad. Math. Bull. Vol. 4 4), 1999 pp Ramanujan and the Modular j-invariant Bruce C. Berndt and Heng Huat Chan Abstract. A new infinite product t n was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan s assertions about t n by establishing new connections between the modular j- invariant and Ramanujan s cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, t n generates the Hilbert class field of Q n). This shows that t n is a new class invariant according to H. Weber s definition of class invariants. 1 Introduction Except for four entries, the last two pages in Ramanujan s third notebook, pages 39 and 393 in the pagination of [1, vol. ], are devoted to values of the modular j-invariant. Recall [14, p. 81], [15, p. 4] that the invariants Jτ )and jτ), for τ H := {τ :Imτ>0},are defined by 1.1) Jτ) = g3 τ) τ) and jτ ) = 178 Jτ), where 1.) τ) = g 3 τ) 7g 3τ), g τ ) = 60 m,n= m,n) 0,0) mτ + n) 4, and g 3 τ ) = 140 m,n= m,n) 0,0) mτ + n) 6. Furthermore, the function γ τ) is defined by [15, p. 49] 1.3) Received by the editors November 1, γ τ) = 3 jτ), The first author is supported by a grant from the National Security Agency. The second author was supported by NSF grant DMS during the initial stage of this work. AMS subject classification: 33C05, 33E05, 11R0, 11R9. Keywords: modular functions, the Borweins cubic theta-functions, Hilbert class fields. c Canadian Mathematical Society

2 48 Bruce C. Berndt and Heng Huat Chan where that branch which is real when τ is purely imaginary is chosen. At the top of page 39 in [1, vol. ], which inexplicably is printed upside down, Ramanujan defines J := J n and u := u n by 1.4) J n = 1 16α n1 α n ) 8 4α n 1 α n ) ) 1/3 and J n = 3 4un 3, where n is a natural number. Here, if, as usual in the theory of elliptic functions, k, 0< k<1, denotes the modulus, then α n := k n is the singular modulus. To help identify J n, first recall Ramanujan s definition of the class invariant G n, 1.5) G n := 1/4 q 1/4 n 1 + q n j 1 ), j=1 where n is a positive rational number and q n = exp π n). In the notation of Weber [6], G n =: 1/4 f n). Furthermore [4], 1.6) G n = {4α n 1 α n )} 1/4. Now, from 1.4) and 1.6), we easily find that 1.7) J n = 1 8 G8 n1 4G 4 n ). We now identify J n with γ. First, following Ramanujan, set 1.8) f q) := 1 q n ) = q 1/4 ητ), n=1 where q = expπiτ ), τ H, andητ) denotes the Dedekind eta-function. From Cox s text [15, p. 57, Theorem 1.17], 1.9) γ τ ) = f τ ) f τ) 8, where f τ) = q 1/4 f q )/ f q) andq=e πiτ. Hence, by 1.9), 1.10) γ τ ) = 8 q/3 f 16 q ) f 16 q) + f 8 q) q 1/3 f 8 q ). Setting τ = 3 + n)/, we deduce from 1.5) and 1.10) that 1.11) γ 3+ n )= 8 6 G 4 n 4 G 16 n = 4G 8 n1 4G 4 n ).

3 Ramanujan and the Modular j-invariant 49 Hence, from 1.7), 1.11), and 1.3), 1.1) J n = 1 3 γ ) 3+ n = 1 3 j 3 3+ n Ramanujan records fifteen values of J n, n 3 mod 4), although some are not given very explicitly. He also records factors of certain polynomials in J n. A complete discussion of this material can be found in Berndt s book [, Chap. 34, Sect. 11]. Up to this point, all of Ramanujan s claims about the j-invariant are well known, but on page 393, Ramanujan records the following interesting result. ). Theorem ) Then 1.14) For q = exp π n), define t n = t := t n := 3q 1/18 f q1/3 ) f q 3 ) f. q) 64 J n 4 J n J n 3)) 1/6. Ramanujan then gives a table of polynomials satisfied by t n,forfivevaluesofn. Theorem 1. For the values of n given below, we have the following table of polynomials p n t) satisfied by t n. n p n t) 11 t 1 35 t + t 1 59 t 3 +t 1 83 t 3 +t +t t 3 t +4t 1 The simplicity of these polynomials is remarkable, since the corresponding well-known polynomials of the same degree) satisfied by J n are considerably more complicated, especially in the latter three instances. The form of Theorem 1.1 suggests hitherto unknown connections between the j-invariant and Ramanujan s cubic theory of elliptic functions to alternative bases developed by Berndt, S. Bhargava, and F. G. Garvan [3]. In Section, we offer some of these connections and show how they can be exploited to calculate particular values of γ τ ). We also use this connection to give a proof of Theorem 1.1 in Section 3. In Section 4, we derive the polynomials in Theorem 1. from Theorem 1.1 and results in papers by A. G. Greenhill [17], [18]. In view of 1.14), there is clearly a relation between t n and the Hilbert class field of Q n). We will show that in fact t n generates the Hilbert class field of Q n) for certain integers n. In this final section we also provide some historical remarks on what Ramanujan likely knew about the j invariant from the literature.

4 430 Bruce C. Berndt and Heng Huat Chan Connections with Ramanujan s Alternative Cubic Theory Recall that the ordinary hypergeometric function F 1 a, b; c; z) is defined by F 1 a, b; c; z) := k=0 a) k b) k c) k where a) k = a)a +1) a+k 1), and z < 1. In the alternative cubic theory, the base q is defined by.1) z k k!, q := q 3 := exp π F 1 1 3, 3 ;1;1 α) ) 3 F 1 1 3, 3 ;1;α), 0<α<1. From the fundamental inversion formula for the cubic theta functions [3, Lemma.9], we find that.) α := αq) = c3 q) a 3 q), where.3) aq) := m,n= q m +mn+n and.4) cq) := q m+1/3) +m+1/3)n+1/3)+n+1/3). m,n= Following Ramanujan, define Mq) andnq)by g τ)= 4π4 3 Mq) and g 3τ) = 8π6 7 Nq), where q = expπiτ ), τ H. Then, it was shown by Berndt, Bhargava, and Garvan [3, Theorems ] see also [11]) that.5).6) Mq) = z α), Nq) = z 6 1 0α 8α ), Mq 3 ) = z ) 9 α, and Nq 3 ) = z α + 8 ) 7 α, where z := F 1 1 3, 3 ;1;α).

5 Ramanujan and the Modular j-invariant 431 It follows from 1.1) and 1.) that.7) M 3 q) jτ ) = 178 M 3 q) N q). Now, employing.5) and.6) in.7), we deduce that, respectively,.8) 1 + 8α)3 jτ ) = 7 α1 α) 3 and.9) j3τ ) = α) ) 3 1 α)α 3. Alternatively, we show how to derive.9) from.1) and.8). In.1), set q = expπiτ ). Then, replacing α by 1 α in.1), we find that.10) exp π 3 F 1 1 3, 3 ;1;α) ) F 1 1 3, 3 ;1;1 α) =exp πi/3τ ) ). It is well known that jτ ) is a modular function on the full modular group [14, p. 84], [15, p. 1]. In particular,.11) j 1/τ ) = jτ ). Hence, by.11),.10), and.8), j3τ ) = j 1 ) ) α) = 7 3τ 1 α)α 3, and thus.9) has been established again. Identity.8) is truly a cubic analogue of the well-known identity [8, p. 115, Theorem 4.4] where with ψq) := jτ ) = 56 1 λ + λ ) 3 λ 1 λ), λ := λq) = q ψ4 q ) ϕ 4 q), q nn+1)/ and ϕq) := n=0 q n, q = e πiτ. n=

6 43 Bruce C. Berndt and Heng Huat Chan We remark here that the discovery of identities.8) and.9) has recently motivated a proof of Ramanujan s cubic transformation via Goursat s transformation formulas. For more details, see [11]. Next, we use.8) and.9) to calculate some values for γ τ ). We emphasize that we could utilize 1.11) in our procedures. However, our calculations avoid the need for the values of class invariants. Instead, we employ some simple modular equations from the alternative cubic theory. Examples.1).13).14).15).16).17) γ 3 1) = ) 3 + 1) + 3) /3, γ 3 ) = ) )5 + 6) /3, γ 5/3) = )7 ) 10/ ), γ 15) = )7 + ) 10/ ), γ 7/3) = ) ) ) 1/3, γ 1) = ) ) ) 1/3. Remark Note that if we use 1.11) to evaluate γ τ ), we obtain a product of two rather complicated algebraic numbers. On the other hand, our new method produces much simpler expressions for γ τ ). We will provide proofs of.1) and.13) and say a few words about the proofs of identities.14).17). Proof of.1) We first recallthe definitionofa modularequation of degreen in the cubic theory [3]. Suppose that, for some positive integer n,.18) F 1 1 3, 3 ;1;1 β) F 1 1 3, 3 ;1;β) =n F 1 1 3, 3 ;1;1 α) F 1 1 3, 3 ;1;α). Then a modular equation of degree n in Ramanujan s alternative cubic theory is a relation between α and β induced by.18). As in the classical theory, we say that β has degree n over α.

7 Ramanujan and the Modular j-invariant 433 If n is a positive rational number, then the singular modulus 3 αn in the alternative cubic theory is that unique positive number such that.19) F 1 1 3, 3 ;1;1 α n) F 1 1 3, 3 ;1;α = n. n) Now suppose that β = 1 α. Then, from.18), F 1 1 3, 3 ;1;1 β) F 1 1 3, 3 ;1;β) = n, i.e., by.19), β = α n, the cube of the singular modulus. We now demonstrate how this principle can be used to calculate values of γ τ ). Suppose that β has degree 3 over α in the cubic theory. Then [3, Lemma 7.4].0) β 1/3 = α α. Assuming that β = 1 α, we conclude, as above, that β = α 3. Hence, from.0), which implies that.1) 3 = 1 α1/3 3, 1+α 1/3 α 1/3 α 1/ =. 3 From.1), note that q = e π. Thus, by.8) and a straight-forward calculation, we find that γ 1) = 1, which is well known [15, p. 61]. From.9) and.1), we also find that ) γ 3i) = ) 3 Upon simplifications, we arrive at.1). Next, assume that β has degree over α and degree 3 over 1 α. Then, from.18) we can deduce that F 1 1 3, 3 ;1;1 β) F 1 1 3, 3 ;1;β) = 6. Thus, β = α 6.Sinceβhas degree over α, from [3, Theorem 7.1],.) αβ) 1/3 + {1 α)1 β)} 1/3 = 1.

8 434 Bruce C. Berndt and Heng Huat Chan Since β has degree 3 over 1 α, by.0), from which it follows that β 1/3 = 1 α1/3 1+α 1/3,.3) α 1/3 = 1 β1/3 1+β 1/3. If x = α 1/3 6, substitution of.3) into.) yields, after a lengthy calculation,.4) 1 + x + x )1 8x +10x )=0. Checking numerically the two real roots, we find that the appropriate root of.4) is α 1/3 6 = 4 6.5). 10 From.1), q = e π, and so from.8), we easily find that γ i) = 0, which is wellknown [15, p. 61]. Next, substituting.5) into.9) and simplifying, we deduce.13). Unlike.13), the proofs of identities.14).17) require only a single modular equation in each case. For.14) and.15), we use Ramanujan s modular equation of degree 5 [3, Theorem 5.6], namely, and show that αβ) 1/3 + {1 α)1 β)} 1/3 +3{αβ1 α)1 β)} 1/6 = 1, α 5 = ) 5. 5 This corresponds to q = e π 5/3 and gives us the value for γ 5/3). Next, observe that 1 α 5 = ) 5 5 differs from the value α 5 by only a sign change in 5. Since.9) relates γ 15) and 1 α 5 in exactly the same way as γ 5/3) and α 5, we can simply replace 5by 5 in the expression for γ 5/3) to obtain γ 15). For the proofs of identities.16) and.17), we use a new modular equation of degree 7 recently discovered by Chan and W.-C. Liaw [13]. Suppose β has degree 7 over α. Let w = x + y 1)x + y +1)ands=xy,wherex= αβ and y = 1 α)1 β). Then w sw s s)w s s s)w 3 5 s 6 s+3 ) = 0. From the modular equation above, we deduce that α 7 = This value corresponds to q = e π 7/3 and gives us the value for γ 7/3). The value for γ 1) can be obtained by replacing 3by 3 in the expression for γ 7/3).

9 Ramanujan and the Modular j-invariant Theorem 1.1 and the Alternative Cubic Theory In this section, we will derive the following q-identity from the alternative cubic theory and deduce Theorem 1.1 as a corollary of this identity. Theorem 3.1 Set hq) = where f q) is given as in 1.8). Then f 1 q 3 ) qf 6 q)f 6 q 9 ), 3.1) hq) 7hq) 1 = γ 3τ )+6, where q = expπiτ ). Proof of Theorem 3.1 From [3, Lemma 5.1] or [10], 3.) cq 3 ) = 3q f 3 q 9 ) f q 3 ). By combining Entry 1v) in Chapter 0 of Ramanujan s second notebook [1, p. 346] with a Lambert series representationforaq) found by both Ramanujan and the Borweins [9], we find that 3.3) aq 3 ) = f 3 q)+3qf 3 q 9 ) f q 3. ) For several historical remarks on this Lambert series, see [3, p. 4167].) Hence, by 3.) and 3.3), we deduce that 3.4) 1 α 1/3 q 3 ) = f 3 q) 3qf 3 q 9 ) +1= s 3 +1, where 3.5) s = f 3 q) qf 3 q 9 ). Substituting 3.4) into.8) with q replaced by q 3 ), we find that 3.6) γ 3τ ) = s +3)s+9)s + 7) ss. +9s+ 7) On the other hand, if we cube Entry 1iv) in Chapter 0 of Ramanujan s second notebook with q replaced by q 3 [1, p. 345], we find that 3.7) hq) = s s,

10 436 Bruce C. Berndt and Heng Huat Chan with s given by 3.5). Now, from 3.6) and 3.7), γ 3τ ) = s +9s+7 3s)s +7+9s 9s) ss +9s+ 7) shq)+3s)shq) 9s) = s hq) = hq) 6 7hq) 1, which is 3.1). This completes the proof of Theorem 3.1. As a corollary, we deduce Theorem 1.1. Proof of Theorem 1.1 deduce that 3.8) In view of 1.1) and 1.13), we set τ = 3 + n)/6 in 3.1) to t 6 n 7t 6 n = 3 J n +6, since h e π n/3 ) = 7t 6 n. Solving the quadratic equation 3.8) in t 6 n yields 1.14) immediately. This completes the proof of Theorem 1.1. Remark Theorem 3.1 can be proved using the theory of modular forms. From M. Newman s criterion [0] or [1]), it is easy to show that hq) generates the function field associated with Γ 0 9). On the other hand, γ 3τ ) is a modular function for the group Γ 0 9) [15, p. 50, Proposition 1.3]. Hence, γ 3τ ) is a rational function of hq). For detailed computations of identities similar to 3.1), see [1]. 4 Calculating the Polynomials in Theorem 1. Proof of Theorem 1. It is well known that J 11 = 1 [15, p. 61]. Thus, we find that t 11 = 7 13) 1/6 = 1, as desired. Secondly, from a paper of W. E. Berwick [6], Hence, J 35 = ) ) t 35 = ) ) 4 ) /6 = /6 5).

11 Ramanujan and the Modular j-invariant 437 It is easy to verify that if a db = c,then a±b a+c a c 4.1) d= ±. Now, since = 589, we find that = + = = , by 4.1). Hence, t 35 = ) / ) = 9 4 5) 1/6 =. Hence, t 35 is a root of t + t 1, and the second result is established. For n = 59, Greenhill [18] showed that u 59, defined by 1.4), is a root of the equation u 39 1/3 u / /3 u 1/3 816 = 0. Since, by 1.4) and 1.1), u 59 = γ 3 /56, where γ = γ ), it follows that γ satisfies 4.) γ γ γ = 0. Set x := t 6 := t 6 n. From 3.8), we find that 4.3) γ = x 7x 1 6. Substituting 4.3) into 4.) and turning to Mathematica,wefindthat x x x x x + 730x = x 3 +13x + 115x 1)x x 9477x ) = 0. Numerically checking the roots, we see that x is a root of the first factor above, i.e., t t t 6 1 = 0. We use Mathematica to factor this polynomial and find that t t t 6 1 = t 3 +t 1)t 3 +t+1)t 6 t 4 +t 3 +4t t+1) t 6 t 4 t 3 +4t +t+1).

12 438 Bruce C. Berndt and Heng Huat Chan Again, numerically calculating the roots, we find that as claimed by Ramanujan. For n = 83, u 83 is a root of [17] t 3 +t 1=0, u /3 u / /3 u 1/ = 0, and for n = 107, u 107 is a root of [18, p. 405] u /3 u / /3 u 1/ = 0. The derivations of p 83 t)andp 107 t) from the relations above are similar to the case n = 59 and we omit the details here. This completes the proof of Theorem 1.. Our derivations of p n t) forn = 59, 83, and 107 are not satisfactory as they involve Greenhill s polynomials. Greenhill discovered these polynomials using Russell s modular equations and relations between u n and Weber s class invariants. His computations are laborious. An effective and direct method for evaluating p n t) is highly desirable. Let n be squarefree and set ω = 3 + n)/. It is well known that if K n = Q n), 3 n, andn 3 mod 4), then K n γ ω) ) is the Hilbert class field of K n. This means that the degree K n γ ω) ) ) : K n = h Kn, the class number of K n. In view of 3.8), we conclude that K n γ ω) ) is a subfield of K n t n ). Now, the class numbers of K n for n = 11, 35, 59, 83 and 107 are respectively 1,, 3, 3, and 3. These class numbers coincide with the degrees of p n t) and hence, K n t n ) = K n γ ω) ) for these integers. It is therefore natural to conjecture that for certain integers n, t n generates the Hilbert class field of K n. Indeed, we have the following result: Theorem 4.1 Let n be squarefree, n 11 mod 4), and suppose that the class number of K n is odd. Then t n is a real unit generating the Hilbert class field of K n. In [7, p. 91, Section 8], B. J. Birch relates a function g suggested by A. O. L. Atkin satisfying 4.4) g 6 τ ) 7g 6 τ) = γ τ)+6, where gτ) is real and positive when z is purely imaginary. He then established the following [7, p. 9]: Lemma 4. K n. If n 11 mod 4), theng 3 + n)/ ) generates the Hilbert class field of Birch did not indicate the explicit form of gτ ), and so no examples were given to verify his results. In [5], H. M. Stark successfully determined an expression for gτ ). Using Kronecker s limit formula, he computed the first few numerical examples for 4.4).

13 Ramanujan and the Modular j-invariant 439 In view of 3.1) and 4.4), we conclude, by solving x 7/x = y 7/y, that g 6 τ)=he πiτ/3 ) or 7 he πiτ/3 ). By considering the q-expansion of hτ )andγ τ), we conclude that 4.5) g 6 τ ) = he πiτ/3 ). Combining the identification 4.5) and Lemma 4., we conclude that tn = 3 g 3 + n)/ )) generates the Hilbert class field of Kn. We emphasize here that this result is independent of the condition on the class number of K n. To obtain Theorem 4.1, we use the idea given in Birch s paper. We quote the following simple but useful lemma see [7, Lemma 5]): Lemma 4.3 If Kθ) is a quadratic extension of K with θ K, and if α Kθ) with α K, then either α Korθα K. Proof of Theorem 4.1 By Söhngen s Theorem [4], t n is abelian over K n. Hence, t n lies in a quadratic extension K n θ), where θ K n. This implies that t n H n θ), where H n denotes the Hilbert class field of K n. Now, the hypotheses of Lemma 4.3 are satisfied with K = H n and α = t n note that θ K n H n ). Thus, t n H n or θt n H n. If t n H n, we are done. Otherwise, suppose θt n H n.let Squaring both sides, we obtain θt n =ζ, with ζ H n. θ t n = ζ. Next, since all the quantities above are in H n, we deduce that 4.6) Nθ )Nt n) = Nζ), where N denotes the H n K n norm. Now, Nt n) = ±1, since t n is a real unit by Deuring s Theorem [16], and Nθ ) = θ h θ K n ), where h is the class number of K n.sincehis odd, we may rearrange 4.6) and deduce that θ = ± Nζ) θ h 1 or ± Nζ) iθ h 1. If θ = ±Nζ)θ 1 h,thenθ K n,andsot n H n.ifθ=±nζ)iθ h 1 ) 1,thenθ K n i). This means that we can replace θ by i. But now, if t n H n i), then t n is fixed by complex conjugation since t n is real. Hence, t n H n. This completes the proof of Theorem 4.1. Historical Remarks It has often been conjectured that Ramanujan had seen Greenhill s book [19] on elliptic functions. For example, see a letter by K. Ananda Rau and the following commentary in the book by R. A. Rankin and one of the authors [5, pp. 89, 90]. On

14 440 Bruce C. Berndt and Heng Huat Chan References pages 37 39, Greenhill [19] briefly summarizes much of R. Russell s work on modular equations. There are several modular equations of the type found by Russell in Ramanujan s notebooks, and this is further evidence that Ramanujan had seen both of Russell s papers [], [3], and therefore also both of Greenhill s papers [17], [18], since all four of these papers appear in volumes 19 and 1 of the Proceedings of the London Mathematical Society. [1] B. C. Berndt, Ramanujan s Notebooks, Part III. Springer-Verlag, New York, [], Ramanujan s Notebooks, Part V. Springer-Verlag, New York, [3] B.C.Berndt,S.BhargavaandF.G.Garvan,Ramanujan s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc ), [4] B. C. Berndt, H. H. Chan and L.-C. Zhang, Ramanujan s class invariants, Kronecker s limit formula, and modular equations. Trans. Amer. Math. Soc ), [5] B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary. American Mathematical Society, Providence, RI; London Mathematical Society, London, [6] W.E.Berwick,Modular invariants expressible in terms of quadratic and cubic irrationalities. Proc. London Math. Soc. 8197), [7] B. J. Birch, Weber s class invariants. Mathematika ), [8] J.M.BorweinandP.B.Borwein,Pi and the AGM. John Wiley, New York, [9], A cubic counterpart of Jacobi s identity and the AGM. Trans. Amer. Math. Soc ), [10] J. M. Borwein, P. B. Borwein and F. G. Garvan, Some cubic identities of Ramanujan. Trans. Amer. Math. Soc ), [11] H. H. Chan, On Ramanujan s cubic transformation formula for F 1 1/3, /3; 1; z). Math. Proc. Camb. Phil. Soc ), [1] H. H. Chan and M.-L. Lang, On Ramanujan s modular equations and Atkin-Lehner involutions. Israel J. Math ), [13] H. H. Chan and W.-C. Liaw, On Russell type modular equations. Canad. J. Math., to appear. [14] K. Chandrasekharan, Elliptic Functions. Springer-Verlag, Berlin, [15] D. A. Cox, Primes of the form x + ny. Wiley, New York, [16] M. Deuring, Die Klassenkörper der komplexen Multiplikation. Enz. Math. Wiss. Band I, Heft 10 Teil II, Stuttgart, [17] A. G. Greenhill, Complex multiplication moduli of elliptic functions. Proc. London Math. Soc ), [18], Table of complex multiplication moduli. Proc. London Math. Soc ), [19], The Applications of Elliptic Functions. Dover, New York, [0] M. Newman, Construction and application of a class of modular functions II. Proc. London Math. Soc ), [1] S. Ramanujan, Notebooks. vols., Tata Institute of Fundamental Research, Bombay, [] R. Russell, On κλ κ λ modular equations. Proc. London Math. Soc ), [3], On modular equations. Proc. London Math. Soc ), [4] H. Söhngen, Zur komplexen Multiplikation. Math. Ann ), [5] H. M. Stark, Class numbers of complex quadratic fields. In: Modular functions of one variable I ed. W. Kuijk), Lecture Notes in Math ), Springer-Verlag, [6] H. Weber, Lehrbuch der Algebra, dritter Band. Chelsea, New York, Department of Mathematics University of Illinois 1409 West Green Street Urbana, IL USA Department of Mathematics National University of Singapore Kent Ridge Singapore Republic of Singapore