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Exceptional congruences for the coefficients of certain eta-product newforms
Journal of Number Theory 98 (2003) 377–389
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Exceptional congruences for the coefficients of certain eta-product newforms Matthew Boylan Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA Received 14 February 2002; revised 29 April 2002 Communicated by D. Goss
Abstract P n Let F ðzÞ ¼ N n¼1 aðnÞq denote the unique weight 16 normalized cuspidal eigenform on SL2 ðZÞ: In the early 1970s, Serre and Swinnerton-Dyer conjectured that aðpÞ2 p15 0; 1; 2; 4 ðmod 59Þ; when pa59 is prime. This was proved in 1983 by Haberland. We describe a general method for proving congruences for the coefficients of eigenforms which arise from odd octahedral complex two-dimensional Galois representations, of which this congruence is the prototypical example. In particular, we prove all such congruences for the coefficients of eta-product newforms. r 2002 Elsevier Science (USA). All rights reserved. MSC: primary 11F33; 11F80; 11P76
1. Introduction and statement of results In this note, we prove congruences for certain kinds of partition functions. If r is a non-zero integer, we define the functions pr ðnÞ by N X
pr ðnÞqn :¼
n¼0
N Y
ð1 qn Þr :
n¼1
E-mail address: [email protected]. 0022-314X/02/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 3 1 4 X ( 0 2 ) 0 0 0 3 7 - 9
ð1Þ
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Letting pe;r ðnÞ and po;r ðnÞ denote the number of r-colored partitions into an even (respectively, odd) number of distinct parts, it is easy to see that pr ðnÞ ¼ pe;r ðnÞ po;r ðnÞ;
ð2Þ
when r is a positive integer. Moreover, it is well-known that p1 ðnÞ ¼ pðnÞ; the unrestricted partition function. The functions pr ðnÞ have been studied extensively by Ramanujan [B-O], Newman [N1], Atkin [A], and Serre [Se1]. These functions enjoy certain congruence properties. For example, the coefficients of Ramanujan’s Delta-function, DðzÞ ¼ PN n n¼1 tðnÞq ; are given by tðnÞ ¼ p24 ðn 1Þ when nX1; and they satisfy the congruence p24 ðn 1Þ s11 ðnÞ ðmod 691Þ; where sk ðnÞ ¼
P
djn d40
ð3Þ
dk:
Here, we prove congruences of a much different flavor for p12 ðnÞ and the function vðnÞ which we now define: N X
vðnÞqn :¼
n¼0
N Y
ð1 qn Þ4 ð1 q2n Þ4 :
ð4Þ
n¼1
It follows that vðnÞ ¼ ve ðnÞ vo ðnÞ;
ð5Þ
where ve ðnÞ and vo ðnÞ are the number of 8-colored partitions into an even (respectively, odd) number of distinct parts, where each of the parts of the last four colors are even. Theorem 1. If p is an odd prime, then v
p1 2 0; p3 ; 2p3 ; or 4p3 ðmod 11Þ; 2
p12
p1 2
2
0; p5 ; 2p5 ; or 4p5 ðmod 19Þ:
Remark. If for 1pip4 we let dðiÞ denote the Dirichlet density of primes p for which v
p1 2
2
ip3 ðmod 11Þ
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then 8 3=8 > > > < 1=3 dðiÞ ¼ > 1=4 > > : 1=24
if i ¼ 0; if i ¼ 1; if i ¼ 2;
ð6Þ
if i ¼ 4:
The Dirichlet density of primes p for which p1 2 p12 ip5 ðmod 19Þ 2 is also given by (6). To study congruences like (3), Serre and Swinnerton-Dyer used modular Galois P n representations. If F ðzÞ ¼ N n¼1 aðnÞq AMk ðG0 ðNÞ; wÞ is an eigenform for the action of the Hecke algebra and has c-adic integer coefficients, then by a theorem of % Deligne [De], there is a representation rc;F : GalðQ=QÞ-GL 2 ðZc Þ for each prime c; depending on F and unramified outside Nc; for which Trðrc;F ðFrobp ÞÞ ¼ aðpÞ and detðrc;F ðFrobp ÞÞ ¼ wðpÞpk1 for all primes p[Nc: The projectivization of rc;F reduced modulo c is denoted by r* c;F : If the image of r* c;F in PGL2 ðFc Þ is A4 ; S4 ; or A5 ; following Swinnerton-Dyer, we say that c is exceptional of type (iii) for F : In particular, when the image is S4 ; we have aðpÞ2 0; wðpÞpk1 ; 2wðpÞpk1 ; or 4wðpÞpk1 ðmod cÞ
ð7Þ
for primes p[Nc: Moreover, the Dirichlet density of primes p for which aðpÞ2 iwðpÞpk1 ðmod cÞ is given by (6). If we let L be the kernel field of r* c;F ; then aðpÞ2 0; wðpÞpk1 ; 2wðpÞpk1 ; or 4wðpÞpk1 ðmod cÞ precisely when oðFrobp Þ ¼ 2; 3; 4; or 1 in GalðL=QÞ ¼ S4 ; respectively, where oðgÞ denotes the order of gAS4 : Therefore, the density result follows from the Chebotarev Density Theorem. In the Serre and Swinnerton-Dyer theory of congruences for the coefficients of modular forms on SL2 ðZÞ; exceptional congruences modulo c for f ðzÞASk ðG0 ð1ÞÞ of types other than type (iii) follow from congruences between two specific modular forms in certain spaces of modular forms of weight no greater than k þ c þ 1 with coefficients reduced modulo c: Therefore, these types of exceptional congruences may be verified simply by checking that the nth coefficients of the two modular forms in question agree modulo c for npkþcþ1 12 : It is particularly easy to explicitly calculate many coefficients of the modular forms in these cases. Exceptional congruences of type (iii) are also dictated by congruences between modular forms in spaces of forms
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with coefficients reduced modulo c but are much harder to verify since one of these modular forms is always a weight one form whose explicit construction is difficult. To clarify, observe the following example of a type (iii) congruence that Serre and Swinnerton-Dyer conjectured, but did not prove [SwD, Corollary (iii) to Theorem 4]. Conjecture (Serre and Swinnerton-Dyer). The prime 59 is exceptional of type (iii) for the unique normalized eigenform FSwD AS16 ðG0 ð1ÞÞ: In 1983, Haberland gave a constructive proof of the conjecture in a series of three papers [H]. His methods relied heavily on Galois cohomology. To prove Theorem 1, we exhibit all exceptional congruences of type (iii) for etaquotient newforms. However, there are a variety of intervening technical questions. Although it is straightforward to eliminate all but finitely many primes c as possible exceptional primes, it is harder to verify that any given prime is indeed exceptional. In particular, there could be rogue primes c which appear to be exceptional based on limited numerical evidence. More precisely, we do not benefit from the existence of effective forms of the Chebotarev Density Theorem since at the outset we do not have a convenient restriction on the image of the corresponding Galois representation. Therefore, to prove an exceptional congruence of type (iii) modulo the prime c for the coefficients of f ðzÞAMk ðG0 ðNÞ; wÞ; we first explicitly construct the number field L (e.g., an S4 -field if the congruence has form (7)) that seems to be dictating the alleged exceptional congruence. The proof then follows in a purely computational way from recent works of Crespo [Cp1,Cp2], Quer [Q], and Bayer and Frey [B-F] which show how to explicitly construct from L an odd irreducible complex octahedral twodimensional Galois representation and the corresponding weight one Hecke newform whose existence comes from the well-known theorems of Weil–Langlands and Langlands–Tunnell [T]. The coefficients of the weight one newform constructed this way satisfy an exceptional congruence of type (iii) modulo c; so to complete the proof we employ standard facts about modular forms to compare the well-known properties of the coefficients of the weight one form to the coefficients of f ðzÞ: In particular, this strategy may also be applied to prove the Serre and Swinnerton-Dyer example. I. Kiming and H. Verrill [K-V] have also proved the Serre and SwinnertonDyer example as well as the exceptional congruences for the coefficients of the modular forms F1 and F3 below using methods somewhat different than ours. It should also be noted that B. Gordon proved exceptional congruences of this type in unpublished notes in the early 1990s. Consider the following eta-product newforms: F1 ðzÞ ¼ Z4 ð2zÞZ4 ð4zÞAS4 ðG0 ð8ÞÞ; F2 ðzÞ ¼ Z4 ð2zÞZ16 ð4zÞZ4 ð8zÞAS4 ðG0 ð128ÞÞ; F3 ðzÞ ¼ Z12 ð2zÞAS6 ðG0 ð4ÞÞ; F4 ðzÞ ¼ Z12 ð2zÞZ36 ð4zÞZ12 ð8zÞAS6 ðG0 ð64ÞÞ: It is easy to verify that F2 and F4 are w1 -twists of F1 and F3 ; respectively.
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Theorem 2. (1) The prime 11 is exceptional of type (iii) for F1 and F2 : (2) The prime 19 is exceptional of type (iii) for F3 and F4 : Remark. (1) By checking the complete list of eta-product newforms [Ma, Table I], it is straightforward to use (7) to find that the only possible congruences of type (iii) are those given in Theorem 2. (2) There are congruences for the coefficients of eigenforms arising from odd irreducible complex two-dimensional Galois representations for which Artin’s Conjecture is true. In particular, we expect many congruences arising from tetrahedral, octahedral, and icosahedral representations. In view of the works of Buhler, Frey, Taylor, and others, one can find newforms of high weight possessing such congruences. However, checking the list of eta-product newforms, pffiffi and using (7) with 0,1,2,4 replaced by 0,1,4 and by 0; 1; 4; 372 5 in the tetrahedral and icosahedral cases, respectively, we find that there are no congruences for the coefficients of eta-product newforms arising from tetrahedral or icosahedral representations. Since the only published proof of an exceptional congruence of type (iii) is [H] (to the best of our knowledge), we feel it is important to give a separate and elementary explanation of how any exceptional congruence of type (iii) coming from an octahedral representation may be proved. Moreover, it is interesting to see the Serre and Swinnerton-Dyer theory applied to the eta-product newforms, yielding concrete and elegant number theoretic statements in line with the earlier works cited above.
2. Preliminaries In this section, we give the important preliminary facts regarding certain kinds of complex two-dimensional Galois representations % r : GalðQ=QÞ-GL 2 ðCÞ; and their connections to weight one newforms. Throughout, we will denote by r0 the projectivization of r: A representation r is octahedral if the image of r0 in PGL2 ðCÞ is S4 : We begin by explicitly constructing r; an odd irreducible complex octahedral two-dimensional Galois representation. The projectivization r0 is uniquely determined by its kernel field which we denote by K: The Galois group of K is S4 ; and K is the splitting field of some irreducible quartic polynomial gðxÞ having roots fx1 ; x2 ; x3 ; x4 g: Hence, such a gðxÞ explicitly defines r0 : Throughout this section, we will assume that K contains a quartic subfield with negative discriminant d which we may take to be K1 ¼ Qðx1 Þ without loss of generality. We will also assume that the A4 -subfield of K is pffiffiffi M ¼ Qð d Þ: Later we will discuss how to choose gðxÞ so that K enjoys certain properties.
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Before we can explicitly define r; it is necessary to determine whether or not K ˜ where S˜ 4 is isomorphic to GL2 ðF3 Þ; the may be extended to an S˜ 4 -extension K; double cover of S4 in which transpositions lift to involutions. Assuming the hypotheses on K and K˜ given above, the solvability of this embedding problem may be determined by a simple calculation. Theorem 3 ([B-F, Proposition 1.2]). The field K may be extended to K˜ if and only if the local invariant eK;p is trivial in the Brauer group of Qp for each odd prime p which ramifies in K: The invariants eK;p may be calculated from the following table, where pi ; p0i and p00i denote factors of residue degree i in K1 and ðd; pÞp ¼ ðapÞ; where a ¼ ð1Þordp ðdÞ pordp ðdÞ d is the Hilbert Symbol. pOK1
eK;p
p41
ð1Þ
p2 1 p1 8 ð1Þ 2
p31 p01
1
p21 p01 p001
1
p21 p02
1
p21 p02 1
ð1Þ
p1 2 ðd; pÞ
p22
ð1Þ
pþ1 2
p
Assuming that the embedding problem is solvable, explicit solutions lAK for pffiffiffi which K˜ ¼ Kð lÞ are given by a formula of Crespo. Theorem 4 ([Cp1, Theorem 5]). Let T be the matrix of the bilinear form pffiffiffi TrK1 ðpffiffid Þ=M ðX 2 Þ with respect to the basis f1; x1 ; x21 ; x31 g for K1 ð d Þ=M: Let P be a matrix in GL4 ðQÞ satisfying
0
1
0 0
0
1
B0 B Pt TP ¼ B @0
1 0 0 2
0 0
C C C A
0
0 0
2d
ð8Þ
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383
and define 20 1 6B 6B 1 g :¼ det6 6B 4@ 1 1
x1 x2
x12 x22
x3 x4
x23 x24
1 0 1 x31 3C B x2 C B 0 CPB x33 A B @0 x34
0
0
0
1 0
0
0
1 2 1 ffiffi p 2 d
0
1
3
7 C 0 C 7 C þ I 7a0; 1 C 7 5 2 A 1 ffiffi p
ð9Þ
2 d
where I is the identity. Then all solutions of the embedding problem are given by pffiffiffiffi Kð rgÞ as r runs over Qn =Q * 2 : The field K˜ is the kernel field of an irreducible complex octahedral twodimensional Galois representation pffiffiffiffiffiffiffi ˜ % r : GalðQ=QÞ-Galð K=QÞ ¼ S˜ 4 -GL2 ðZ½ 2 ÞCGL2 ðCÞ; ð10Þ where we have chosen one of the two faithful irreducible representations of S˜ 4 in GL2 ðCÞ: If we suppose that r is odd, then by the theorems of Weil–Langlands and Langlands–Tunnell mentioned in the introduction, there exists a weight one Hecke P n newform GðzÞ ¼ N n¼1 bðnÞq AS1 ðG0 ðNÞ; wÞ which is the inverse Mellin transform of the Artin L-function Lðs; rÞ: Here N is the Artin conductor of r; and w is the character detðrÞ: (For background on Artin L-functions and connections to weight one newforms, see [Me] or [Se2], for example). One may compute N directly from its algebraic definition. We verify numerically that N is the conductor of r by checking that GðzÞ is an eigenform for the action of the Atkin–Lehner involution oðNÞ ¼ 0 1 N 0 with level N as required by the Atkin–Lehner–Li-Miyake theory of newforms. We note that the image of r consists of matrices whose entries are algebraic pffiffiffiffiffiffiffi integers in Z½ 2 : For an arbitrary rational prime c; the reduction of r modulo a pffiffiffiffiffiffiffi prime above c in Z½ 2 has projective image S4 ; so every prime c is exceptional of type (iii) for G: It remains to actually calculate the coefficients of GðzÞ: We briefly describe the algorithm of Bayer and Frey enabling us to explicitly calculate the coefficients bðpÞ for primes p[N: By (10), the coefficients bðpÞ for primes p[N are given by the character table for the chosen representation of S˜ 4 (see [Df, Chapter 28], for example). The factorization of p in K1 corresponds to a particular conjugacy class of Frobp in S4 ; denoted by pp : We denote the conjugacy class of Frobp in S˜ 4 by p* p : This information is summarized in Table 1, where the lower index on the primes in K1 indicates the residue degree, as before. pffiffiffi Since we know explicitly the factor l for which K˜ ¼ Kð lÞ by Theorem 4, we can determine the conjugacy class in S˜ 4 when the factorization types 1, 4, and 5 occur by using Propositions 5 and 6. In what follows, let ls denote the action of sAS4 on l: Proposition 5 ([B-F, Proposition 2.4]). Suppose P is a prime in K above p: If pp ¼ 1A; (resp. 3A) then bðpÞ ¼ 2; (resp. 1) if and only if l is a square modulo P:
M. Boylan / Journal of Number Theory 98 (2003) 377–389
384 Table 1 Factorization type
pOK1
pp
p* p
bðpÞ
1
p1 p01 p001 p000 1
1A
2 3 4
p2 p02 p1 p01 p2 p1 p3
2A 2B 3A
5
p4
4A
1A˜ 2A˜ 4A˜ 2B˜ 3A˜ 6A˜ 8A˜ 8B˜
2 2 0 0 1 1 pffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffi 2
Proposition 6 ([B-F, Proposition 2.5]). Suppose p is an odd prime and pOK1 ¼ p4 : pffiffiffiffiffiffiffi Then bðpÞ ¼ 2 if and only if l
p1 2
1 lð3;2;1Þ lð1;2;3Þ ðmod p4 Þ: þ 2 2l
ð11Þ
3. The proof of Theorem 2 To prove Theorem 2, we will also need certain facts about modular forms modulo p when pX5 is prime. Let Mk;p ðG0 ðNÞÞ denote the Fp -vector space of modular forms in Mk ðG0 ðNÞÞ-Z½½q
with coefficients reduced modulo p; and suppose that F ðzÞ ¼ PN n n¼0 aðnÞq AMk ðG0 ðNÞÞ-Z½½q
: Then the action of the Ramanujan theta-operator is given by yp ðF Þ :
N X
naðnÞqn ðmod pÞAMkþpþ1;p ðG0 ðNÞÞ:
ð12Þ
n¼0
Propositions 7 and 8 are also useful tools for studying modular forms modulo p: Proposition 7. (1) If p is an odd prime and Ap ðzÞ :¼ p1 2 p
Then Ap ðzÞAMp1 ðG0 ðpÞ; ðð1Þ
Zp ðzÞ : ZðpzÞ
ÞÞ and Ap ðzÞ 1 ðmod pÞ:
2
(2) [Sw-D] Suppose kX4 is an even integer. Then the normalized Eisenstein series of weight k with respect to SL2 ðZÞ is given by Ek ðzÞ ¼ 1
2k X sk1 ðnÞqn : Bk nX1
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Furthermore, if pX5 is prime, then Ep1 ðzÞ 1 ðmod pÞ: Proof. (1) See [G-H], [N2,N3] for the first part. The second part follows from the Children’s Binomial Theorem in characteristic p: (i.e., that 1 xp ð1 xÞp ðmod pÞ when p is a prime.) (2) By the Von Staudt–Claussen Theorem, the power of a prime p dividing the denominator of Bk ; the kth Bernoulli number, is one if p 1 j k and zero otherwise. The congruence for Ep1 follows. If F is a number field with ring of algebraic integers OF ; if g ¼ PN n n¼0 aðnÞq AOF ½½q
; and if p is a prime ideal in OF ; then ( minfn : aðnÞepg if aðnÞep for some n; ordp ðgÞ :¼ þN otherwise: With this definition we can state Sturm’s Theorem on the congruence of modular forms. Proposition 8 ([St, Theorem 1]). If f ; gAMk ðG0 ðNÞÞ-OF ½½q
and if kN Y 1 1þ ; ordp ðf gÞ4 12 pjN p then ordp ðf gÞ ¼ þN: i.e., f g ðmod pÞ: The constant on the right-hand side of the inequality in Proposition 8 is called Sturm’s bound for f g: To prove that a prime c is exceptional of type (iii) for an eigenform F ðzÞ ¼ PN n n¼0 aðnÞq ASk ðG0 ðNÞ; wÞ; we first build a specific odd irreducible complex % octahedral two-dimensional Galois representation r : GalðQ=QÞ-GL 2 ðCÞ with certain properties depending on F ; as described in Section 2. In this general setting, we want to construct K so that it is isomorphic to the kernel field L of r* c;F if c were exceptional of type (iii) for F : The comments immediately following (7) indicate that we can determine the splitting of primes in L from the coefficients aðpÞ modulo c: Any irreducible quartic polynomial defining K must split completely modulo primes which are known to split completely in L: The field K must also be unramified outside Nc; which restricts the values of the discriminants of the defining quartics. These requirements allow us to quickly verify whether or not a given quartic is a good candidate for defining K: Using the facts about modular forms modulo p stated at the beginning of this section, we then show that the weight one newform obtained from r is congruent modulo c to a certain power of the theta-operator acting on F ; thus proving a
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M. Boylan / Journal of Number Theory 98 (2003) 377–389
nontrivial congruence for the coefficients of F : Therefore, c is exceptional of type (iii) for F : Proof of Theorem 2. We show that 19 is exceptional of type (iii) for F3 ðzÞ ¼ Z12 ð2zÞ: We first choose an appropriate irreducible quartic defining the S4 -field K: Our calculations are based on gðxÞ ¼ x4 þ 76x2 722x þ 1444
ð13Þ
with roots fx1 ; x2 ; x3 ; x4 g obtained from the arithmetic of the CM elliptic curve 361A [Cr] by a formula of Bayer and Frey [Ba-F, p. 401]. The discriminant of K1 ¼ Qðx1 Þ pffiffiffiffiffiffiffiffiffi is 193 22 ; and the A4 -subfield is M ¼ Qð 19Þ: The prime factorization of 19 in K1 is given by 19 ¼ P4 ; so eK;19 ¼ 1; implying that K may be extended to an S˜ 4 -field K˜ by Theorem 3. Next we calculate g19 ¼ 72x31 x22 þ 513x21 x22 1368x1 x22 4332x22 þ 1026x31 x2 4104x21 x2 þ 64980x1 x2 217683x2 þ 3420x31 þ 36100x21 þ 488433x1 2249752
ð14Þ
pffiffiffiffiffiffi by Theorem 4. The S*4 -field Kð g19 Þ defines an irreducible complex octahedral twodimensional Galois representation r: Following the Bayer–Frey algorithm, we explicitly construct the coefficients of the weight one Hecke newform HðzÞ coming pffiffiffiffiffiffiffiffi from r: The other solutions Kð rg19 Þ to this embedding problem define the Galois representations r#wr ; where wr ¼ ðr Þ: These representations determine the weight one modular forms HðzÞ#wr : Hence, the weight one form that seems to be dictating the exceptional congruence for the coefficients of F3 ðzÞ is HðzÞ or some twist of HðzÞ: In particular, the weight one newform we need is GðzÞ ¼ HðzÞ#w38 ¼
N X
bðnÞqn ¼ q þ
pffiffiffiffiffiffiffi 3 2q þ q5 þ q7 q9
n¼1
þ ?AS1 G0 ð22 192 Þ; : 19
ð15Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coming from the field K˜ ¼ Kð 38g19 Þ having discriminant 218 1921 : The Artin conductor is 22 192 and the determinant character is ð19 Þ; so r is also odd. As mentioned earlier, every prime p is exceptional of type (iii) for G: In particular, the prime p ¼ 19 is exceptional of type (iii) for G; so bðpÞ2 0; p9 ; 2p9 ; or 4p9 ðmod 19Þ:
ð16Þ
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387
By Proposition 7, GðzÞ GðzÞE18 ðzÞ2 A19 ðzÞ ðmod 19Þ
ð17Þ
and by (12), y219 ðF3 ðzÞÞ
N X
n2 aðnÞqn ðmod 19Þ;
ð18Þ
n¼0
where GðzÞE18 ðzÞ2 A19 ðzÞ ðmod 19Þ and y219 ðF3 ðzÞÞ ðmod 19Þ lie in the space M46;19 ðG0 ð22 192 ÞÞ whose Sturm bound is 8740. Using Propositions 5 and 6 to calculate the coefficients of GðzÞ; we verify that GðzÞ y219 ðF3 ðzÞÞ ðmod 19Þ
ð19Þ
by Proposition 8. In light of (16), this shows that 19 is exceptional of type (iii) for F3 ; and hence, for F4 ; since F4 is a w1 -twist of F3 ; proving part (2) of Theorem 2. Table 2 contains data for the remaining case of Theorem 2 (when c ¼ 11) and for the example of Serre and Swinnerton-Dyer. Here N is the conductor of r; and S is the Sturm bound for the space Mk;p ðG0 ðNÞÞ in which the relevant congruence of modular forms is proved. We observe that in each case F ðzÞAMcþ5 ðG0 ðMÞÞ;
ð20Þ
4
where M is the part of N prime to c and that c3
yc 8 ðF ðzÞÞ GðzÞ ðmod cÞAMc2 1
8 þ1;c
ðG0 ðNÞÞ:
ð21Þ
A finite computation completes the proof of each exceptional congruence of type (iii). Remark. (1) We determine GðzÞ; the appropriate twist of HðzÞ in the example above, so that GðzÞ will satisfy (19). In particular, since the pth coefficients of such a GðzÞ pffiffiffiffiffiffiffi (when p is prime) are in the set f0; 71; 72; 7 2g; these pth coefficients must be determined by the reduction of the pth coefficients of y219 ðF3 ðzÞÞ: We use this criteria to check which twist HðzÞ#wr for squarefree r j 38 is the correct one. Table 2 c
F ðzÞ
gðxÞ
N
w
S
11 19 59
Z4 ð4zÞZ4 ð2zÞ Z12 ð2zÞ FSwD
x4 16x3 þ 30x2 þ 30x 74 x4 þ 76x2 722x þ 1444 x4 x3 7x2 þ 11x þ 3
23 112 22 192 592
ð11 Þ ð19 Þ ð59 Þ
2112 8740 2180
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388
(2) We note that the choice of gðxÞ in these examples is certainly not unique. By performing a brute force search on quartic polynomials in the example above, we are immediately led to gðxÞ ¼ x4 x3 2x2 6x 2; having roots x1 ; x2 ; x3 ; x4 : In this pffiffiffiffiffiffiffiffiffiffiffi case, we apply the Bayer-Frey algorithm to K˜ ¼ Kð 38g019 Þ; where g019 ¼ 6x31 x22 þ 18x21 x22 51x1 x22 33x22 þ 42x31 x2 105x21 x2 þ 33x1 x2 þ 3x2 10x31 7x21 þ 95x1 þ 190 to obtain the same weight one form GðzÞ as before. (3) The author notes that the Artin L-function constructed in connection with showing that 11 is exceptional of type (iii) for F1 is also constructed by Bayer and Frey [B-F, Example 3], but not in the context of exceptional congruences. In this case, gðxÞ is obtained from the arithmetic of the CM elliptic curve 121B [Cr]. (4) In Swinnerton-Dyer’s original discussion of the exceptional prime 59 for FSwD ; the coefficients 11 and 7 in our gðxÞ are interchanged, a typographical error.
Acknowledgments I am thankful to Jordi Quer for helping me implement Propositions 5 and 6 with Mathematica, and for calculating the minimal polynomials of the generators of ˜ Thanks also to Teresa Crespo for advice on implementing the relevant fields K: formula (9).
References [A] [B-F] [B-O]
[Cr] [Cp1] [Cp2] [De] [Df] [H] [K-V]
A.O.L. Atkin, Ramanujan congruences for pk ðnÞ; Canad. J. Math. 20 (1968) 67–78. P. Bayer, G. Frey, Galois representations of octahedral type and 2-coverings of elliptic curves, Math. Z. 207 (1991) 395–408. B. Berndt, K. Ono, Ramanujan’s unpublished manuscript on the partition and tau functions with commentary, in: D. Foata, G.-N. Han (Eds.), The Andrews Festschrift, Springer, Berlin, 2001, pp. 39–110. J. Cremona, Algorithms for Modular Elliptic Curves, Cambridge University Press, Cambridge, 1997. T. Crespo, Advances in Number Theory (Kingston, ON, 1991), Oxford Scientific Publication, Oxford University Press, Oxford, 1993, pp. 255–260. T. Crespo, Explicit construction of 2Sn -Galois extensions, J. Algebra 129 (1990) 312–319. P. Deligne, Formes modulaire et repre´sentations c-adiques, Se´m. Bourbaki 355 (1969) 139–172. L. Dornhoff, Group Representation Theory, Part A, Dekker, New York, 1971. K. von Haberland, Perioden von Modulformen einer Variablen und Gruppencohomologie I,II,III, Math. Nachr. 112 (1983) 245–282, 283–295, 297–315. I. Kiming, H. Verrill, On modular mod c Galois representations with exceptional images, preprint (2000), to appear.
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[N1] [N2] [N3] [Q] [Se1] [Se2]
[St] [SwD]
[T]
389
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996) 4825–4856. J. Martinet, Algebraic Number Fields: L-functions and Galois Properties (Proceedings of the Symposium of University of Durham, Durham, 1975), Academic Press, London, 1977, pp. 1–87. M. Newman, Analytic number Theory (Allerton Park, IL, 1989), Birkhauser, Boston, 1990, pp. 405–411. M. Newman, Construction and application of a certain class of modular functions, Proc. London Math. Soc. 7 (3) (1956) 334–350. M. Newman, Construction and application of a certain class of modular functions, II Proc. London Math. Soc. 9 (3) (1959) 373–387. J. Quer, private communication J.-P. Serre, Sur la lacunarite´ des puissance de Z; Glasgow Math. J. 27 (1985) 203–221. J.-P. Serre, Algebraic Number Fields: L-functions and Galois Properties (Proceedings of the Symposium of University of Durham, Durham, 1975), Academic Press, London, 1977, pp. 193–268. J. Sturm, On the Congruence of Modular Forms, Lecture Notes in Mathematics, Vol. 1240, Springer, Berlin, 1984, pp. 275–280. H.P.F. Swinnerton-Dyer, On c-adic representations and congruences for coefficients of modular forms, in: Modular Functions of One Variable, Lecture Notes in Mathematics, Vol. III 350, Springer, Berlin, 1973, pp. 1–55. J. Tunnell, Artin’s Conjecture for representations of octahedral type, Bull. Amer. Math. Soc. (N.S.) 5 (1981) 173–175.
http://www.elsevier.com/locate/jnt
Exceptional congruences for the coefficients of certain eta-product newforms Matthew Boylan Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA Received 14 February 2002; revised 29 April 2002 Communicated by D. Goss
Abstract P n Let F ðzÞ ¼ N n¼1 aðnÞq denote the unique weight 16 normalized cuspidal eigenform on SL2 ðZÞ: In the early 1970s, Serre and Swinnerton-Dyer conjectured that aðpÞ2 p15 0; 1; 2; 4 ðmod 59Þ; when pa59 is prime. This was proved in 1983 by Haberland. We describe a general method for proving congruences for the coefficients of eigenforms which arise from odd octahedral complex two-dimensional Galois representations, of which this congruence is the prototypical example. In particular, we prove all such congruences for the coefficients of eta-product newforms. r 2002 Elsevier Science (USA). All rights reserved. MSC: primary 11F33; 11F80; 11P76
1. Introduction and statement of results In this note, we prove congruences for certain kinds of partition functions. If r is a non-zero integer, we define the functions pr ðnÞ by N X
pr ðnÞqn :¼
n¼0
N Y
ð1 qn Þr :
n¼1
E-mail address: [email protected]. 0022-314X/02/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 3 1 4 X ( 0 2 ) 0 0 0 3 7 - 9
ð1Þ
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378
Letting pe;r ðnÞ and po;r ðnÞ denote the number of r-colored partitions into an even (respectively, odd) number of distinct parts, it is easy to see that pr ðnÞ ¼ pe;r ðnÞ po;r ðnÞ;
ð2Þ
when r is a positive integer. Moreover, it is well-known that p1 ðnÞ ¼ pðnÞ; the unrestricted partition function. The functions pr ðnÞ have been studied extensively by Ramanujan [B-O], Newman [N1], Atkin [A], and Serre [Se1]. These functions enjoy certain congruence properties. For example, the coefficients of Ramanujan’s Delta-function, DðzÞ ¼ PN n n¼1 tðnÞq ; are given by tðnÞ ¼ p24 ðn 1Þ when nX1; and they satisfy the congruence p24 ðn 1Þ s11 ðnÞ ðmod 691Þ; where sk ðnÞ ¼
P
djn d40
ð3Þ
dk:
Here, we prove congruences of a much different flavor for p12 ðnÞ and the function vðnÞ which we now define: N X
vðnÞqn :¼
n¼0
N Y
ð1 qn Þ4 ð1 q2n Þ4 :
ð4Þ
n¼1
It follows that vðnÞ ¼ ve ðnÞ vo ðnÞ;
ð5Þ
where ve ðnÞ and vo ðnÞ are the number of 8-colored partitions into an even (respectively, odd) number of distinct parts, where each of the parts of the last four colors are even. Theorem 1. If p is an odd prime, then v
p1 2 0; p3 ; 2p3 ; or 4p3 ðmod 11Þ; 2
p12
p1 2
2
0; p5 ; 2p5 ; or 4p5 ðmod 19Þ:
Remark. If for 1pip4 we let dðiÞ denote the Dirichlet density of primes p for which v
p1 2
2
ip3 ðmod 11Þ
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379
then 8 3=8 > > > < 1=3 dðiÞ ¼ > 1=4 > > : 1=24
if i ¼ 0; if i ¼ 1; if i ¼ 2;
ð6Þ
if i ¼ 4:
The Dirichlet density of primes p for which p1 2 p12 ip5 ðmod 19Þ 2 is also given by (6). To study congruences like (3), Serre and Swinnerton-Dyer used modular Galois P n representations. If F ðzÞ ¼ N n¼1 aðnÞq AMk ðG0 ðNÞ; wÞ is an eigenform for the action of the Hecke algebra and has c-adic integer coefficients, then by a theorem of % Deligne [De], there is a representation rc;F : GalðQ=QÞ-GL 2 ðZc Þ for each prime c; depending on F and unramified outside Nc; for which Trðrc;F ðFrobp ÞÞ ¼ aðpÞ and detðrc;F ðFrobp ÞÞ ¼ wðpÞpk1 for all primes p[Nc: The projectivization of rc;F reduced modulo c is denoted by r* c;F : If the image of r* c;F in PGL2 ðFc Þ is A4 ; S4 ; or A5 ; following Swinnerton-Dyer, we say that c is exceptional of type (iii) for F : In particular, when the image is S4 ; we have aðpÞ2 0; wðpÞpk1 ; 2wðpÞpk1 ; or 4wðpÞpk1 ðmod cÞ
ð7Þ
for primes p[Nc: Moreover, the Dirichlet density of primes p for which aðpÞ2 iwðpÞpk1 ðmod cÞ is given by (6). If we let L be the kernel field of r* c;F ; then aðpÞ2 0; wðpÞpk1 ; 2wðpÞpk1 ; or 4wðpÞpk1 ðmod cÞ precisely when oðFrobp Þ ¼ 2; 3; 4; or 1 in GalðL=QÞ ¼ S4 ; respectively, where oðgÞ denotes the order of gAS4 : Therefore, the density result follows from the Chebotarev Density Theorem. In the Serre and Swinnerton-Dyer theory of congruences for the coefficients of modular forms on SL2 ðZÞ; exceptional congruences modulo c for f ðzÞASk ðG0 ð1ÞÞ of types other than type (iii) follow from congruences between two specific modular forms in certain spaces of modular forms of weight no greater than k þ c þ 1 with coefficients reduced modulo c: Therefore, these types of exceptional congruences may be verified simply by checking that the nth coefficients of the two modular forms in question agree modulo c for npkþcþ1 12 : It is particularly easy to explicitly calculate many coefficients of the modular forms in these cases. Exceptional congruences of type (iii) are also dictated by congruences between modular forms in spaces of forms
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with coefficients reduced modulo c but are much harder to verify since one of these modular forms is always a weight one form whose explicit construction is difficult. To clarify, observe the following example of a type (iii) congruence that Serre and Swinnerton-Dyer conjectured, but did not prove [SwD, Corollary (iii) to Theorem 4]. Conjecture (Serre and Swinnerton-Dyer). The prime 59 is exceptional of type (iii) for the unique normalized eigenform FSwD AS16 ðG0 ð1ÞÞ: In 1983, Haberland gave a constructive proof of the conjecture in a series of three papers [H]. His methods relied heavily on Galois cohomology. To prove Theorem 1, we exhibit all exceptional congruences of type (iii) for etaquotient newforms. However, there are a variety of intervening technical questions. Although it is straightforward to eliminate all but finitely many primes c as possible exceptional primes, it is harder to verify that any given prime is indeed exceptional. In particular, there could be rogue primes c which appear to be exceptional based on limited numerical evidence. More precisely, we do not benefit from the existence of effective forms of the Chebotarev Density Theorem since at the outset we do not have a convenient restriction on the image of the corresponding Galois representation. Therefore, to prove an exceptional congruence of type (iii) modulo the prime c for the coefficients of f ðzÞAMk ðG0 ðNÞ; wÞ; we first explicitly construct the number field L (e.g., an S4 -field if the congruence has form (7)) that seems to be dictating the alleged exceptional congruence. The proof then follows in a purely computational way from recent works of Crespo [Cp1,Cp2], Quer [Q], and Bayer and Frey [B-F] which show how to explicitly construct from L an odd irreducible complex octahedral twodimensional Galois representation and the corresponding weight one Hecke newform whose existence comes from the well-known theorems of Weil–Langlands and Langlands–Tunnell [T]. The coefficients of the weight one newform constructed this way satisfy an exceptional congruence of type (iii) modulo c; so to complete the proof we employ standard facts about modular forms to compare the well-known properties of the coefficients of the weight one form to the coefficients of f ðzÞ: In particular, this strategy may also be applied to prove the Serre and Swinnerton-Dyer example. I. Kiming and H. Verrill [K-V] have also proved the Serre and SwinnertonDyer example as well as the exceptional congruences for the coefficients of the modular forms F1 and F3 below using methods somewhat different than ours. It should also be noted that B. Gordon proved exceptional congruences of this type in unpublished notes in the early 1990s. Consider the following eta-product newforms: F1 ðzÞ ¼ Z4 ð2zÞZ4 ð4zÞAS4 ðG0 ð8ÞÞ; F2 ðzÞ ¼ Z4 ð2zÞZ16 ð4zÞZ4 ð8zÞAS4 ðG0 ð128ÞÞ; F3 ðzÞ ¼ Z12 ð2zÞAS6 ðG0 ð4ÞÞ; F4 ðzÞ ¼ Z12 ð2zÞZ36 ð4zÞZ12 ð8zÞAS6 ðG0 ð64ÞÞ: It is easy to verify that F2 and F4 are w1 -twists of F1 and F3 ; respectively.
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Theorem 2. (1) The prime 11 is exceptional of type (iii) for F1 and F2 : (2) The prime 19 is exceptional of type (iii) for F3 and F4 : Remark. (1) By checking the complete list of eta-product newforms [Ma, Table I], it is straightforward to use (7) to find that the only possible congruences of type (iii) are those given in Theorem 2. (2) There are congruences for the coefficients of eigenforms arising from odd irreducible complex two-dimensional Galois representations for which Artin’s Conjecture is true. In particular, we expect many congruences arising from tetrahedral, octahedral, and icosahedral representations. In view of the works of Buhler, Frey, Taylor, and others, one can find newforms of high weight possessing such congruences. However, checking the list of eta-product newforms, pffiffi and using (7) with 0,1,2,4 replaced by 0,1,4 and by 0; 1; 4; 372 5 in the tetrahedral and icosahedral cases, respectively, we find that there are no congruences for the coefficients of eta-product newforms arising from tetrahedral or icosahedral representations. Since the only published proof of an exceptional congruence of type (iii) is [H] (to the best of our knowledge), we feel it is important to give a separate and elementary explanation of how any exceptional congruence of type (iii) coming from an octahedral representation may be proved. Moreover, it is interesting to see the Serre and Swinnerton-Dyer theory applied to the eta-product newforms, yielding concrete and elegant number theoretic statements in line with the earlier works cited above.
2. Preliminaries In this section, we give the important preliminary facts regarding certain kinds of complex two-dimensional Galois representations % r : GalðQ=QÞ-GL 2 ðCÞ; and their connections to weight one newforms. Throughout, we will denote by r0 the projectivization of r: A representation r is octahedral if the image of r0 in PGL2 ðCÞ is S4 : We begin by explicitly constructing r; an odd irreducible complex octahedral two-dimensional Galois representation. The projectivization r0 is uniquely determined by its kernel field which we denote by K: The Galois group of K is S4 ; and K is the splitting field of some irreducible quartic polynomial gðxÞ having roots fx1 ; x2 ; x3 ; x4 g: Hence, such a gðxÞ explicitly defines r0 : Throughout this section, we will assume that K contains a quartic subfield with negative discriminant d which we may take to be K1 ¼ Qðx1 Þ without loss of generality. We will also assume that the A4 -subfield of K is pffiffiffi M ¼ Qð d Þ: Later we will discuss how to choose gðxÞ so that K enjoys certain properties.
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Before we can explicitly define r; it is necessary to determine whether or not K ˜ where S˜ 4 is isomorphic to GL2 ðF3 Þ; the may be extended to an S˜ 4 -extension K; double cover of S4 in which transpositions lift to involutions. Assuming the hypotheses on K and K˜ given above, the solvability of this embedding problem may be determined by a simple calculation. Theorem 3 ([B-F, Proposition 1.2]). The field K may be extended to K˜ if and only if the local invariant eK;p is trivial in the Brauer group of Qp for each odd prime p which ramifies in K: The invariants eK;p may be calculated from the following table, where pi ; p0i and p00i denote factors of residue degree i in K1 and ðd; pÞp ¼ ðapÞ; where a ¼ ð1Þordp ðdÞ pordp ðdÞ d is the Hilbert Symbol. pOK1
eK;p
p41
ð1Þ
p2 1 p1 8 ð1Þ 2
p31 p01
1
p21 p01 p001
1
p21 p02
1
p21 p02 1
ð1Þ
p1 2 ðd; pÞ
p22
ð1Þ
pþ1 2
p
Assuming that the embedding problem is solvable, explicit solutions lAK for pffiffiffi which K˜ ¼ Kð lÞ are given by a formula of Crespo. Theorem 4 ([Cp1, Theorem 5]). Let T be the matrix of the bilinear form pffiffiffi TrK1 ðpffiffid Þ=M ðX 2 Þ with respect to the basis f1; x1 ; x21 ; x31 g for K1 ð d Þ=M: Let P be a matrix in GL4 ðQÞ satisfying
0
1
0 0
0
1
B0 B Pt TP ¼ B @0
1 0 0 2
0 0
C C C A
0
0 0
2d
ð8Þ
M. Boylan / Journal of Number Theory 98 (2003) 377–389
383
and define 20 1 6B 6B 1 g :¼ det6 6B 4@ 1 1
x1 x2
x12 x22
x3 x4
x23 x24
1 0 1 x31 3C B x2 C B 0 CPB x33 A B @0 x34
0
0
0
1 0
0
0
1 2 1 ffiffi p 2 d
0
1
3
7 C 0 C 7 C þ I 7a0; 1 C 7 5 2 A 1 ffiffi p
ð9Þ
2 d
where I is the identity. Then all solutions of the embedding problem are given by pffiffiffiffi Kð rgÞ as r runs over Qn =Q * 2 : The field K˜ is the kernel field of an irreducible complex octahedral twodimensional Galois representation pffiffiffiffiffiffiffi ˜ % r : GalðQ=QÞ-Galð K=QÞ ¼ S˜ 4 -GL2 ðZ½ 2 ÞCGL2 ðCÞ; ð10Þ where we have chosen one of the two faithful irreducible representations of S˜ 4 in GL2 ðCÞ: If we suppose that r is odd, then by the theorems of Weil–Langlands and Langlands–Tunnell mentioned in the introduction, there exists a weight one Hecke P n newform GðzÞ ¼ N n¼1 bðnÞq AS1 ðG0 ðNÞ; wÞ which is the inverse Mellin transform of the Artin L-function Lðs; rÞ: Here N is the Artin conductor of r; and w is the character detðrÞ: (For background on Artin L-functions and connections to weight one newforms, see [Me] or [Se2], for example). One may compute N directly from its algebraic definition. We verify numerically that N is the conductor of r by checking that GðzÞ is an eigenform for the action of the Atkin–Lehner involution oðNÞ ¼ 0 1 N 0 with level N as required by the Atkin–Lehner–Li-Miyake theory of newforms. We note that the image of r consists of matrices whose entries are algebraic pffiffiffiffiffiffiffi integers in Z½ 2 : For an arbitrary rational prime c; the reduction of r modulo a pffiffiffiffiffiffiffi prime above c in Z½ 2 has projective image S4 ; so every prime c is exceptional of type (iii) for G: It remains to actually calculate the coefficients of GðzÞ: We briefly describe the algorithm of Bayer and Frey enabling us to explicitly calculate the coefficients bðpÞ for primes p[N: By (10), the coefficients bðpÞ for primes p[N are given by the character table for the chosen representation of S˜ 4 (see [Df, Chapter 28], for example). The factorization of p in K1 corresponds to a particular conjugacy class of Frobp in S4 ; denoted by pp : We denote the conjugacy class of Frobp in S˜ 4 by p* p : This information is summarized in Table 1, where the lower index on the primes in K1 indicates the residue degree, as before. pffiffiffi Since we know explicitly the factor l for which K˜ ¼ Kð lÞ by Theorem 4, we can determine the conjugacy class in S˜ 4 when the factorization types 1, 4, and 5 occur by using Propositions 5 and 6. In what follows, let ls denote the action of sAS4 on l: Proposition 5 ([B-F, Proposition 2.4]). Suppose P is a prime in K above p: If pp ¼ 1A; (resp. 3A) then bðpÞ ¼ 2; (resp. 1) if and only if l is a square modulo P:
M. Boylan / Journal of Number Theory 98 (2003) 377–389
384 Table 1 Factorization type
pOK1
pp
p* p
bðpÞ
1
p1 p01 p001 p000 1
1A
2 3 4
p2 p02 p1 p01 p2 p1 p3
2A 2B 3A
5
p4
4A
1A˜ 2A˜ 4A˜ 2B˜ 3A˜ 6A˜ 8A˜ 8B˜
2 2 0 0 1 1 pffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffi 2
Proposition 6 ([B-F, Proposition 2.5]). Suppose p is an odd prime and pOK1 ¼ p4 : pffiffiffiffiffiffiffi Then bðpÞ ¼ 2 if and only if l
p1 2
1 lð3;2;1Þ lð1;2;3Þ ðmod p4 Þ: þ 2 2l
ð11Þ
3. The proof of Theorem 2 To prove Theorem 2, we will also need certain facts about modular forms modulo p when pX5 is prime. Let Mk;p ðG0 ðNÞÞ denote the Fp -vector space of modular forms in Mk ðG0 ðNÞÞ-Z½½q
with coefficients reduced modulo p; and suppose that F ðzÞ ¼ PN n n¼0 aðnÞq AMk ðG0 ðNÞÞ-Z½½q
: Then the action of the Ramanujan theta-operator is given by yp ðF Þ :
N X
naðnÞqn ðmod pÞAMkþpþ1;p ðG0 ðNÞÞ:
ð12Þ
n¼0
Propositions 7 and 8 are also useful tools for studying modular forms modulo p: Proposition 7. (1) If p is an odd prime and Ap ðzÞ :¼ p1 2 p
Then Ap ðzÞAMp1 ðG0 ðpÞ; ðð1Þ
Zp ðzÞ : ZðpzÞ
ÞÞ and Ap ðzÞ 1 ðmod pÞ:
2
(2) [Sw-D] Suppose kX4 is an even integer. Then the normalized Eisenstein series of weight k with respect to SL2 ðZÞ is given by Ek ðzÞ ¼ 1
2k X sk1 ðnÞqn : Bk nX1
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385
Furthermore, if pX5 is prime, then Ep1 ðzÞ 1 ðmod pÞ: Proof. (1) See [G-H], [N2,N3] for the first part. The second part follows from the Children’s Binomial Theorem in characteristic p: (i.e., that 1 xp ð1 xÞp ðmod pÞ when p is a prime.) (2) By the Von Staudt–Claussen Theorem, the power of a prime p dividing the denominator of Bk ; the kth Bernoulli number, is one if p 1 j k and zero otherwise. The congruence for Ep1 follows. If F is a number field with ring of algebraic integers OF ; if g ¼ PN n n¼0 aðnÞq AOF ½½q
; and if p is a prime ideal in OF ; then ( minfn : aðnÞepg if aðnÞep for some n; ordp ðgÞ :¼ þN otherwise: With this definition we can state Sturm’s Theorem on the congruence of modular forms. Proposition 8 ([St, Theorem 1]). If f ; gAMk ðG0 ðNÞÞ-OF ½½q
and if kN Y 1 1þ ; ordp ðf gÞ4 12 pjN p then ordp ðf gÞ ¼ þN: i.e., f g ðmod pÞ: The constant on the right-hand side of the inequality in Proposition 8 is called Sturm’s bound for f g: To prove that a prime c is exceptional of type (iii) for an eigenform F ðzÞ ¼ PN n n¼0 aðnÞq ASk ðG0 ðNÞ; wÞ; we first build a specific odd irreducible complex % octahedral two-dimensional Galois representation r : GalðQ=QÞ-GL 2 ðCÞ with certain properties depending on F ; as described in Section 2. In this general setting, we want to construct K so that it is isomorphic to the kernel field L of r* c;F if c were exceptional of type (iii) for F : The comments immediately following (7) indicate that we can determine the splitting of primes in L from the coefficients aðpÞ modulo c: Any irreducible quartic polynomial defining K must split completely modulo primes which are known to split completely in L: The field K must also be unramified outside Nc; which restricts the values of the discriminants of the defining quartics. These requirements allow us to quickly verify whether or not a given quartic is a good candidate for defining K: Using the facts about modular forms modulo p stated at the beginning of this section, we then show that the weight one newform obtained from r is congruent modulo c to a certain power of the theta-operator acting on F ; thus proving a
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M. Boylan / Journal of Number Theory 98 (2003) 377–389
nontrivial congruence for the coefficients of F : Therefore, c is exceptional of type (iii) for F : Proof of Theorem 2. We show that 19 is exceptional of type (iii) for F3 ðzÞ ¼ Z12 ð2zÞ: We first choose an appropriate irreducible quartic defining the S4 -field K: Our calculations are based on gðxÞ ¼ x4 þ 76x2 722x þ 1444
ð13Þ
with roots fx1 ; x2 ; x3 ; x4 g obtained from the arithmetic of the CM elliptic curve 361A [Cr] by a formula of Bayer and Frey [Ba-F, p. 401]. The discriminant of K1 ¼ Qðx1 Þ pffiffiffiffiffiffiffiffiffi is 193 22 ; and the A4 -subfield is M ¼ Qð 19Þ: The prime factorization of 19 in K1 is given by 19 ¼ P4 ; so eK;19 ¼ 1; implying that K may be extended to an S˜ 4 -field K˜ by Theorem 3. Next we calculate g19 ¼ 72x31 x22 þ 513x21 x22 1368x1 x22 4332x22 þ 1026x31 x2 4104x21 x2 þ 64980x1 x2 217683x2 þ 3420x31 þ 36100x21 þ 488433x1 2249752
ð14Þ
pffiffiffiffiffiffi by Theorem 4. The S*4 -field Kð g19 Þ defines an irreducible complex octahedral twodimensional Galois representation r: Following the Bayer–Frey algorithm, we explicitly construct the coefficients of the weight one Hecke newform HðzÞ coming pffiffiffiffiffiffiffiffi from r: The other solutions Kð rg19 Þ to this embedding problem define the Galois representations r#wr ; where wr ¼ ðr Þ: These representations determine the weight one modular forms HðzÞ#wr : Hence, the weight one form that seems to be dictating the exceptional congruence for the coefficients of F3 ðzÞ is HðzÞ or some twist of HðzÞ: In particular, the weight one newform we need is GðzÞ ¼ HðzÞ#w38 ¼
N X
bðnÞqn ¼ q þ
pffiffiffiffiffiffiffi 3 2q þ q5 þ q7 q9
n¼1
þ ?AS1 G0 ð22 192 Þ; : 19
ð15Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coming from the field K˜ ¼ Kð 38g19 Þ having discriminant 218 1921 : The Artin conductor is 22 192 and the determinant character is ð19 Þ; so r is also odd. As mentioned earlier, every prime p is exceptional of type (iii) for G: In particular, the prime p ¼ 19 is exceptional of type (iii) for G; so bðpÞ2 0; p9 ; 2p9 ; or 4p9 ðmod 19Þ:
ð16Þ
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387
By Proposition 7, GðzÞ GðzÞE18 ðzÞ2 A19 ðzÞ ðmod 19Þ
ð17Þ
and by (12), y219 ðF3 ðzÞÞ
N X
n2 aðnÞqn ðmod 19Þ;
ð18Þ
n¼0
where GðzÞE18 ðzÞ2 A19 ðzÞ ðmod 19Þ and y219 ðF3 ðzÞÞ ðmod 19Þ lie in the space M46;19 ðG0 ð22 192 ÞÞ whose Sturm bound is 8740. Using Propositions 5 and 6 to calculate the coefficients of GðzÞ; we verify that GðzÞ y219 ðF3 ðzÞÞ ðmod 19Þ
ð19Þ
by Proposition 8. In light of (16), this shows that 19 is exceptional of type (iii) for F3 ; and hence, for F4 ; since F4 is a w1 -twist of F3 ; proving part (2) of Theorem 2. Table 2 contains data for the remaining case of Theorem 2 (when c ¼ 11) and for the example of Serre and Swinnerton-Dyer. Here N is the conductor of r; and S is the Sturm bound for the space Mk;p ðG0 ðNÞÞ in which the relevant congruence of modular forms is proved. We observe that in each case F ðzÞAMcþ5 ðG0 ðMÞÞ;
ð20Þ
4
where M is the part of N prime to c and that c3
yc 8 ðF ðzÞÞ GðzÞ ðmod cÞAMc2 1
8 þ1;c
ðG0 ðNÞÞ:
ð21Þ
A finite computation completes the proof of each exceptional congruence of type (iii). Remark. (1) We determine GðzÞ; the appropriate twist of HðzÞ in the example above, so that GðzÞ will satisfy (19). In particular, since the pth coefficients of such a GðzÞ pffiffiffiffiffiffiffi (when p is prime) are in the set f0; 71; 72; 7 2g; these pth coefficients must be determined by the reduction of the pth coefficients of y219 ðF3 ðzÞÞ: We use this criteria to check which twist HðzÞ#wr for squarefree r j 38 is the correct one. Table 2 c
F ðzÞ
gðxÞ
N
w
S
11 19 59
Z4 ð4zÞZ4 ð2zÞ Z12 ð2zÞ FSwD
x4 16x3 þ 30x2 þ 30x 74 x4 þ 76x2 722x þ 1444 x4 x3 7x2 þ 11x þ 3
23 112 22 192 592
ð11 Þ ð19 Þ ð59 Þ
2112 8740 2180
M. Boylan / Journal of Number Theory 98 (2003) 377–389
388
(2) We note that the choice of gðxÞ in these examples is certainly not unique. By performing a brute force search on quartic polynomials in the example above, we are immediately led to gðxÞ ¼ x4 x3 2x2 6x 2; having roots x1 ; x2 ; x3 ; x4 : In this pffiffiffiffiffiffiffiffiffiffiffi case, we apply the Bayer-Frey algorithm to K˜ ¼ Kð 38g019 Þ; where g019 ¼ 6x31 x22 þ 18x21 x22 51x1 x22 33x22 þ 42x31 x2 105x21 x2 þ 33x1 x2 þ 3x2 10x31 7x21 þ 95x1 þ 190 to obtain the same weight one form GðzÞ as before. (3) The author notes that the Artin L-function constructed in connection with showing that 11 is exceptional of type (iii) for F1 is also constructed by Bayer and Frey [B-F, Example 3], but not in the context of exceptional congruences. In this case, gðxÞ is obtained from the arithmetic of the CM elliptic curve 121B [Cr]. (4) In Swinnerton-Dyer’s original discussion of the exceptional prime 59 for FSwD ; the coefficients 11 and 7 in our gðxÞ are interchanged, a typographical error.
Acknowledgments I am thankful to Jordi Quer for helping me implement Propositions 5 and 6 with Mathematica, and for calculating the minimal polynomials of the generators of ˜ Thanks also to Teresa Crespo for advice on implementing the relevant fields K: formula (9).
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