RAMANUJAN WEBER CLASS INVARIANT G n AND WATSON S EMPIRICAL PROCESS

Transcription

1 AND WATSON S EMPIRICAL PROCESS HENG HUAT CHAN Dedicated to Professor J P Serre ABSTRACT In an attempt to prove all the identities given in Ramanujan s first two letters to G H Hardy, G N Watson devoted a paper to the evaluation of G At the end of his paper, Watson remarked that his proof was not rigorous as he had assumed certain identities (see (11) (1)) which he found empirically In this paper, we use class field theory, Galois theory Kronecker s limit formula to justify Watson s assumptions We shall then use our results to compute some new values of G n 1 Introduction Let (a; q) (1aqn) n= χ(q) (q; q) for q 1 S Ramanujan first introduced the class invariant G n /e π n/χ(e π n) in his famous paper Modular equations approximations to π [13; 1, pp 3 39] In H Weber s notation, G n /f(n) A table consisting of 50 values of G n was first constructed by Weber [19, pp 71 76] In the aforementioned paper, Ramanujan added another 31 values to Weber s list For these obvious reasons, we shall call G n a Ramanujan Weber class inariant In his second letter to G H Hardy, Ramanujan recorded the identity [3, p 6, no 3], G (311) (533) / Received November 1995; revised 7 March Mathematics Subject Classification 11Y60 This work was supported by NSF, grant number DMS J London Math Soc () 57 (199)

2 1 56 HENG HUAT CHAN In an attempt to prove all the identities given in Ramanujan s two letters to Hardy [3], G N Watson devoted a paper to the evaluation of G [17] At the end of his paper [17, p 13], Watson remarked that his proof was not rigorous, as he had assumed that G G / G / G / (11) G G G G / / / G G / G / G / (1333) (1) G G G G / / / Despite the lack of rigour in his empirical process, Watson [1] employed it further to verify other values of G n stated explicitly in [13; 1, pp 3 39] In particular, he succeeded in verifying the values of G n, for n 65, 777, 97, 165, 1677 based on the assumptions of identities similar to (11) (1) The main purpose of this paper is to provide rigorous proofs of the aforementioned identities for G n We achieve this by justifying Watson s assumptions We shall then use our results to compute some new values of G n To fix notations, we first recall some facts in algebraic number theory Let K (m) (where m is squarefree) be an imaginary quadratic field K be its ring of integers Two non-zero K ideals are said to be equivalent if there exists an α K0 such that α It is known that the set of equivalence classes forms a group under multiplication of ideals we call this group, denoted by C K, the ideal class group of K The order of C K, denoted by h K, is known as the class number of K In Section, we shall prove the following THEOREM 11 Let ν 3 if 3 pqr, 1 if 3 pqr Suppose that K (pqr) satisfies the conditions (i) p, q r are distinct primes such that pqr 1(mod), (ii) the class number h K 16 Then α α p,q,r 0 G pqr 1 0 ν G pqr 1 ν, β β p,q,r 0 G pqr 1 0 ν G pqr 1 ν, γ γ p,q,r 0 G pqr 1 0 ν G pqr 1 ν, δ δ p,q,r (G pqr ) ν (G pqr ) ν are algebraic integers belonging to a real quadratic field R K, where K (m)m p ε q ε r ε, ε i 0 or 1 for i 1,, 3 contains fields such that none of the primes (), (p), (q) or (r) is inert Theorem 11 provides an explanation of the existence of identities such as (11) (1) In Section 3, we shall prove the following

3 AND WATSON S EMPIRICAL PROCESS 57 THEOREM 1 Let R (m) be the field which contains α, β, γ δ If α a a m, β b b m, γ c c m, δ d d m, then a,a,b,b,c,c,d d are positie integers Theorem 1 implies that the determinations of α, β, γ δ can be done in a finite number of steps since the number of real quadratic fields to be considered is finite the set umu, is discrete in [m] Using Theorems 11 1, we conclude that Watson s assumptions are valid As corollaries to these theorems, we evaluate G n for n 5, 9, 65, 561, 609, 65, 777, 05, 97, 957, 1005, 105, 1065, 1105, 1113, 115, 1353, 1605, 165, 1653, 1677, 1705, 15, 013, A list of the G n for n 65, 777, 97, 1353, 165, or 1677 is given in Section The number G was first evaluated by D Shanks [15, p 399, ()] using Epstein Zeta Functions Readers are encouraged to read Shanks paper for a different proof of G Theorems 11 1 are natural extensions of results given in [, Theorems 7 75] They can be extended to evaluate class invariants G n whenever the class group of K (n) is isomorphic to For example, when n , we have C ( ) we find that G / (101791)/ In [9], S Chowla proved that the number of imaginary quadratic fields with k classes in the principal genus is finite Applying this result with k indicates that the number of G n which we can compute using the methods mentioned in this paper is finite In fact, D Buell s table [7], which gives a list of those n less than satisfying the hypotheses in the previous paragraph, shows that we can compute six more G n The associated imaginary quadratic fields for these six n have the same group structure as that of (175) For the sake of completion, we shall list these values in Section 5 Proof of Theorem 11 From class field theory, we know that there exist an everywhere unramified extension K() of K such that Gal(K()K) C K This field is usually known as the Hilbert class field or the absolute class field of K One can describe the isomorphism explicitly using the j-invariant function (see Lemma 1)

4 1 5 HENG HUAT CHAN Let [τ, τ ]bea K -ideal Define g() j() 17 g()7g(), where 1 1 g () 60 g () 10 (mτ nτ ) (mτ nτ ) m,n= (m,n) (,) m,n= (m,n) (,) It is clear from the definitions of g ([τ, τ ]) g ([τ, τ ]) that j([τ, τ ]) j([1, τ]) : j(τ), where τ τ τ We will also let γ (τ) (j(τ))/ with the cube root being real-valued when j() is real It is well known that K() K(j( K )) [10, Theorem 111, p 0] If D K is the discriminant of K 3 D K, then K() K(γ (τ K )) [10, Theorem 1, p 9], where m if D K 0 (mod ), τ K 3 3m if D K 1 (mod ) These are the facts which lead to the definition of ν given in Theorem 11 We are now ready to state the following lemmas LEMMA 1 Let be two K -ideals Define σ (j()) by σ (j()) j( ), (1) where is a principal ideal Then σ is a well-defined element of Gal(K() K), σ induces an isomorphism C K Gal(K() K) Proof See [10, Corollary 1137, p 0] LEMMA Let K (pqr), where p, q, r are three distinct primes satisfying pqr 1 (mod ), let ν be as defined in Theorem 11 Then G ν is a real pqr unit generating the field K() Proof From [5, p 90], we find that G pqr is a real unit of K() Since [10, Theorem 117, p 57] j( K ) j(pqr) (16G pqr ), () G pqr we conclude that K() K(G ) (3) pqr Next, suppose that 3 pqr Then 3 D K γ (τ K ) generates K() From the equality [10, Theorem 117, p 57] γ (pqr) 16G pqr G pqr (3), we find that G pqr K() Hence, G pqr K(), by (3)

5 AND WATSON S EMPIRICAL PROCESS 59 REMARK In [5, p 90], B J Birch quoted M Deuring s results [11, p 3], indicated that G n is a real unit when n n 1 (mod ) A more elaborate proof of this statement can be found in [, Corollary 5] In fact, from the treatment given in [], one can show that, are units These facts will be needed in the proof of Theorem 11 Proof of Theorem 11 From the hypotheses, we deduce that [, 1pqr], [p, pqr] [q, pqr] are K -ideals lying in distinct equivalence classes This implies that C K contains a group generated by the ideal classes [], [] [] Using the isomorphism in Lemma 1, we conclude that Gal(K() K) contains the group U σ, σ, σ To show that α, β, γ δ belong to a field with degree over K, it suffices to show that σ, σ σ fix these elements The fact that they are algebraic integers follows from Lemma the remark after the lemma Now, assume that 3 pqr From [10, Theorem 117, p 57], we have j() j(qrp) (16 ) () G qr/p By Lemma 1, we find that σ (j( K )) j() (5) From (), () (5), we find that (16σ (G pqr )) σ (G pqr ) (16 ) (6) G qr/p Simplifying (6), we deduce that (ab)(ab) 6(ab) abab10, where a σ (G ) b G pqr qr/p But 6(ab) abab1 0, for otherwise it would contradict the fact that a b are algebraic integers Thus, σ (G pqr ) (7) By a similar argument, we have σ (G pqr ) () From () (7), we deduce that both σ (G pr/q ) σ (G ) are defined, qr/p respectively Using the same argument as in the proof of (6), we find that σ (G pr/q ) G (9) qp/r σ (G qr/p ) (10) By (7), (), (9) (10), we conclude that σ σ fix α, β, γ δ Next, from [10, p 63], we have j() j0 3 pqr 1 G pqr0 16 G 1 (11) pqr Applying Lemma 1 again, we deduce that σ (j( K )) j() (1) By (), (11) a similar argument to that in the proof of (6), we find that σ (G pqr ) G pqr (13)

6 1 550 HENG HUAT CHAN Using instead of K in (1) the corresponding expression for j(), we find that σ (G qr/p ) G or G qr/p qr/p, that is, σ (G qr/p ) may have the same or opposite sign as σ (G ) We shall show that pqr the latter case is inadmissible If σ (G qr/p ) G pqr, σ (G qr/p ) G qr/p (13) holds, then σ σ (G pqr ) G σ qr/p σ (G pqr ) G qr/p This clearly contradicts the fact that σ σ σ σ Similar arguments show that, corresponding to (13), we have σ (G pr/q ) G pr/q, σ (G pq/r ) G pq/r Collecting our results, we conclude that σ fixes α, β, γ δ This proves the first part of Theorem 11 when 3 pqr The proof when 3 pqr is similar In this case, G pqr generates K() σ (G pqr )is well defined for,, Hence, we may deduce from (6) that 16σ (G pqr ) 16G σ (G pqr ) qr/p (1) Simplifying (1), we have (ab)(abab1) 0, where a σ (G pqr ) b But abab1 0, for otherwise it would contradict the fact that a b are algebraic integers Hence, we deduce that σ (G pqr ) Now, since σ Gal(K() K) G pqr generates K(), we find that σ (G pqr ) The rest of the arguments are analogous to those of the previous case, we shall omit them We have already seen that α, β, γ δ lie in a real quadratic field R Now, if one of the primes, say (u) (), (p), (q), or (r) is inert in R, then the corresponding Frobenius automorphism σ Gal(K() K), where (u) in K, is trivial This contradicts the fact that σ has order For more details, see [, Theorem 73] This gives the necessary condition for R completes the proof of Theorem 11 3 Proof of Theorem 1 Let C K denote the subgroup of squares in C K, let G K be the genus group C K C K A group homomorphism χ:g K 1 is known as a genus character One can show that a genus character arises from a certain decomposition of D K, where D K is the discriminant of K More precisely, if χ is a genus character, then there exist d d satisfying D K d d with d 0 d i 0 or 1 (mod ), such that, for any prime ideal in K, we have χ() 3 0 d N()1, if N() d, 0 d N()1, if N() d,

7 AND WATSON S EMPIRICAL PROCESS 551 where N() is the norm of the ideal 0 1 is an ideal class in C K α, then we set χ([]) χ() α denotes the Kronecker symbol If [] LEMMA 31 Let χ be a genus character arising from the decomposition D K d d Let h i be the class number of the field (d i ), w be the number of roots of unity in (d ), ε be the fundamental unit of (d ) Let with F([]) (N([1, τ]))/ η(τ), η(z) e π iz/ (1e π inz) n= τ τ, Imτ 0, where [τ, τ ] τ Then εh h /w F([]) χ ([ ]) (31) [ ] C K Proof See [16, p 7, Theorem 6] From now on, we set K (pqr) with h K 16 In this case, G K Suppose that the principal genus contains [], that is, G [ K ], [] Corresponding to this assumption, we know that there exist three genera, say G [], [], G [], [], G [], [] Note that G G G G forms a subgroup of index in C K Let be the set of genus characters such that χ(g ) 1, χ(g ) 1 χ(g ) 1 Note that For any ideal class [], we have which implies that χ([]) 0, χ F([]) χ ([ ]) 1 χ Hence, by (31), we conclude that χ εh h /w F([]) χ ([ ]) F([]) χ ([ ]) χ [ ] C K [ ] χ F([]) χ ([ ]), (3) [ ] since If we substitute F([]) by its infinite product representation use the definition of G n (see (1)), we can rewrite (3) as 0 G pqr G qp/r 1 εh h /w (33) χ

8 1 55 HENG HUAT CHAN Next, let be the set of genus characters such that χ(g ) 1, χ(g ) 1 χ(g ) 1, then by following the previous argument, we conclude that Since 0 G pqr 1 χ w 3 or if 3 pqr,, or 6 if 3 pqr, we deduce from (33) (3) the following lemma εh h /w (3) LEMMA 3 Suppose that [] is in the principal genus Then 0 G pqr 1 ν εe /εe / 0 G pqr 1 ν εe /εe /, where the ε i are fundamental units in some real quadratic fields the e i are positie integers Next, if we assume that [] is in the principal genus, then by using similar arguments to those in the proof of Lemma 3, we obtain the following lemma LEMMA 33 Suppose that [] is in the principal genus Then 0 G pqr 1 ν εe /εe / (G pqr ) ν εe /εe / where the ε i are fundamental units in some real quadratic fields the e i are positie integers If we replace in Lemma 3 Lemma 33 by or, we obtain four further lemmas LEMMA 3 Suppose that [] is in the principal genus Then 0 G pqr 1 ν εe /εe / 0 G pqr 1 ν εe /εe /, where the ε i are fundamental units in some real quadratic fields the e i are positie integers LEMMA 35 Suppose that [] is in the principal genus Then 0 G pqr 1 ν εe /εe / (G pqr ) ν εe / εe /, where the ε i are fundamental units in some real quadratic fields the e i are positie integers LEMMA 36 Suppose that [] is in the principal genus Then 0 G pqr 1 ν εe /εe / 0 G pqr 1 ν εe /εe /, where the ε i are fundamental units in some real quadratic fields the e i are positie integers

9 AND WATSON S EMPIRICAL PROCESS 553 LEMMA 37 Suppose that [] is in the principal genus Then 0 G pqr 1 ν εe /εe / (G pqr ) ν εe / εe /, where the ε i are fundamental units in some real quadratic fields the e i are positie integers REMARKS Lemmas 3 37 are natural extensions of theorems discovered by L Kronecker [19, pp 55 57], K G Ramanathan [1] L-C Zhang [3, Theorems 31, 3; 1] These authors have used their results explicitly to evaluate certain infinite products related to either the class invariant G n or the Rogers Ramanujan continued fraction Here, we shall use Lemmas 3 37 implicitly without having to determine the fundamental units involved From class field theory, we know that if H is a subgroup of C K, then there exists an abelian everywhere unramified extension LK such that Gal(K()K) H In particular, when H C K, the corresponding field MK is known as the genus field of K One can show that M is the maximal unramified extension of K which is abelian over [10, p 1] Proof of Theorem 1 Let [ ] [ ] be two ideal classes in, let H [ ], [ ] From the previous paragraph, we know that there exists an abelian everywhere unramified extension LK such that Gal(K()K) H In fact, from the isomorphism of Lemma 1, we find that L Fix(σ, σ ) Since Gal(K()K), the group Gal(LK) is isomorphic to either or In the former case, L is a subfield of M, the genus field of K, one of the ideal classes in H must be in the principal genus As for the latter case, Gal(L) D, where D is the dihedral group of eight elements, since L is generalized dihedral over [10, p 191] Hence, Gal(L) is non-abelian Now, let α ηη Then, by assumption, we have where η 0 G pqr 1 ν (ηη ) a a m (35) Note that σ, σ σ fix η σ (η) η Therefore, the field L Fix(σ, σ ) is of degree over K Suppose that Gal(LK) Then, from the beginning of our proof, we conclude that either [], []or[] is in the principal genus Using either Lemma 3, Lemma 3 or Lemma 37, which correspond to each of the possibilities, we deduce that η εe/εe /, where ε ε are fundamental units in certain quadratic fields, e, e Itis known that a fundamental unit of a real quadratic field takes the form ud with

10 55 HENG HUAT CHAN u, 0 [6, p 133] Furthermore, if ud ud d, then u, 0 Collecting these observations, we deduce that η is of the form u u l u l u l l, where u i 0 for each i Hence, if α (ηη ) a a m, then a a must be positive integers since the u i are positive Next, suppose that Gal(LK) Recall that Gal(L) D is non-abelian We claim that there exists σ Gal(LK) such that σ(η) is complex Suppose that the contrary holds Then L (η) would be Galois over, hence Gal(L(η)) is a normal subgroup of Gal(L) On the other h, Gal((η)) Gal(LK), is a normal subgroup of Gal(L) [10, p 191] Hence, Gal(L) is isomorphic to the direct sum of Gal(L(η)) Gal((η)) is therefore an abelian group This contradicts our initial assumption Next, we shall show that σ(m) m Suppose that the contrary holds Then σ(η)σ(η) ηη, therefore σ(η) is equal to η or η This shows that σ(η) is real, which contradicts our choice of σ Now, applying σ to (35), we deduce that (σ(η)σ(η) ) a a m (36) From (35), (36) the fact that σ(η) is complex, we find that (a a m) 16 (a a m) 16 This implies that a a m 0 Since η 0, we deduce that a a are positive The integrality of a a follows easily from Theorem 11 In a similar way, we can show that b, b, c, c, d, d are positive integers Of course, we have to use Lemmas 3 37, as appropriate REMARK The argument given here for the case when Gal(L) is non-abelian is due to Weber [10, p 69] Some alues of G n In this section, we list explicit values of G n which are mentioned in the Introduction Our computations are done with the aid of Mathematica MAPLE V G G / (519)/ /

11 AND WATSON S EMPIRICAL PROCESS 555 G / / (79)/ (13970)/ G G G 35 1 / / (1753)/ ( )/ / / G / / (13970)/ ( )/ G / / (1753)/ ( )/

12 556 HENG HUAT CHAN G G 3 53 (713095)/ ( )/ G G (3) 337 (5531)/ (11753)/ G (3) (1753)/ (579)/ G (30556)/ ( )/ / G ( )/ (13970)/ / / /

13 AND WATSON S EMPIRICAL PROCESS 557 G G (3 965) G ( )/ G / G Values of G n with C ( n) From Buell s table [7], we find that for n, C ( n) when n 175, 15, 305, 3705, 305, 5, 5005 Note that n pqrs where p, q, r, s are distinct primes To compute the G pqrs for these seven n, we consider the

14 55 HENG HUAT CHAN eight expressions analogous to α, β, γ δ defined in Theorem 11 One such example is 0 G pqrs s s G ps/qr G pqr/s G prs/q G pqs/r G qrs/p 1 ν 0 G pqr/s G prs/q G pqs/r G qrs/p G pqrs s s G ps/qr 1 ν (51) Using similar reasonings as in Theorems 11 1, we conclude that the eight expressions, one of which is given in (51), are of the form abm with a, b 0 We then proceed to compute G pqrs The following list contains the remaining six values of G n : G / (5131)/ G (6315)/ (797135)/ G (19915)/ 0 57 (5) /

15 AND WATSON S EMPIRICAL PROCESS G (5) ( ) G ( )/ (1537)/ (5) (519) (15) ( ) G (3511)

16 560 HENG HUAT CHAN REMARK It is interesting to note that our methods can be used to compute G n whenever the class groups of (n) has one class per genus Assuming the generalized Riemann hypothesis, P J Weinberger [0] showed that Q(1365) is the only imaginary quadratic field with C ( ) To compute G, we simply follow the computations of G n in this section The only difference is that we obtain rational integers for expressions such as (51) instead of real quadratic numbers Of course, the above value can be obtained directly from Kronecker s Limit Formula Acknowledgements The author is greatly indebted to B C Berndt, D A Cox S Ullom for their keen interest valuable suggestions He also wishes to express his gratitude to D Buell for kindly providing a table of the pqr [7] which satisfy the conditions of Theorem 11, S L Tan for evaluating some of these invariants, the referee for his suggestions for bringing the author s attention to Weinberger s work References 1 B C BERNDT, Ramanujan s notebooks, Part III (Springer, New York, 1991) B C BERNDT, Ramanujan s notebooks, Part IV (Springer, New York, 199) 3 B C BERNDT R A RANKIN, Ramanujan: letters commentary, History of Mathematics, 9 (American Mathematical Society, Providence, 1995) B C BERNDT, HHCHAN L-C ZHANG, Ramanujan s class invariants, Kronecker s limit formulas modular equations, Trans Amer Math Soc 39 (1997) B J BIRCH, Weber s class invariants, Mathematika 16 (1969) Z I BOREVICH I R SHAFAREVICH, Number theory (Academic, New York, 1966) 7 D A BUELL, Personal communication H H CHAN S S HUANG, On the Ramanujan Go llnitz Gordon continued fraction, Ramanujan J 1 (1997) S CHOWLA, An extension of Heilbronn s class number theorem, Quart J Math Oxford Ser 5 (193) D A COX, Primes of the form xny (Wiley, New York, 199) 11 M DEURING, Die Klassenko rper der komplexes Multiplication, Enz Math Wiss B I, Heft 10, Tail II (Stuttgart) (195) 1 K G RAMANATHAN, Some applications of Kronecker s limit formula, J Indian Math Soc 5 (197) S RAMANUJAN, Modular equations approximations to π, Quart J Math Oxford Ser 5 (191) S RAMANUJAN, Collected papers (Chelsea, New York, 196) 15 D SHANKS, Dihedral quadratic approximations series for π, J Number Theory 1 (19) C L SIEGEL, Adanced analytic number theory (Tata Institute of Fundamental Research, Bombay, 190) 17 G N WATSON, Theorems stated by Ramanujan (XIV): a singular modulus, J London Math Soc 6 (1931) G N WATSON, Some singular moduli (I), Quart J Math Oxford Ser H WEBER, Lehrbuch der Algebra, B 3 (Chelsea, New York, 1961)

17 AND WATSON S EMPIRICAL PROCESS P J WEINBERGER, Exponents of the class groups of complex quadratic fields, Acta Arith (1973) L C ZHANG, Ramanujan s class invariants, Kronecker s limit formula modular equations (II), Analytic number theory: Proceedings of a conference in honor of Heini Halberstam, to appear, Birkha user Department of Mathematics National University of Singapore Kent Ridge Singapore 11960