Arithmetic of singular moduli and class polynomials

Transcription

1 Comositio Math. 141 (2005) DOI: /S X Arithmetic of singular moduli and class olynomials Scott Ahlgren and Ken Ono Abstract We investigate divisibility roerties of the traces and Hecke traces of singular moduli. In articular we rove that, if is rime, these traces satisfy many congruences modulo owers of which are described in terms of the factorization of in imaginary quadratic fields. We also study generalizations of Lehner s classical congruences j(z) U 744 (mod ) (where 11 and j(z) is the usual modular invariant), and we investigate connections between class olynomials and suersingular olynomials in characteristic. 1. Introduction and statement of results Let j(z) =q q q q Z[[q]] denote the usual ellitic modular function on SL 2 (Z) (q := e 2πiz throughout). The values of j(z) at imaginary quadratic arguments in the uer half of the comlex lane are known as singular moduli; two imortant examles are the evaluations ( ) 1+ 3 j(i) = 1728 and j =0. (1.1) 2 Singular moduli are algebraic integers which lay many imortant roles in classical and modern number theory. For examle, they generate ring class field extensions of imaginary quadratic fields. Also, the work of Deuring [Deu58, Deu46] highlights their dee connections with the theory of ellitic curves with comlex multilication. In recent work, Borcherds [Bor95a, Bor95b] used them to define an imortant class of automorhic forms ossessing certain striking infinite roduct exansions. There is a vast amount of literature on the comutation of singular moduli which dates back to the works of Kronecker; this includes the classical calculations of Berwick [Ber28] andweber [Web61]. In more recent work, Gross and Zagier [GZ85] comuted exactly the rime factorization of the absolute norm of suitable differences of singular moduli (further work in this direction has been carried out by Dorman [Dor89, Dor88]). In this aer we investigate the divisibility roerties of the traces and Hecke traces of singular moduli in terms of the factorization of rimes in imaginary quadratic fields. We begin by fixing notation. Throughout, d denotes a ositive integer congruent to 0 or 3 modulo 4 (so that d is the discriminant of an order in an imaginary quadratic field). Denote by Q d the set of ositive definite integral binary quadratic forms Q(x, y) =ax 2 + bxy + cy 2 Received 29 Aril 2003, acceted in final form 14 May 2004, ublished online 10 February Mathematics Subject Classification 11F33, 11F37 (rimary). Keywords: singular moduli, class olynomials, modular forms. The first author thanks the National Science Foundation for its suort through grant DMS The second author is suorted by the National Science Foundation, a Guggenheim Fellowshi, a Packard Research Fellowshi, and an H. I. Romnes Fellowshi. This journal is c Foundation Comositio Mathematica 2005.

2 S. Ahlgren and K. Ono with discriminant d = b 2 4ac. ForeachQ let α Q be the unique comlex number in the uer half-lane which is a root of Q(x, 1); the singular modulus j(α Q ) deends only on the equivalence class of Q under the action of Γ := PSL 2 (Z). We define the class olynomial of discriminant d as H d (x) := (x j(α Q )) Z[x]. Q Q d /Γ This is a mild dearture from the customary definition since the roduct above is taken over all Q with discriminant d, not just the rimitive ones (note also the mild dearture from the definition in [Zag02]). Nevertheless, if d is a fundamental discriminant, then H d (x) is the minimal olynomial of each j(α Q )overq. Define ω Q {1, 2, 3} by 2 if Q Γ [a, 0,a], ω Q := 3 if Q Γ [a, a, a], 1 otherwise. Let J(z) be the Hautmodul J(z) :=j(z) 744 = q q q 2 +. Then, following Zagier [Zag02], define the trace of the singular moduli of discriminant d by t(d) := J(α Q ) = j(α Q ) 744. ω Q ω Q Q Q d /Γ Q Q d /Γ We also consider more general Hecke traces; these are defined in terms of a natural sequence of modular functions. Let J 0 (z) := 1, and for ositive integers m define J m (z) by J m (z) =J(z) T 0 (m), (1.2) where T 0 (m) is the normalized weight zero Hecke oerator of index m. EachJ m (z) isamonic olynomial in j(z) of degree m with integer coefficients; the first few are J 0 (z) =1, J 1 (z) =j(z) 744 = J(z), J 2 (z) =j(z) j(z) = q q +. Then, for each ositive integer m, we define the mth Hecke trace of the singular moduli of discriminant d as the integer t m (d) := J m (α Q ). ω Q Q Q d /Γ Zagier [Zag02] showed that these traces t m (d) are determined as the coefficients of a certain sequence of meromorhic modular forms on Γ 0 (4) (see 3 for details). Note that t 1 (d) =t(d); for convenience, we define t m (n) := 0 for every ositive integer n 1, 2(mod4). Remark. For fundamental discriminants d, let h( d) denote the class number of rimitive ositive definite binary quadratic forms of discriminant d. Then the values t m (d) for0 m h( d) determine the resective ower sums in the j(α Q ). Therefore, these Hecke traces comletely determine the olynomial H d (x). Our first result shows that these traces satisfy many striking congruences based on the factorization of rimes in certain imaginary quadratic fields. 294

3 Singular moduli and class olynomials Theorem 1.1. Suose that is an odd rime and that s and m are ositive integers with m. Then the following are true. (1) If n is a ositive integer for which slits in Q( n), then t m ( 2 n) 0 (mod ). (2) A ositive roortion of the rimes l have the roerty that t m (l 3 n) 0 (mod s ) for every ositive integer n corime to l such that is inert or ramified in Q( nl). (3) A ositive roortion of the rimes l have the roerty that ( ( )) n t m (l 2 n) t m (n) 2 (mod s ) l for every ositive integer n with l 2 n such that is inert or ramified in Q( n). Remarks. (1) It would be interesting to find a natural descrition of the rimes l which arise in the second and third arts of Theorem 1.1. (2) If d is a fundamental discriminant and l is a rime, then the class numbers h( d)andh( l 2 d) are related by the formula ( ( )) d h( l 2 d)=h( d) l. l Note the resemblance between this formula and the congruences in the third art of Theorem 1.1. (3) The roof of the second (resectively, third) art of Theorem 1.1 shows that the rimes l can be chosen from the arithmetic rogression 1 (mod4 s ) (resectively, 1 (mod 4 s )). Here we give some examles of the henomena described in Theorem 1.1. Examle 1.2. For each 11, the maximal congruence modulus in the first art of Theorem 1.1 exceeds ; these moduli are 729, 125, 49, and 121. If = 7, for examle, then for every non-negative integer n we have t 1 (343n + 147) t 1 (343n + 245) t 1 (343n + 294) 0 (mod 49). Examle 1.3. As an examle of the henomenon described in the second art of Theorem 1.1, we have t 1 (13 3 n) 0 (mod 7) for every ositive integer n corimeto13with ( ) 13n 7 1. Examle 1.4. As an examle of the third art of Theorem 1.1, we have, for each ositive integer n such that ( n 7 ) 1and292 n, the congruence t 1 (29 2 n) t 1 (n) ( 2 ( )) n 29 (mod 7). In the second art of the aer we turn to a study of the arithmetic roerties of the class olynomial H d (x). If is rime, then let F [x] denote the olynomial ring over the finite field with elements. For fundamental discriminants d, it is natural to study the factorizations of H d (x) over F [x]. This question is of articular interest for those rimes which are inert or ramified in Q( d); for such a theorem of Deuring (see, for examle, Theorem 12 of [Lan87, 13.4]) imlies 295

4 S. Ahlgren and K. Ono that each root of H d (x) overf [x] isthej-invariant of a suersingular ellitic curve in characteristic (this fact is an essential ingredient in Elkies roof [Elk87] that every ellitic curve over Q has infinitely many suersingular rimes). In view of this, it is natural to seek a general descrition, for arbitrary rimes, ofthesetof d for which the distinct roots of H d (x) inf form the comlete set of suersingular j-invariants in characteristic. Using a result of Koike (which is related to work of Dwork and Deligne on the -adic rigidity of j(z)) we show in Theorem 1.5 below that this question is closely related to the roblem of classifying certain congruences for the Fourier coefficients of nearly holomorhic modular functions. We recall that a modular form on a congruence subgrou Γ is called nearly holomorhic if its oles (if there are any) are suorted at the cuss of Γ. Every non-zero nearly holomorhic modular function f(z) onsl 2 (Z) is a olynomial in j(z) and has a Fourier exansion of the form f(z) = a(n)q n, n n 0 where n 0 0anda(n 0 ) 0. We define the oerator U by f(z) U := a(n)q n. n= For nearly holomorhic modular forms with integral coefficients, we consider congruences of the form f(z) U a(0) (mod ). (1.3) Perhas the most famous congruences of the form (1.3) are due to Lehner [Leh49], who roved that if 11 is rime, then j(z) U 744 (mod ). (1.4) To state our first result in this context, it is convenient to make the following definition. If 11 is rime, then let S (x) :=1,andfor>11 define S (x) F [x] by S (x) := (x j(e)), (1.5) E/F suersingular j(e) {0,1728} where the roduct is taken over F -isomorhism classes of suersingular ellitic curves E. Itiswell known that the degree of S (x) is /12. With this notation we have the following general result. Theorem 1.5. Let F (x) Z[x] be a olynomial of degree m, andlet>mbe a rime for which F (x) 0(mod). IfS (x) 2 divides F (x) in F [x], then F (j(z)) U a(0) (mod ), where a(0) is the constant term in the Fourier exansion of F (j(z)). Remarks. (1) The converse is false in view of the fact that if the conclusion of the theorem holds for F (x), then it holds for F (x)+α for every α F. (2) Since S (x) =1for 11, Lehner s congruences (1.4) follow from Theorem 1.5. For fundamental discriminants d, define integers c d (n) by H d (j(z)) = c d (n)q n. n= h( d) 296

5 Singular moduli and class olynomials After Theorem 1.5, it would be desirable to find a characterization of those rimes for which H d (j(z)) U c d (0) (mod ). (1.6) Comutations reveal many uniform sets of examles. For examle, we have the following corollary. Corollary 1.6. If 239 < d <0 is a fundamental discriminant and h( d) <<6h( d) 1 is a rime which is inert or ramified in Q( d), thens (x) 2 divides H d (x) in F [x] and H d (j(z)) U c d (0) (mod ). The uniformity of the range of rimes in Corollary 1.6 suggests that this henomenon might hold in generality. However, this is not true; when d = 239, we have H 239 (j(z)) U q +62q 2 + (mod 79), although 79 is inert in Q( 239) and h( 239) = 15. In this case S 79 (x) divides H 239 (x) inf 79 [x], but the suersingular j-invariant j = 15 is a root of multilicity only 1. Another natural question to ask is whether or not, for those rimes >h( d) which are inert in Q( d), the condition that S (x) 2 divides H d (x) inf [x] is imlied by a congruence of the form (1.6) (recall the first remark following Theorem 1.5). Comutations reveal that the answer is affirmative for all fundamental discriminants d with 700 < d < 0. However, further calculation reveals soradic counterexamles (there are only three counterexamles with d<1000; these are the cases when ( d, ) =( 707, 47), ( 731, 79), and ( 767, 101)). Nevertheless, this condition aears to be necessary for the vast majority of d; forthese d it is clear by comaring the degrees of S (x) andh d (x) thath d (j(z)) can satisfy the congruence (1.6) only for those inert rimes with 6h( d)+r, where r {1, 5, 7, 11} has r (mod 12). (1.7) In view of (1.7), it is natural to seek an unconditional uer bound for those rimes which admit a congruence of the form (1.6). Here, as a secial case of a result for nearly holomorhic modular forms on SL 2 (Z), we show, for all d, that(1.6) can only hold for those rimes 12h( d)+1. To state our general result, some notation is required. If k 4 is an even integer, then let M k denote the sace of weight k holomorhic modular forms on SL 2 (Z). Furthermore, if is rime, then let M k, denote the set of reductions modulo of those forms f(z) M k with integral coefficients. The filtration of a ower series f(z) whose reduction belongs to M k, for some k is defined by ω (f) :=inf{k : f(z) (mod) M k, }. Let (z) :=q n=1 (1 qn ) 24 be the unique normalized cus form of weight 12 on SL 2 (Z). We define the usual theta-oerator on ower series by ( ) Θ a(n)q n = na(n)q n. (1.8) n n 0 n n 0 Then we have the following theorem. Theorem 1.7. Suose that f(z) = n=m a(n)qn is a nearly holomorhic modular form of integral weight k on SL 2 (Z) with integral coefficients. Furthermore, suose that max(5,k 12m) is a rime for which a(m) and m. If there is a non-negative integer s for which s + m>0 and ω (Θf,s ) 1, 2 (mod ), where f,s (z) :=f(z) (z) s, then f(z) U 0(mod). 297

6 As a corollary, we obtain the following. S. Ahlgren and K. Ono Corollary 1.8. Suose that F (x) Z[x] is a olynomial of degree D 1. If>12D +1is a rime which does not divide the leading coefficient of F (x), then F (j(z)) U 0 (mod ). In articular, if d is a fundamental discriminant and is a rime for which then 12h( d)+1. H d (j(z)) U c d (0) (mod ), We remark that, since j(z) U (mod 13), Corollary 1.8 imlies that Lehner s list of congruences of the form (1.4) is comlete. This recovers an observation of Serre (see (6.16) of [Ser76]). In 2 we record some reliminaries on modular forms. In 3 we recall recent work of Zagier, and in 4 we use the theory of integral and half-integral weight modular forms to rove Theorem 1.1. In 5 we rove Theorem 1.5 and Corollary 1.6, andin 6 we rove Theorem 1.7 and Corollary Preliminaries on modular forms We begin by collecting some facts which we require on half-integral weight modular forms. For details on many of these facts one may consult, for examle, [Kob84] or[koh82]. If f(z) is a function of the uer half-lane, λ 1 2 Z,and( ab cd) GL + 2 (R), then we define the usual slash oerator by ( ) ( ) a b az + b f(z) λ := (ad bc) λ/2 (cz + d) λ f (2.1) c d cz + d (we always take the branch of the square root having non-negative real art). If γ = ( ab cd) Γ0 (4), then define ( c j(γ,z) := ɛ d) 1 d cz + d, (2.2) where { 1 if d 1(mod4), ɛ d := (2.3) i if d 1(mod4). If k is an integer and N is an odd ositive integer, then we denote by M k+1/2 (Γ 0 (4N)) the infinitedimensional vector sace of nearly holomorhic modular forms of weight k +1/2 onγ 0 (4N); these are functions f(z) which are holomorhic on the uer half-lane, meromorhic at the cuss, and which satisfy f(γz) =j(γ,z) 2k+1 f(z) for all γ Γ 0 (4N). (2.4) As usual, we denote by M k+1/2 (Γ 0 (4N)) (resectively, S k+1/2 (Γ 0 (4N))) the finite-dimensional subsace of M k+1/2 (Γ 0 (4N)) consisting of those forms which are holomorhic (resectively, vanish) at the cuss. Finally, denote by M + k+1/2 (Γ 0(4N)) and M + k+1/2 (Γ 0(4N)) the Kohnen lus-saces of holomorhic and nearly holomorhic forms which transform according to (2.4) and which, in addition, have a Fourier exansion of the form a(n)q n. (2.5) ( 1) k n 0,1 (mod4) We next recall some facts on the exansions of modular forms at the cuss of a congruence subgrou; many of these can be found, for examle, in [Mar96]. The cuss are reresented by rational numbers a/c together with the oint at infinity. We say that the cuss s 1 and s 2 are 298

7 Singular moduli and class olynomials equivalent under the congruence subgrou Γ if there exists γ Γ such that s 1 = γ s 2 ;byacus of Γ we tyically mean an equivalence class of cuss under this relation. If N is a ositive integer, then a comlete set of reresentatives for the cuss of Γ 0 (N) is { ac } c Q : c N,1 a c N,gcd(a c,n)=1, a c distinct (mod gcd(c, N/c)). (2.6) If γ 1 is a cus of Γ 0 (4N) andg M k+1/2 (Γ 0 (4N)) is not identically zero, then at the cus γ 1 we have an exansion of the form g k+1/2 γ 1 = c q α + for some α Q and c 0. (2.7) Moreover, if γ 1 and γ 2 are equivalent under Γ 0 (4N), then the first term in the exansion of g k+1/2 γ 2 differs from (2.7) only by multilication by some root of unity. We now consider the cusidal behavior of roducts and quotients of Dedekind s eta-function η(z) :=q 1/24 n=1 (1 q n ) (although only the integral weight case is discussed in [Mar96], similar facts hold in the half-integral weight case). Suose that f(z) is the eta-quotient f(z) := δ N ηr δ(δz), and set λ := 1 2 δ N r δ 1 2 Z.Thenforγ = ( ab cd) SL2 (Z), we have an exansion of the form f(z) λ γ = ξ q (1+ τ(c) α(n)q ), n/β (2.8) where ξ 0 is an algebraic number, the coefficients α(n) are algebraic, β N, and τ(c) := 1 (δ, c) 2 r δ. (2.9) 24 δ δ N Finally, we remark that to exlicitly comute the exansion (2.8), we can use the fact that, for each γ SL 2 (Z), we have the transformation formula n=1 η(z) 1/2 γ = ɛ γ η(z), (2.10) where ɛ γ is a root of unity. Also, for any δ N and γ SL 2 (Z), there exist a matrix γ SL 2 (Z)anda matrix ( ) ( ) AB 0 D,whereB is an integer and A and D are ositive integers, such that δ 0 01 γ = γ ( ) AB 0 D. By using this fact together with (2.10), one can comute, u to a root of unity, the value of ξ aearing in (2.8). We briefly recall some roerties of the Hecke oerators on the saces M + k+1/2 (Γ 0(4N)) (these definitions are given in [Koh82] for holomorhic forms, but the situation is the same if we allow oles at the cuss). Suose that l is a rime with l N (note that this does not necessarily exclude l = 2). Then the action of the Hecke oerator T k+1/2,4n (l 2 ) on a modular form f(z) := a(n)q n M + k+1/2 (Γ 0(4N)) (2.11) is given by f(z) T k+1/2,4n (l 2 ):= ( 1) k n 0,1 (mod4) ( 1) k n 0,1 (mod4) ( a(l 2 n)+ ( ( 1) k n l ) ( )) n l k 1 a(n)+l 2k 1 a l 2 q n. (2.12) Then, for each ositive m corime to N, the oerator T k+1/2,4n (m 2 )onm + k+1/2 (Γ 0(4N)) can be exressed as a olynomial in the oerator T k+1/2,4n (l 2 ) via the usual multilicative relations. 299

8 S. Ahlgren and K. Ono If f(z) isasin(2.11) andχ is a quadratic Dirichlet character with conductor α, thenwehave the quadratic twist f χ := χ(n)a(n)q n M + k+1/2 (Γ 0(4Nα 2 )). (2.13) ( 1) k n 0,1 (mod4) Suose now that f(z) isasin(2.11) andthatχ is a quadratic character with conductor α, wherem is corime to Nα. Then, combining (2.12)with(2.13) and the fact that the oerators T k+1/2,4nα 2(l 2 ) with l Nα generate the Hecke algebra on M + k+1/2 (Γ 0(4Nα 2 )), we see that (f χ) T k+1/2,4nα 2(m 2 )=(f T k+1/2,4n (m 2 )) χ. (2.14) 3. Zagier s formulas To rove Theorem 1.1, we make use of recent work of Zagier [Zag02] on the traces of singular moduli. We begin by recalling Zagier s results on two articular sequences of nearly holomorhic modular forms. Let θ 1 (z) ande 4 (z) be defined by θ 1 (z) := η2 (z) η(2z) = E 4 (z) := n= n=1 n 3 q n 1 q n, ( 1) n q n2 =1 2q +2q 4 2q 9 +. Then let g 1 (z) M + 3/2 (Γ 0(4)) be the nearly holomorhic modular form defined by g 1 (z) := θ 1(z)E 4 (4z) η 6 = q 1 2+ B(1,d)q d = q q 3 492q 4 +. (3.1) (4z) 0<d 0,3 (mod4) More generally, if D 0, 1 (mod 4) is a ositive integer, then let g D (z) denote the unique element of M + 3/2 (Γ 0(4)) whose Fourier exansion has the form g D (z) =q D + B(D, 0) + B(D, d)q d. 0<d 0,3 (mod4) (The existence and uniqueness of these forms is discussed in 4of[Zag02]). Similarly, for a non-negative integer d 0, 3(mod4),letf d (z) be the unique form in M + 1/2 (Γ 0 (4)) whose exansion has the form f d (z) =q d + A(D, d)q D. 0<D 0,1 (mod4) Existence and uniqueness follow from Lemma 14.2 of [Bor95a]; see also 4of[Zag02]. All of the coefficients of each f d (z) andg D (z) are integers. For each m 1 and each air of integers D 0, 1(mod4)and0 d 0, 3 (mod 4), define integers A m (D, d) andb m (D, d) in the following manner: A m (D, d) := the coefficient of q D in f d (z) T 1/2,4 (m 2 ), B m (D, d) := the coefficient of q d in g D (z) T 3/2,4 (m 2 ). For D =1andm 1 (see (19) of [Zag02]) we have A m (1,d)= n m na(n 2,d). (3.2) 300

9 Singular moduli and class olynomials Using this notation, Zagier [Zag02, Theorem 5] roved the following result. Theorem 3.1. The following are true. (1) If m 1 and 0 <d 0, 3(mod4),then t m (d) = B m (1,d). (2) If m 1, 0 <D 0, 1(mod4),and0 d 0, 3(mod4),then A m (D, d) = B m (D, d). Examle. As an illustration of the first art of Theorem 3.1, wecomarethecoefficientsonq 3 and q 4 in (3.1) with the following values (which are comuted using (1.1)): t 1 (3) = j((1 + 3)/2) 744 = 248, 3 j(i) 744 t 1 (4) = = ProofofTheorem1.1 We now turn to the roof of Theorem 1.1. The roof of the first art follows easily from Zagier s work. Proof of Theorem 1.1(1). Since m, a calculation using the definition (2.12) andthefirstartof Theorem 3.1 shows that it suffices to rove Theorem 1.1(1) in the case where m = 1. Suose that n is a ositive integer for which ( n )=1.By(2.12) andtheorem3.1, wehave t 1 ( 2 n)= B 1 (1, 2 n) ( ) ( n = B (1,n)+ B 1 (1,n)+B 1 1, n ) 2 B (1,n)+B 1 (1,n) (mod ). Then, using the second art of Theorem 3.1 and (3.2), we obtain t 1 ( 2 n) A (1,n)+B 1 (1,n) A 1 (1,n)+A 1 ( 2,n)+B 1 (1,n) A 1 (1,n)+B 1 (1,n) B 1 (1,n)+B 1 (1,n) 0 (mod ). This establishes the first claim in Theorem 1.1. The roofs of the second and third arts of Theorem 1.1 are more involved. To begin, for each odd rime we define ( ) ( ) 1 h 1, := g 1 g 1 M + 3/2 (Γ 0(4 2 )). (4.1) Using (3.1) andtheorem3.1, we find that h 1, (z) = 2 t 1 (d)q d 2 t 1 (d)q d. (4.2) 0<d 0,3 (mod4) d 0<d 0,3 (mod4) ( d )= 1 We next define, for each ositive integer m with m, the modular form h m, (z) :=h 1, T 3/2,4 2(m 2 ) M + 3/2 (Γ 0(4 2 )). (4.3) 301

10 S. Ahlgren and K. Ono Using (2.14), (4.2), (4.3) andtheorem3.1, we see that for some integer c m, we have ( ) ( ) 1 h m, (z) =g 1 T 3/2,4 (m 2 ) (g 1 T 3/2,4 (m 2 )) = c m, t m (d)q d 2 t m (d)q d. (4.4) 0<d 0,3 (mod4) d 0<d 0,3 (mod4) ( d )= 1 We require the following crucial result. Theorem 4.1. Suose that 3 is rime, and that s and m are ositive integers with m. Then there exists an integer β s 1 and a holomorhic modular form f m,,s (z) M + 3/2+( β ( 2 1))/2 (Γ 0 (4 2 )) with the roerty that f m,,s (z) h m, (z) (mod s ). To rove Theorem 4.1, we emloy the following lemma. Lemma 4.2. The form h 1, (z) defined in (4.1) is a nearly holomorhic modular form on Γ 0 (4 2 ) which is holomorhic on the uer half-lane and at the cuss 1/ 2 and 1/4 2, and which vanishes at the cus 1/2 2. Proof. Using (4.1) and(2.13), we see that h 1, (z) is a modular form on Γ 0 (4 2 ). From (3.1) and (4.2) it is clear that h 1, is holomorhic on the uer half-lane and at 1/4 2 (which is equivalent to under Γ 0 (4 2 )). Therefore, we only have to rove that h 1, (z) is holomorhic at 1/ 2 and vanishes at 1/2 2 ; to this end we consider the series ( ) ( ) h 1, (z) 3/2 2 and h 1 1, (z) 3/ We begin by noting that η 2 (z) g 1 (z) = η(2z)η 6 (4z) E 4(4z). Using this descrition together with (2.8), (2.9) (and the discussion thereafter) and the fact that E 4 (z) is a modular form on SL 2 (Z), we conclude (after a lengthy comutation) that there exist roots of unity ζ 1 and ζ 2 such that ( ) 1 0 g 1 (z) 3/2 2 1 ( ) 1 0 g 1 (z) 3/ = ζ 1 2 3/2 + O(q1/4 ), (4.5) = ζ 2 2 q 1/4 + O(q 3/4 ). (4.6) Let g := 1 v=1 ( v )e2πiv/ be the usual Gauss sum. Then we have ( ) g 1 (z) = g 1 ( ) ( ) v 1 v/ g 1 (z) 3/2. (4.7) 0 1 v=1 The exansion of g 1 ( )at1/22 is given by ( ( g 1 )) 3/2 ( ) (4.8) 1 We now make the crucial observation that if, for each v aearing in (4.7), we choose an integer k v satisfying 4k v 3v (mod ), (4.9) 302

11 then we have where γ v = ( 1 v/ 0 1 Singular moduli and class olynomials )( ) = γ v ( )( 1 4v/ +4kv / 0 1 ( 8v +8kv 16 2 vk v v /(3v 4k v 8v 2 ) +8vk v ) 16 3 v Γ k v 8v 8k v +1 0 (4). Using (2.1) (2.4), we see that (g 1 3/2 γ v )(z) = ( 16 3 v k v 8v 8k v +1 ), ) ( ) v + kv g 1 (z) = g 1 (z) =g 1 (z). 8(v k v )+1 Using these facts, together with (4.6), (4.7) and(4.8), we see that the only term in (4.8) witha non-ositive exonent on q is the term g ζ 2 1 ( ) v 2 q 1/4 e 2πi(v kv)/. (4.10) v=1 A comutation shows that if N is defined by 4N 1(mod), then the exression in (4.10) isequal to g ζ 2 1 ( ) ( ) v 2 q 1/4 e (2πi/)Nv = g2 ζ 2 1 ζ2 2 q 1/4 = 2 q 1/4. (4.11) v=1 Using (4.11) together with (4.1) and(4.6) we conclude that ( ) 1 0 h 1, (z) 3/2 2 2 = O(q 3/4 ). 1 The situation is similar at the cus 1/ 2 ;hereweusethefactthatifk v is defined by (4.9), then we have ( )( ) ( )( ) 1 v/ = γ v v/ +4kv / 1 2, where, as in the revious case, γ v Γ 0 (4) has (g 1 3/2 γ v)(z) =g(z). It follows from this together with (4.5) that the first term in the exansion of ( ) ( ) 1 0 g 1 (z) 3/2 2 1 is ζ 1 2 3/2 g 1 v=1 ( ) v =0. We conclude from this together with (4.1) thath 1, (z) is indeed holomorhic at 1/ 2. This finishes the roof of Lemma 4.2. Proof of Theorem 4.1. Suose first that the result has been roved in the case when m =1,and let f 1,,s be the form given by the theorem in this case. Using (2.12) together with the facts that φ( s ) β ( 2 1)/2 and that the oerators of rime index generate the Hecke algebra, we conclude, for each m corime to, that f 1,,s T 3/2+ β ( 2 1)/2,4 2(m2 ) h 1, T 3/2,4 2(m 2 ) (mod s ). Since the form on the left is again an element of M + 3/2+( β ( 2 1))/2 (Γ 0(4 2 )), we conclude that it suffices to rove the theorem in the case when m =1. 303

12 S. Ahlgren and K. Ono To accomlish this task, we first consider the modular form η 2 (4z) η(4 2 z) =1 2 q 4 + O(q 8 ) if 5, F (z) := η 27 (4z) η 3 (36z) =1 27q4 + if =3. (4.12) By a well-known criterion (see, for examle, [GH93]) together with (2.8) and(2.9), it follows, for each rime 3, that F (z) is a holomorhic form of integral weight on Γ 0 (4 2 ) which vanishes at each cus a/c Q for which 2 c (when 5 the weight is ( 2 1)/2, and when =3the weight is 12). Moreover, it is clear from the definition of the eta-function that F (z) 1(mod) for each. Suose that we are in the case when 5. By the receding discussion, we see that if β s 1 is sufficiently large, then the modular form f 1,,s := h 1, F β h 1, (mod s ) is a form of weight 3/2+( β ( 2 1))/2 onγ 0 (4 2 ) which vanishes at all cuss a/c for which 2 c. Using (2.6), we see that the equivalence classes of cuss of Γ 0 (4 2 )forwhich 2 c are reresented by 1/ 2,1/2 2,and1/4 2, and after Lemma 4.2 we know that h 1, is holomorhic on the uer half-lane and at these cuss. This gives Theorem 4.1. In the case when = 3, the situation is essentially the same; the only difference is that the form f 1,,s defined above has weight 3/2 +( β+1 ( 2 1))/2. Therefore, Theorem 4.1 follows exactly as before. We roceed with the roof of Theorem 1.1. Fix, s, andm with m, andlet f m,,s (z) := a m,,s (n)q n n=0 c m, 0<d 0,3 (mod4) d t m (d)q d 2 0<d 0,3 ( ) (mod4) d = 1 t m (d)q d (mod s ) (4.13) be the form given by Theorem 4.1. Theorem1.1 is an easy consequence of the following result. Lemma 4.3. Let f m,,s (z) M + 3/2+( β ( 2 1))/2 (Γ 0(4 2 )) be the form given in (4.13). Then we have the following. (a) A ositive roortion of the rimes l 1(mod4 s ) have the roerty that f m,,s (z) T 3/2+( β ( 2 1))/2,4 2(l2 ) 0 (mod s ). (b) A ositive roortion of the rimes l 1(mod4 s ) have the roerty that f m,,s (z) T 3/2+( β ( 2 1))/2,4 2(l2 ) 2f m,,s (z) (mod s ). Deduction of Theorem 1.1(2) and (3) from Lemma 4.3. Suose for the moment that the lemma has been roved, and let l be a rime as in the first art of the lemma. Then, using (2.12), we obtain f m,,s T 3/2+( β ( 2 1))/2,4 2(l2 ) n=0 ( a m,,s (l 2 n)+ 0 (mod s ). 304 ( n l ) a m,,s (n)+la m,,s ( n l 2 ) ) q n

13 Singular moduli and class olynomials Relacing n by nl in the last equation, we conclude that a m,,s (l 3 n) 0 (mod s ) for all n with l n. By (4.13) itfollowsthatt m (l 3 n) 0(mod s ) for all n corime to l such that is inert or ramified in Q( nl). This establishes the second assertion in Theorem 1.1. We turn to the third assertion. If l is a rime as in the second art of Lemma 4.3, then for all n we have (( ) ( ) n nl a m,,s (l 2 n)+ 2 )a m,,s (n)+la m,,s l 2 0 (mod s ). Therefore, if l 2 n and is inert or ramified in Q( n), we conclude from (4.13) that ( ( )) n t m (l 2 n) 2 t m (n) (mod s ). l This gives the statement in the third art of Theorem 1.1. Lemma 4.3 is a consequence of the next result, which was roved by Serre [Ser76, 6.4] using the Chebotarev Density Theorem and modular Galois reresentations. Lemma 4.4. Suose that λ and N are ositive integers. Suose that K is an algebraic number field with ring of integers O K,andthatM is a ositive integer. Then let M λ (Γ 0 (N)) M denote the set of reductions modulo M of those forms in M λ (Γ 0 (N)) with coefficients in O K.Ifl is rime then let T λ,n (l) be the usual integral weight Hecke oerator of index l on M λ (Γ 0 (N)). Then we have the following. (1) A ositive roortion of the rimes l 1(modNM) have the roerty that for all f M λ (Γ 0 (N)) M we have f T λ,n (l) 0 (mod M). (2) A ositive roortion of the rimes l 1(modNM) have the roerty that for all f M λ (Γ 0 (N)) M we have f T λ,n (l) 2f (mod M). Proof of Lemma 4.3. To begin, recall that for all m corime to we may take f m,,s = f 1,,s T 3/2+( β ( 2 1))/2 (m2 ). By the commutativity of the Hecke oerators, we see that it suffices to rove the lemma in the case when m =1. We begin by recalling some facts about the Shimura corresondence [Shi73]. In articular, if g(z) := n=1 b(n)qn S k+1/2 (Γ 0 (4N)) with k 1, then for every ositive squarefree integer t, we have the Shimura lift S t (g)(z) := n=1 B t(n)q n,wherethecoefficientsb t (n) aregivenby B t (n)n s = L(s k +1,χ t χ k 1 ) b(tn 2 )n s (4.14) n=1 (here χ 1 and χ t denote the Kronecker characters for the fields Q(i) andq( t), resectively). For each such t we have S t (g)(z) M 2k (Γ 0 (4N)). Moreover, if k 2, then S t (g)(z) S 2k (Γ 0 (4N)). This corresondence commutes with the action of the Hecke oerators T (l 2 )andt(l) on the relevant half-integral and integral weight saces. 305 n=1

14 For simlicity, we define S. Ahlgren and K. Ono f(z) :=f 1,,s (z) =h 1, (z) F β (z) M + 3/2+( β ( 2 1))/2 (Γ 0(4 2 )) (4.15) with some sufficiently large β s 1. Unfortunately, we cannot aly the Shimura corresondence directly to f(z), since f(z) isnot a cus form. Overcoming this obstacle requires some additional work. To begin, note that, using Lemma 4.2 and the roerties of the form F (z), we may assume that f(z) vanishes at each cus a/c which is equivalent neither to 1/ 2 nor to 1/4 2 under Γ 0 (4 2 ). We now recall some imortant roerties of half-integral weight Eisenstein series on Γ 0 (4) as develoed, for examle, in 4.2 of [Kob84]. For each integer k 2 there are Eisenstein series E k+1/2 (z) andf k+1/2 (z) M k+1/2 (Γ 0 (4)) (these Eisenstein series are not in the lus sace) with the following roerties. (i) E k+1/2 (z) has constant term equal to 1 and vanishes at the cuss 1 and 1 2 of Γ 0(4). F k+1/2 (z) has value ζ 3 ( 2) 2k 1 at the cus 1 (where ζ 3 is some root of unity) and vanishes at and 1 2 (recall that the value of a modular form at an equivalence class of cuss is defined only u to multilication by roots of unity). (ii) For each rime l 2, the forms E k+1/2 (z) andf k+1/2 (z) are eigenforms of the oerator T k+1/2,4 (l 2 ) defined in (2.12) (where the domain of definition is extended to all of M k+1/2 (Γ 0 (4))). Moreover, the eigenvalue of each of these forms under T k+1/2,4 (l 2 )is1+l 2k 1 (this fact follows from the Euler factors given in (2.30) and (2.32) in 4.2 of [Kob84]). Using (4.15), (4.12), (4.2), and item (i) above, we see that, for any constant c, the modular form f := f +2E 3/2+( β ( 2 1))/2 + cf 3/2+( β ( 2 1))/2 (4.16) vanishes at the cus. Using the discussion following (2.9), we comute that the first non-vanishing term in the exansion of F β at the cus 1/ 2 is(utoarootofunity)anintegralowerof2. It follows from this together with item (i) above, (4.15), and (4.5) that,ifwesetc = ζ 4 2 λ/2 with some aroriate root of unity ζ 4 and integer λ, then the modular form f defined in (4.16) vanishes at the cuss 1/ 2,1/2 2,and. Of course, it will now be the case that f does not vanish at the other cuss of Γ 0 (4 2 )which are equivalent to 1/ 2 and under Γ 0 (4). However, this situation is easily remedied using the form F (z) defined in (4.12). In articular, since F (z) s 1 1(mod s )vanishesateachcusa/c for which 2 c, we see that the modular form f := (f +2E 3/2+( β ( 2 1))/2 + cf 3/2+( β ( 2 1))/2) F s 1 is in fact a cus form of weight 3/2+(( β + s 1 )( 2 1))/2 onγ 0 (4 2 ). Before we roceed we need the following fact. Lemma 4.5. Suose that 3 and that β is a non-negative integer (with β 1 if =3). Then the Eisenstein series E 3/2+( β ( 2 1))/2 and F 3/2+( β ( 2 1))/2 have -integral Fourier exansions at. Proof. Write E 3/2+( β ( 2 1))/2 = n=0 a(n)qn, F 3/2+( β ( 2 1))/2 = n=0 b(n)qn. Using the Euler factors (2.30) and (2.32) in 4.2 of [Kob84], we see that it suffices to rove that if n is squarefree, then a(n) andb(n) are-integral. Moreover, formula (2.18) of the same section shows that it is enough to rove that b(n) is integral for each square-free n. Setλ := 1 + ( β ( 2 1))/2. 306

15 Singular moduli and class olynomials Then from (2.16) in that section we obtain, for each such n, 2 2λ 1/2 b(n) = L(χ n, 1 λ) (1 i)ζ(1 2λ) 2 λ if n 1(mod4), + χ n (2) 2 1/2+λ 2 2λ if n 2, 3(mod4). 1 The desired result now follows after a lengthy case by case comutation (in articular, one must searate the case when = 3; recall that in this case we have β 1). In each case, the L-andζ-values in this formula can be written in terms of Bernoulli numbers and generalized Bernoulli numbers. The desired conclusion follows after alying standard results on the divisibility roerties of these numbers (see, in articular, Theorems 1 and 3 of [Car59], and Proosition and Theorem 5 of [IR90, 15.2]). We do not include the details here. Together with item (ii) above and (2.12), this result imlies that for each rime l 1(mod s ) we have f T 3/2+(( β + s 1 )( 2 1))/2,4 2(l2 ) 0 (mod s ) f T 3/2+( β ( 2 1))/2,4 2(l2 ) 0 (mod s ), (4.17) and that for each rime l 1(mod s )wehave f T 3/2+(( β + s 1 )( 2 1))/2,4 2(l2 ) 2f (mod s ) f T 3/2+( β ( 2 1))/2,4 2(l2 ) 2f (mod s ). (4.18) After this discussion, Lemma 4.3 follows from Lemma 4.4. Here we rove only the first case. Lemma 4.4 shows that a ositive roortion of the rimes l 1(mod4 s ) have the roerty that (S t f ) T 2+( β + s 1 )( 2 1),4 2(l) 0 (mod s ) for all squarefree t. Since the Shimura corresondence commutes with the Hecke oerators, we conclude that for such arimel we have S t (f T 3/2+(( β + s 1 )( 2 1))/2,4 2(l2 )) 0 (mod s ) for all squarefree t. (4.19) A comutation using (4.14) shows that(4.19) imlies that f T 3/2+(( β + s 1 )( 2 1))/2,4 2(l2 ) 0 (mod s ), which, together with (4.17) and(4.15), gives the first assertion in Lemma 4.3. Using (4.18), the roof in the second case roceeds in a similar manner, and we do not include it here. This roves Lemma 4.3, and so finishes the roof of Theorem 1.1. Remark. There are other congruences, similar to those aearing in the second art of Theorem 1.1, which corresond to instances where g m (z) (or some related modular form) is an eigenform modulo s of the relevant Hecke oerator T (l 2 ), and the eigenvalue has some secial roerty. As examles of such congruences, we have t 1 (81n +9) t 1 (81n + 36) t 1 (81n + 63) 0 (mod 3), t 1 (135n + 63) t 1 (135n + 90) t 1 (135n + 117) 0 (mod 5). 5. U -congruences and the roof of Theorem 1.5 We begin by recalling a result of Koike [Koi73] which is closely related to work of Dwork and Deligne [Dwo69] onthe-adic rigidity of the ma j(z) j(z). If 5isrime,thenletS denote the set of those suersingular j-invariants in characteristic which are in F \{0, 1728}, andletm 307

16 S. Ahlgren and K. Ono denote the set of monic irreducible quadratic olynomials in F [x] whose roots are suersingular j-invariants. If ɛ ρ () andɛ i () are defined by { 0 if 1(mod3), ɛ ρ () := 1 if 2(mod3), ɛ i () := { 0 if 1(mod4), 1 if 3(mod4), then it is well known (see, for examle, [Sil86]) that, with S (x) as defined in (1.5), we have (x j(e)) = x ɛρ() (x 1728) ɛ i() (x α) g(x) α S g M E/F suersingular = x ɛρ() (x 1728) ɛ i() S (x). (5.1) Koike s result describes the Fourier exansion of j(z) (mod 2 )intermsofj(z) andthecollection of suersingular j-invariants in characteristic. Toberecise,Proosition1of[Koi73] imlies, for rimes 5, that there exist integers A(α), B(g), and C(g), as well as a olynomial D (x) Z[x], such that j(z) j(z) + D (j(z)) + A(α) j(z) α + B(g)j(z)+C(g) g(j(z)) α S g(x) M (mod 2 ). (5.2) Moreover, the integers A(α), B(g), C(g) have the roerty that A(α) and gcd(b(g),c(g)). The work of Deligne and Dwork (see [Dwo69, 7]) considers the full -adic exansion of j(z). Examle. As an examle of (5.2), when =37wehaveM = {x 2 +31x +31}, S 37 = { 29}, and j(37z) j(z) (33j(z) j(z) j(z) j(z)+6) (9j(z) + 10) + + j(z)+29 j(z) 2 (mod 37 2 ). +31j(z)+31 Remark. In [KZ98], Kaneko and Zagier give many descritions of the suersingular olynomials S (x). In 10 of the same aer, they rovide an exlicit descrition of the interlay between suersingular olynomials and certain modular olynomials. Proof of Theorem 1.5. We begin by considering the cases where 11. For these rimes, a calculation shows that for each non-negative r<,we have j(z) r U a r (0) (mod ), where a r (0) is the constant in the Fourier exansion of j(z) r.sinceu is a linear oerator, it follows that every F (x) Z[x] withdegreem<has the roerty that F (j(z)) U() 0(mod). The theorem holds since S (x) =1fortheserimes. In the case when 13, denote the Fourier exansion of F (j(z)) by F (j(z)) = a(n)q n. n m The action of the normalized weight zero Hecke oerator T 0 () onf (j(z)) is given by F (j(z)) T 0 () =F (j(z)) U + F (j(z)). 308

17 Since >m, it follows that Singular moduli and class olynomials F (j(z)) U = F (j(z)) T 0 () F (j(z)) = a(n)q n. (5.3) Using the modular functions J m (z) defined in (1.2), we see that F (j(z)) T 0 () isaolynomial in j(z) with integral coefficients (this also follows from the fact that F (j(z)) T 0 () isanearly holomorhic modular function on SL 2 (Z)). Suose now that F (j(z)) (mod 2 ) is congruent to an integral olynomial in j(z). Then (5.3) imlies that there is a olynomial F (x) Z[x] forwhich F (j(z)) U F (j(z)) a(n)q n n=0 n=0 (mod ). However, the constants in F [j(z)] are the only olynomials whose Fourier exansions modulo have no terms with negative exonents; it follows that F (j(z)) U a(0) (mod ). Therefore, to rove Theorem 1.5 it suffices to rove that if S (x) 2 divides F (x) inf [x], then F (j(z)) is congruent modulo 2 to a olynomial in j(z). To this end, suose that S (x) 2 divides F (x)inf [x], and write F (x) = m s=1 (x r s). Then (5.2) imlies that m F (j(z)) = (j(z) r s ) α S s=1 m s=1 ( j(z) r s + D (j(z)) + m (j(z) r s )+D (j(z)) s=1 ( + α S A(α) j(z) α + m α S s=1 1 t s m A(α) j(z) α + (j(z) r t ) B(g)j(z)+C(g) g(j(z)) g(x) M ) B(g)j(z)+C(g) g(j(z)) g(x) M m s=1 1 t s m ) (j(z) r t ) (mod 2 ). (5.4) Since m s=1 1 t s m (j(z) r t ) is a olynomial in j(z) with integer coefficients (note that this olynomial is symmetric in the r t ), the first two summands in the last exression above are integral olynomials in j(z). Moreover, it follows that for the last summand we have ( A(α) j(z) α + ) B(g)j(z)+C(g) m (j(z) r t ) g(j(z)) g(x) M s=1 1 t s m ( α S A(α) j(z) α + B(g)j(z)+C(g) g(j(z)) g(x) M ) m s=1 1 t s m (j(z) r t ) (mod ), which is a olynomial in j(z) overf (in view of (5.1) and the fact that S (x) 2 divides F (x) in F [x]). Consequently, (5.4) showsthatf (j(z)) (mod 2 ) is an integral olynomial in j(z), and this comletes the roof. Proof of Corollary 1.6. Corollary 1.6 is obtained by verifying that S (x) 2 divides H d (x) inf [x] for those rimes and fundamental discriminants d satisfying the given hyotheses. The factorizations of H d (x) were comuted using MAGMA. 309

18 S. Ahlgren and K. Ono 6. Filtrations and the Proof of Theorem 1.7 and Corollary 1.8 Here we rove Theorem 1.7 and Corollary 1.8 using the theory of modular forms modulo as develoed by Serre and Swinnerton-Dyer (see, for examle, [Swi73]). Recall the definition (1.8) of the theta-oerator. Then we have the following facts. Proosition 6.1. Suose that f(z) M k and g(z) M k have integral coefficients. If 5 is rime and 0 f(z) g(z) (mod ), then k k (mod 1). Proosition 6.2 [Swi73, Lemma 5]. If 5 is rime and f(z) M k Z[[q]] has f(z) 0(mod), then ω (Θf) ω (f)+2 (mod 1). Moreover, we have ω (Θf) ω (f)+ +1, with equality if and only if ω (f). If f(z) M k, then the valence formula imlies that ord (f) k/12. This imlies the following roosition. Proosition 6.3. Suose that 5 is rime, and that f(z) = n=n 0 a(n)q n S k Z[[q]]. Suose further that a(n 0 ) and that n 0. Then for every non-negative integer m we have ω (Θ m f) 12n 0. We are now in a osition to rove Theorem 1.7. Proof of Theorem 1.7. In the notation of Theorem 1.7, we have f,s (z) =f(z) (z) s = a(m)q s +m + S 12 s +k. (6.1) Suose that the conclusion of the theorem is false (in other words, suose that f(z) U 0 (mod )). Since (z) s ( s z)(mod), we conclude from this that f,s (z) U 0 (mod ). Since we have (f U ) f Θ 1 f (mod ) for all f, it follows from this assumtion that 0 Θ 1 f,s (z) f,s (z) (mod ). (6.2) Suose that ω (Θ 2 f,s ). Then by Proosition 6.2, we would have ω (Θ 1 f,s )=ω (Θ 2 f,s )+ +1. (6.3) However, Proosition 6.3 and (6.1) together give ω (Θ 2 f,s ) 12( s + m), while by (6.1) and (6.2) wehaveω (Θ 1 f,s ) 12 s + k. Taken together with (6.3), these inequalities contradict the assumtion that k 12m. Therefore, we must have ω (Θ 2 f,s ); it follows that there is a smallest ositive integer j 3forwhich ω (Θ j+1 f,s ). For this j, Proosition 6.2 imlies that ω (Θ j+1 f,s )=ω (Θf,s )+j( +1) ω (Θf,s )+j 0 (mod ). Since 1 j 3, this contradicts the assumtion that ω (Θf,s ) 1, 2(mod). This comletes the roof of Theorem 1.7. Proof of Corollary 1.8. Let the notation be as in the statement of Corollary 1.8. Since>12D +1 is rime, we have >12D +3> 5. Set f(z) :=F (j(z)), and consider the modular form f,1 (z) :=f(z) (z) = a( D)q D + S 12. (6.4) 310

19 Singular moduli and class olynomials By Proosition 6.3 and (6.4), we obtain 12( D) ω (f,1 ) 12. Therefore, since >12D+3 and ω (f,1 ) 12 (mod 1), we must have ω (f,1 )=12. It follows by Proositions 6.2 and 6.3 that 12( D) ω (Θf,1 ) (6.5) The facts that ω (Θf,1 ) (mod 1) and that >12D + 3, together with (6.5), show that ω (Θf,1 )= Corollary 1.8 now follows from Theorem 1.7. Acknowledgements The authors thank Masanobu Kaneko, Jean-Pierre Serre, and Don Zagier for their comments on a reliminary version of this manuscrit. References Ber28 W. E. H. Berwick, Modular invariants, Proc. London Math. Soc. (2) 28 (1928), Bor95a R. E. Borcherds, Automorhic forms on O s+2,2 (R) and infinite roducts, Invent. Math. 120 (1995), Bor95b R. E. Borcherds, Automorhic forms on O s+2,2 (R) + and generalized Kac-Moody algebras, inproc. International Congress of Mathematicians (Zürich, 1994) (Birkhäuser, Basel, 1995), Car59 L. Carlitz, Arithmetic roerties of generalized Bernoulli numbers, J. reineangew.math. 202 (1959), Deu46 M. Deuring, Teilbarkeitseigenschaften der singulären Moduln der ellitischen Funktionen und die Diskriminante der Klassengleichung, Comment. Math. Helv. 19 (1946), Deu58 M. Deuring, Die Klassenkörer der komlexen Multilikation, inenzykloädie der Mathematischen Wissenschaften, Band I 2, Heft 10, Teil II (Teubner, Stuttgart, 1958). Dor88 D. Dorman, Secial values of the ellitic modular function and factorization formulae, J. reine angew. Math. 383 (1988), Dor89 D. Dorman, Singular moduli, modular olynomials, and the index of the closure of Z[j(τ)], Math. Ann. 283 (1989), Dwo69 B. Dwork, -adic cycles, Publ. Math. Inst. Hautes Études Sci. 37 (1969), Elk87 N. Elkies, The existence of infinitely many suersingular rimes for every ellitic curve over Q, Invent. Math. 89 (1987), GH93 B. Gordon and K. Hughes, Multilicative roerties of η-roducts, Comment.Math.143 (1993), GZ85 B. Gross and D. Zagier, On singular moduli, J. reine angew. Math. 355 (1985), IR90 K. Ireland and M. Rosen, A classical introduction to modern number theory (Sringer, New York, 1990). Kob84 N. Koblitz, Introduction to ellitic curves and modular forms, Graduate Texts in Mathematics, vol. 97 (Sringer, New York, 1984). Koh82 W. Kohnen, Newforms of half-integral weight, J. reine angew. Math. 333 (1982), Koi73 M. Koike, Congruences between modular forms and functions and alications to the conjecture of Atkin, J.Fac.Sci.Univ.TokyoSect.IAMath.20 (1973), KZ98 M. Kaneko and D. Zagier, Suersingular j-invariants, hyergeometric series, and Atkin s orthogonal olynomials, incomutational ersectives on number theory (Chicago, IL, 1995), AMS/IP Stud. Adv. Math., vol. 7 (American Mathematical Society, Providence, RI, 1998),

20 Singular moduli and class olynomials Lan87 S. Lang, Ellitic functions, second edition (Sringer, New York, 1987). Leh49 J. Lehner, Divisibility roerties of the Fourier coefficients of the modular invariant j(τ), Amer. J. Math. 71 (1949), Mar96 Y. Martin, Multilicative η-quotients, Trans. Amer. Math. Soc. 348 (1996), Ser76 J.-P. Serre, Divisibilité de certaines fonctions arithmétiques, L Enseign. Math. 22 (1976), Shi73 G. Shimura, On modular forms of half-integral weight, Ann. of Math. (2) 97 (1973), Sil86 J. Silverman, The arithmetic of ellitic curves, Graduate Texts in Mathematics, vol. 106 (Sringer, New York, 1986). Swi73 H. P. F. Swinnerton-Dyer, On l-adic reresentations and congruences for coefficients of modular forms, Lecture Notes in Mathematics, vol. 350 (Sringer, Berlin, 1973), Web61 H. Weber, Lehrbuch der Algebra, vol. III, third edition (Chelsea, New York, 1961). Zag02 D. Zagier, Traces of singular moduli, in Motives, olylogarithms and Hodge theory, art I (F. Bogomolov and L. Katzarkov, eds.), International Press Lecture Series (International Press, Somerville, MA, 2002), Scott Ahlgren ahlgren@math.uiuc.edu Deartment of Mathematics, University of Illinois, Urbana, IL 61801, USA Ken Ono ono@math.wisc.edu Deartment of Mathematics, University of Wisconsin, Madison, WI 53706, USA 312