THE PARTITION FUNCTION AND THE ARITHMETIC OF CERTAIN MODULAR L-FUNCTIONS

Transcription

1 THE PARTITION FUNCTION AND THE ARITHMETIC OF CERTAIN MODULAR L-FUNCTIONS Li Guo and Ken Ono International Mathematical Research Notes, 1, 1999, pages Introduction and Statement of Results A partition of a positive integer n is any non-increasing sequence of positive integers whose sum is n. As usual, let pn denote the number of partitions of size n. One of the fundamental tools used for studying pn is Euler s generating function 1 pnq n = 1 + q + q + 3q 3 + 5q 4 + 7q q 6 = n=0 n=1 1 1 q n. Note: By convention we have that p0 = 1. It is well known that partitions and their associated Ferrers-Young diagrams and tableaux play an important role in the study of hypergeometric functions, combinatorics, representation theory, Lie algebras, and statistical mechanics. In some cases the combinatorial properties of the partitions and Ferrers-Young diagrams are important, while in others the content of the theorems rely on identities involving relevant q-series generating functions. In this investigation we show that the values of the partition function pn, viewed as q-coefficients, play a key role in the arithmetic of several infinite families of modular L-functions. In particular, this suggests that there is a correspondence in these special cases between Tate-Shafarevich groups of certain motives of modular forms and sets of partitions. We begin by fixing notation. If l 5 is prime, then let 1 δ l l 1 and 1 r l 3 be the unique pair of integers for which 3 4δ l 1 mod l, r l l mod 4, Both authors are supported by grants from the National Science Foundation, and the second author is also supported by an Alfred P. Sloan Research Fellowship and a grant from the National Security Agency. 1 Typeset by AMS-TEX

2 LI GUO AND KEN ONO and let Dl, n denote the number given by 4 Dl, n def = 1 l 3 4n + rl. Ramanujan proved that if l = 5, 7 or 11, then 5 pln + δ l 0 mod l for every non-negative integer n. There are now many proofs of these congruences which use a variety of techniques. Most notable are those proofs by A. O. L. Atkin and H. P. F. Swinnerton-Dyer [At-SwD], F. Garvan [G], G. E. Andrews and F. Garvan [An-G], and D. Kim, F. Garvan and D. Stanton [G-Ki-St], which are motivated by conjectures of F. Dyson [Dy] on the existence and behavior of combinatorial statistics known as ranks and cranks. These statistics, by design, describe the divisibility of pn in terms of the combinatorial structure of the partitions themselves. If 13 l 31 is prime, then we show that the residue classes pln + δ l mod l play an important role in the arithmetic of certain L-functions. For these primes l, let G l z denote the unique newform in the space S l 3 Γ 0 6, χ 1 whose Fourier expansion at infinity see Appendix begins with the terms 6 G l z = a l nq n := q + n=1 l 5 q + l 3 3 l 5 q here q := e πiz l Here χ 1 denotes the trivial character and l denotes the Legendre symbol modulo l. If D is a fundamental discriminant of a quadratic number field, then let LG l χ D, s denote the L-function associated to the D-quadratic twist of G l z. If D is coprime to 6 and χ D := D denotes the Kronecker character for Q D, then for Res > l we have that LG l χ D, s = n=1 L G l χ Dl,n, l 3 χ D na l n n s. If n 0 is an integer for which Dl, n is square-free, then it turns out that 4n + rl l 4 L G l χ Dl,0, l 3 r l 4 l is the square of an integer. The first result in this paper is a congruence between these quotients and the square of a quotient of values of the partition function. Theorem 1. If 13 l 31 is prime and n 0 is an integer for which Dl, n is square-free, then L G l χ Dl,n, l 3 L G l χ Dl,0, l 3 4n + rl l 4 r l 4 l pln + δ l pδ l mod l.

3 THE PARTITION FUNCTION AND L-FUNCTIONS 3 There have been many important general non-vanishing theorems for the critical values of quadratic twists of modular L-functions by the works of D. Bump, S. Friedberg, J. Hoffstein, H. Iwaniec, N. Katz, M. R. Murty, V. K. Murty, P. Sarnak, among many others. Theorem 1 yields the following curious non-vanishing theorem for these special L-functions. Corollary. If 13 l 31 is prime, n 0 is an integer for which Dl, n is square-free and pln + δ l 0 mod l, then L G l χ Dl,n, l 3 0. Corollary provides a useful criterion for deducing the non-vanishing of these central critical values. To place it in its proper context, we recall that a well known conjecture due to D. Goldfeld [Go] implies that if F S k M, χ 1 is a newform and D denotes the fundamental discriminant of the quadratic field Q D, then 7 #{ D X : LF χ D, k 0} X. Throughout, the notation LX RX shall mean that there is a positive constant c such that for sufficiently large X we have LX c RX. Although N. Katz and P. Sarnak [Ka-Sa] have conditional proofs of 7, at present the best general result is due to C. Skinner and the second author [O-S]. They prove that if F z S k M, χ 1 is a newform, then #{ D X : LF χ D, k 0} X log X. To observe the utility of Corollary, consider the function Ψl, X given by Ψl, X def = # {0 n X : Dl, n square-free and pln + δ l 0 mod l}. # {0 n X : Dl, n square-free} This function denotes the proportion of non-negative integers n X for which Corollary implies the nonvanishing of L G l χ Dl,n, l 3. The authors are indebted to R. Weaver for compiling the following table. X Ψ13, X Ψ17, X Ψ19, X Ψ3, X Ψ9, X Ψ31, X It is well known for example, see [Fr] that if E/Q is an elliptic curve with a rational point of odd prime order l, then there is a close relationship between the l-selmer groups of E D, the D-quadratic twist of E, and the l-part of the ideal class group of Q D. By a theorem of Mazur, this only occurs for primes l 7. In general, apart from some modest results by the second author and W. Kohnen and K. James [Ko-O, J-O], very little is known about the distribution of elements of odd prime order in the Tate-Shafarevich groups of a family of quadratic twists E D.

4 4 LI GUO AND KEN ONO In view of Theorem 1 and the Bloch-Kato conjecture, it is natural to suspect that there is a close relationship between the values of the partition function modulo l, for 13 l 31, and the orders of the Tate-Shafarevich groups of a quadratic twists of certain motives. Here we show that there is indeed such a relation assuming the truth of the Bloch-Kato conjecture. If 13 l 31 is prime, then let M l be the l 3 -th Tate twist of the motive Gl associated to the newform G l by the work of A. Scholl [Sch]. Let M l be any fixed integral structure of M l which satisfies property P at l see section 3 for definitions and a canonical choice. If n is a non-negative integer, then let D l,n denote the motive associated to the Dirichlet character χ Dl,n, and let M l,n denote the twisted motive 8 M l,n def = M l Q D l,n with the integral structure M l,n induced by M l. Let XM l,n denote the Tate- Shafarevich group of M l,n with respect to the integral structure M l,n. Theorem 3. Suppose that 13 l 31 is prime and n 0 is an integer for which [ ] l + 1 i n mod l, 1 ii Dl, n is square-free, iii L G l χ Dl,n, l 3 0. Assume the truth of the Bloch-Kato Conjecture for M l,n and M l,0. Then we have that #XM l,n a ord l 0, #XM l,0 #XM l,n b ord l > 0 pln + δ #XM l,0 l 0 mod l. Note: [ ] denotes the greatest integer function. The Bloch-Kato Conjecture is a precise formula for certain special values of L-function in terms of periods, Tamagawa numbers, and the order of a Tate-Shafarevich group. To obtain Theorem 3 from Theorem 1, it suffices to show that the only factor in the Bloch- Kato formula which might not be an l-adic unit is the term corresponding to the order of the Tate-Shafarevich group. This argument is carried out in 3 and is analogous to results for modular elliptic curves appearing in [J, J-O]. A straightforward generalization of the argments in [J-O, Ko-O] yields the following result.

5 THE PARTITION FUNCTION AND L-FUNCTIONS 5 Corollary 4. Let 13 l 31 be prime. If the Bloch-Kato Conjecture holds for M l,0 and M l,n for every non-negative integer n satisfying conditions i iii in Theorem 3, then { } X # n X : ord l #XM l,n = ord l #XM l,0 l log X. Theorem 3 implies that if the l-part of XM l,0 is non-trivial, then the l-part of XM l,n is non-trivial for every n satisfying i iii. Since it is unlikely that this is ever the case, we record the following corollary. Corollary 5. Let 13 l 31 be prime and suppose that n 0 is an integer satisfying the conditions in Theorem 3. If the Bloch-Kato Conjecture holds for M l,n and M l,0 and the l-part of XM l,0 is trivial, then l #XM l,n pln + δ l 0 mod l. L. Sze and the second author [O-Sz] have defined a correspondence between 4-core partitions of size n and ideal class groups of the imaginary quadratic fields Q 8n 5. In view of this correspondence and recent observations by J. Cremona and B. Mazur [Cr- Ma] on the visualization of elements of Tate-Shafarevich groups of elliptic curves, it is natural to raise the following questions: Questions. 1 Is there a correspondence explaining Theorem 3 and Corollary 5? Do generalizations of ranks and cranks reveal structural properties of XM l,n? Acknowledgements. The authors thank F. Diamond for a number of helpful comments in the preparation of this paper.. Proof of Theorem 1 The proof of Theorem 1 depends on theorems of Shimura and Waldspurger which relate the central critical values of quadratic twists of weight k modular L-functions to the Fourier coefficients of certain weight k + 1 cusp forms. We begin by defining the relevant half integral weight forms. Recall that Dedekind s eta-function is defined by the infinite product 9 ηz def = q 1/4 1 q n. It is well known that η4z is a weight 1/ cusp form in the space S 1 Γ 0576, χ 1. Furthermore, let E 4 z and E 6 z be the usual weight 4 and 6 Eisenstein series for SL Z E 4 z = σ 3 nq n, E 6 z = n=1 n=1 σ 5 nq n, n=1

6 6 LI GUO AND KEN ONO where σ ν n := 1 d n dν. For each prime 13 l 31, define the modular form g l z = n=0 b lnq n Z[[q]] by 10 g l z = n=1 b l nq n def = Note that if n r l mod 4, then b l n is zero. η 11 4z if l = 13, η 7 4zE 4 4z if l = 17, η 5 4zE 6 4z if l = 19, η4ze 4 4zE 6 4z if l = 3, η 19 4zE 4 4z if l = 9, η 17 4zE 6 4z if l = 31. Proposition 6. If 13 l 31 is prime, then g l z is in the space S l Γ 0 576, χ 1. Moreover, g l z is an eigenform of the half integral weight Hecke operators and its image under the Shimura correspondence is G l χ 1, a newform in the space S l 3 Γ 0 144, χ 1. Proof. That each cusp form g l lies in the space S l Γ 0 576, χ 1 follows immediately from the fact that η4z is a cusp form in S 1/ Γ 0 576, χ 1, and the assertion that each g l is an eigenform is easily verified by a straightforward case by case computation for example, see [Prop. 1.3, Fr]. If F l z denotes the image of g l under the Shimura correspondence [Sh], then by a theorem of Niwa [Ni] it turns out that F l z is an eigenform in the space S l 3 Γ 0 88, χ 1. However, one easily checks see Appendix that the initial segments of the q-expansions of F l and G l χ 1 agree for more than l 3[Γ 0 1 : Γ 0 88]/1 terms. Consequently, by the standard dimension counting argument we have that F l = G l χ 1. It is simple to check that G l χ 1 is a newform in the space S l 3 Γ 0 144, χ 1. Q.E.D. By Waldspurger s work on the Shimura correspondence [Wal], it turns out that many of the coefficients of the cusp forms g l interpolate the central values of certain quadratic twists. More precisely, we have the following proposition. Proposition 7. If 13 l 31 is prime and n 0 is an integer for which Dl, n is square-free, then L G l χ Dl,n, l 3 4n + rl l 4 L G l χ Dl,0, l 3 r l 4 l = b l 4n + r l. Proof. This follows from Proposition 6, [Cor., Wal], and the fact that b l r l = 1. Q.E.D.

7 THE PARTITION FUNCTION AND L-FUNCTIONS 7 Proof of Theorem 1. To prove the theorem it suffices to prove that 11g 13 z mod 13 if l = 13, 7g 17 z mod 17 if l = 17, 11 pln + δ l q 4n+r l 5g 19 z mod 19 if l = 19, g 3 z mod 3 if l = 3, n=0 8g 9 z mod 9 if l = 9, 10g 31 z mod 31 if l = 31. Recall that if M is a positive integer, then the UM operator is defined by 1 n=0 Anq n UM def = AMnq n. n=0 Using Euler s generating function from 1, one easily finds that η l 4lz η4z { Ul = pnq 4n+l 1 = n=0 pln + δ l q 4n+βl n=0 } 1 q 4ln l n=1 1 q 4n l, n=1 Ul where β l = l + 4δ l 1 l = l + r l. Since 1 X l l 1 X l mod l, we find that 13 pln + δ l q 4n+r l n=0 l 1 4 4z Ul η l 4z mod l. Therefore, the truth of 11 and Theorem 1 follows from the truth of 14 l 1 4 4z Ul hl z def = 11 4z mod 13 if l = 13, 7 4zE 4 4z mod 17 if l = 17, 5 4zE 6 4z mod 19 if l = 19, 4zE 4 4zE 6 4z mod 3 if l = 3, 8 4zE 4 4z mod 9 if l = 9, 10 4zE 6 4z mod 31 if l = 31. Since the Hecke operator T l and the operator Ul are equivalent on the reduction modulo l of the Fourier expansion of an integer weight modular form with integer Fourier coefficients, for each l the form l 1 4 4z is congruent modulo l to a cusp form with

8 8 LI GUO AND KEN ONO integer coefficients in the space S l 1 Γ 0 4, χ 1. The forms on the right hand side of the alleged congruences in 14 lie in the space S l 1 Γ 0 4, χ 1. For each such l, it is well known that the normalized Eisenstein series E l 1 z satisfies the congruence E l 1 z 1 mod l [SwD]. Therefore, E l 1 l 1 z h lz is a cusp form which is in the space S l 1 Γ 0 4, χ 1, along with l 1 4 4z Ul, and has the same Fourier expansion modulo l as h l z. For each l a simple computation verifies that 14 holds for more than the first l Fourier coefficients. By a theorem of J. Sturm [Theorem 1, St], this implies the truth of 14. Q.E.D. In view of Theorem 1, all of the central critical values of the relevant quadratic twists are uniquely determined by L G l χ Dl,0, l 3. The table below contains approximations for these L-values. These values were calculated using the first few hundred terms of the Fourier expansions of the G l using the following well known result which is easily deduced from the integral representation of modular L-functions and the behavior of newforms under the Atkin-Lehner involution. Theorem 8. Suppose that fz = n=1 anqn S k Γ 0 N, χ 1 is a newform with integer coefficients. If ɛ = ±1 is the eigenvalue of fz with respect to the Atkin-Lehner involution W N, then Here Φx is defined by Lf, k = πk 1 + ɛ n=1 anφπn/ N k 1!N k/. Φx def = k 1! x k e x 1 + x + x! + x k 1. k 1! l L G l χ Dl,0, l 3 pδ l mod l Example. Here we illustrate the first three cases of Theorem 1 when l = 13. Using Theorem 8, one easily obtains the following numerical data using the first 500 Fourier

9 THE PARTITION FUNCTION AND L-FUNCTIONS 9 coefficients of G 13. n b 13 4n + 11 LG 13 χ D13,n, 5 4n / p13n + 6 L G 13 χ 11, / p6 mod mod mod mod Proof of Theorem 3 If is prime and n is a non-negative integer, then let M,n denote the motive defined in 8. By construction, M and each M,n admits a premotivic structure in the sense of Fontaine and Perrin-Riou for details see [F-PR]. This structure consists of Betti, de Rham, and l-adic realizations, together with comparison isomorphisms which relate these realizations. If M is M or one of the M,n, then these realizations and isomorphisms satisfy: R1 The Betti realization M B is a -dimensional Q-vector space with a G R action. R The de Rham realization M dr is a -dimensional Q-vector space with a finite decreasing filtration Fil i. R3 For each finite prime l of Q, the l-adic realization M l is a -dimensional Q l vector space with a continuous pseudo-geometric G Q action. R4 There is an R-linear isomorphism I B : C M B C M dr respecting the G R action and inducing a Q-Hodge structure over R on M B. R5 For each finite prime l of Q there is a Q l -linear isomorphism I B l : Q l Q M B M l respecting the action of G R. R6 Define the integer N M by { l0 3! if M = M l0, N M := 3! 4n + r l0 if M = M l0,n. For each prime l N M there is a B dr,l -linear isomorphism I l : B dr,l Ql M l B dr,l Ql M dr respecting filtrations and the action of G Ql, where B dr,l is the filtered Q l [G Ql ]- algebra constructed by Fontaine.

10 10 LI GUO AND KEN ONO To make sense of Theorem 3, we must first specify the nature of the integral structures M,n for the motives M,n which are used to define the relevant Tate-Shafarevich groups. These structures will be induced from a fixed choice of integral structure M for M. First we begin by defining an essential property for our investigation, motivated by the definition of motivic pairs in [B-K]. If M is a motive, then an integral structure, say M, consists of a Z-lattice M B in M B and a Z-lattice M dr in M dr such that the Z l -lattice M l def = I B l Z l M B M l is invariant under the action of G Q. In addition, there must be a finite set of primes of Q containing the infinite prime, say S, such that for every prime l S we have: C1 M l is a crystalline representation of G l. C There are integers i 0 and j 1 with j i < l such that Fil i DRM l = DRM l and Fil j DRM l = 0. C3 Z l M dr j 1 is a strongly divisible lattice of Q l Q M dr j 1. C4 The isomorphism Fi B crys Ql M dr j 1 F =1 = M l j 1 induced by I l restricts to an isomorphism Fi A crys Zl M dr j 1 F =1 = M l j 1. The ring A crys,l is defined by Fontaine in [F]. For convenience, we make the following definition. Definition Property P. An integral structure M for M has property P at a prime l if C1-C4 hold for l. There is a canonical integral structure M l0 for M l0 that has property P for every prime l 3! see [D-F-G,Ne,Sch] for further details. Specifically, let s : E 6 X 6 be the universal elliptic curve on the modular curve X 6 parameterizing elliptic curves with level 6 structure. One may choose the desired integral structure M l0 to be the integral structure coming from the premotivic structure defined by the parabolic cohomology groups -th Tate twist. For every non-negative integer n, let D l0,n denote for the Dirichlet motive associated to the Dirichlet character χ Dl0,n for example, see [De], and let D l0,n denote the canonical integral structure of D l0,n given by the canonical bases of the realizations. If l 4n + r l0, then D l0,n has property P at l. After all this, one may simply define M l0,n, the integral structure for M l0,n, by associated to the universal elliptic curve, followed by the 3 16 M,n def = M D,n.

11 THE PARTITION FUNCTION AND L-FUNCTIONS 11 By construction, M,n has property P at every prime l 3!4n + r l0. Proof of Theorem 3. Given an integral structure M,n for M,n, let M,n be the dual motive of M,n with the integral structure M,n which is dual to M,n. If n satisfies hypothesis iii in the statement of Theorem 3, then the Bloch-Kato conjecture for M,n with respect to M,n is given by the formula Conjecture Bloch-Kato. 17 LM l0,n, 0 µ M,n R = #XM l0,n p< c pm l0,n #H 0 Q, M,n f /M,n f #H 0 Q, M,n 1 f /M,n 1 f. With respect to the integral structure M,n, recall that µ M,n R is the Bloch-Kato period, XM,n is the Tate-Shafarevich group of M,n, c p M,n is the Tamagawa factor of M,n at p. Moreover, we have that M,n f = l< M,n l and M,n f = l< M,n l. The quantities M l0,n 1 f and M l0,n 1 f are defined in an analogous manner. Note: Although all of the quantities in the conjectured formula except LM,n, 0 depend on the choice of integral structure, Bloch and Kato [B-K] have shown that the truth of the conjectured formula is independent of this choice. It follows from 17 that 18 LM l0,n, 0 -part of µ M,n R = -part of #XM l0,n p< -part of c p M l0,n #H 0 Q, M,n /M,n #H 0 Q, M,n 1 l0 /M,n 1 l0. It turns out that ρ, the residual -Galois representation associated to G l0 [De1] ρ : Gal Q/Q Aut 1 M /M GL F l0, is irreducible. To see this, recall that if p 6 is prime, then tr ρfrobp a l0 p mod, det ρfrobp p 4 mod. Using the formulas in the Appendix and [Prop. 19, Se], it is simple to check that the image of ρ contains SL F l0. Therefore, for every n the mod residual representations of

12 1 LI GUO AND KEN ONO Gal Q/Q on 1 M,n /M,n and 1 M l0,n 1 l0 /M l0,n 1 l0 are both irreducible, and so H 0 Q, M,n /M,n and H 0 Q, M l0,n 1 l0 /M l0,n 1 l0 are both trivial. Hypothesis i in Theorem 3 implies 4n + r l0. So M l0,n has good reduction at [Sch] and M l0,n has P at l =. Therefore, by [Theorem 4.1, B-K], the -part of c p M l0,n is 1 for every prime p <. Consequently, 17 may be replaced by LM,n, 0 19 ord l0 = ord µ M l l0 #XM l0,n. 0,n R Since M l0 is a critical motive in the sense of Deligne [De], the comparison isomorphism I B : C Q M B C Q M dr induces an isomorphism I + M : R Q M B G R R Q M dr /Fi M dr. With respect to the lattices M B M B and M dr M dr given by the integral structure M l0, the determinant of I + M l0 defines c + M l0 R, the Deligne period for M l0, up to a choice of sign. Since M,n = M Q D,n, it is easy to see that I + M,n = I + M I + D,n. Since the determinant of I + D,n = I B D,n : R Q D,n B is gn 1, where g n is the Gauss sum a mod D,n c + M,n = c + M g 1 n. R Q D,n dr πia Dl χ Dl0,nae 0,n, we have A calculation [De] shows that the Bloch-Kato period µ M l0,n R for a critical motive equals the Deligne period up to a power of. Also as is well-known, the motivic L-function LM l0,n, s equals to LG l0 χ Dl0,n, s + 3. So 19 becomes LGl0 χ Dl0,n, 3 0 ord l0 C + M gn 1 = ord l0 #XM l0,n. Since χ Dl0,n is real, the Gauss sum g n is g n = { i Dl0, n = i 4n + r l0 if 1 mod 4, Dl0, n = 4n + r l0 if 3 mod 4.

13 THE PARTITION FUNCTION AND L-FUNCTIONS 13 We then have 1 g 0 g n 4n + rl0 r l0 4 4n + rl0 = r l0 3. In view of Hypothesis i, this fraction is an -unit. By Theorem 1, 19 and 0, for any n satisfying i iii we have p n + δ l0 ord l0 p δ l0 = ord l0 LG χ Dl0,n, 3 4n + r l0 4 LG l0 χ Dl0,0, l l r = ord l0 #XMl0,n g 0 4n + r l0 l0 4 = ord l0 #XM,n #XM,0 #XM,0 g n r 4. Theorem 3 now follows easily from the fact that pδ l0. Q.E.D. Appendix: q-expansions of the G l z We begin with the following proposition whose proof is a standard exercise. Proposition. If 13 l 31 is prime, then let S l denote the set S l := {newforms in S l 3 Γ 0 6, χ 1 } Z[[q]]. i There is exactly one newform in S 13 and its Fourier expansion begins with q 16q + 81q q q ii There is exactly one newform in S 17 and its Fourier expansion begins with q + 64q 79q q q iii There are exactly three newforms in S 19 and their Fourier expansions begin with q 18q 187q q q , q + 18q 187q q q 5..., q + 18q + 187q q q ,

14 14 LI GUO AND KEN ONO iv There are exactly three newforms in S 3 and their Fourier expansions begin with q + 51q q q q , q 51q q q q 5..., q 51q 19683q q q v There are exactly three newforms in S 9 and their Fourier expansions begin with q 4096q q q q , q q q q q 5..., q 4096q q q q vi There are exactly three newforms in S 31 and their Fourier expansions begin with q + 819q q q q 5..., q + 819q q q q , q 819q q q q Remark. It is easy to see that the forms G l are well defined by 6. Now we present formulas for the newforms G l. Since the coefficients a l and a l 3 are well defined, we simplify our formulas by giving closed expressions for some cusp forms G l z = n=1 a l nqn S l 3 Γ 0 6, χ 1. for which gcdn,6=1 a l nq n = gcdn,6=1 a l nq n S l 3 Γ 0 36, χ 1. These identities are verified by checking that the initial segments of the Fourier expansions agree for more than 6l 18 terms. G 13z = 116η 8 zη zη 8 3zη 6z + 50η 7 zη 7 zη 3 3zη 3 6z 450η 3 zη 3 zη 7 3zη 7 6z + η 8 zη zη 8 3zη 6z U

15 THE PARTITION FUNCTION AND L-FUNCTIONS 15 G 17z = η18 zη 3zη 8 6z η14 zη 8 zη 6 3z η13 zη 13 zη3zη6z η 13 zηzη3zη 13 6z η 18 zη 3zη 8 6z U + η 14 zη 8 zη 6 3z U 9 G 19z = η6 zη zη 3zη 6z η 1 zη 3 zη3zη 7 6z η16 zη 16 z η 16 zη 4 zη 1 6z η 14 zη zη 14 3zη 6z + η 6 zη zη 3zη 6z U η 1 zη 3 zη3zη 7 6z U 477 G 3z = η 3 zη 8 z η 8 zη 4 zη 4 3zη 4 6z η 4 zη 8 3zη 8 6z η 3 zη 5 zη 3 3zη 9 6z η 14 zη 14 zη 6 3zη 6 6z + η 3 zη 8 z U η 8 zη 4 zη 4 3zη 4 6z U η 4 zη 8 3zη 8 6z U η 3 zη 5 zη 3 3zη 9 6z U.

16 16 LI GUO AND KEN ONO G 9z = η 4 zη 3zη 8 6z η 38 zη 8 zη 6 3z η 37 zη 13 zη3zη6z η 37 zηzη3zη 13 6z η 4 zη 3zη 8 6z U η 38 zη 8 zη 6 3z U η 37 zη 13 zη3zη6z U η 4 zη 3zη 8 6z U η 38 zη 8 zη 6 3z U η 37 zηzη3zη 13 6z U η 37 zηzη3zη 13 6z U3 η 37 zη 13 zη3zη6z U.

17 THE PARTITION FUNCTION AND L-FUNCTIONS 17 G 31z = η 50 zη zη 3zη 6z η 45 zη 3 zη3zη 7 6z η 40 zη 16 z η 40 zη 4 zη 1 6z η 38 zη zη 14 3zη 6z η 36 zη 4 3zη 16 6z η 50 η zη 3zη 6z U η 40 zη 16 z U3 η 45 zη 3 zη3zη 7 6z U3 η 50 zη zη 3zη 6z U3 η 36 zη 4 3zη 16 6z U η 36 zη 4 3zη 16 6z U3 η 40 zη 4 zη 1 6z U.

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19 THE PARTITION FUNCTION AND L-FUNCTIONS 19 [St] [SwD] [Wal] J. Sturm, On the congruence of modular forms, Lect. Notes Math , Springer-Verlag, H.P.F. Swinnerton-Dyer, On l-adic Galois representations and congruences for coefficients of modular forms, Lecture Notes in Math , Springer-Verlag, J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures et Appl , Dept. of Math. and Comp. Sci., Rutgers University at Newark, Newark, NJ address: Dept. of Math., Penn State University, University Park, Pa address: