CONGRUENCES FOR TRACES OF SINGULAR MODULI

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1 CONGRUENCES FOR TRACES OF SINGULAR MODULI ROBERT OSBURN Abstract. We extend a resut of Ahgren and Ono [1] on congruences for traces of singuar modui of eve 1 to traces defined in terms of Hauptmodu associated to certain groups of genus 0 of higher eves. 1. Introduction Let j(z denote the usua eiptic moduar function on SL 2 (Z with q-expansion (q := e 2πiz j(z = q q q 2 +. The vaues of j(z at imaginary quadratic arguments in the upper haf of the compex pane are known as singuar modui. Singuar modui are important agebraic integers which generate ring cass fied extensions of imaginary quadratic fieds (Theorem 11.1 in [5], are reated to supersinguar eiptic curves ([1], and to Borcherds products of moduar forms ([2], [3]. Let d denote a positive integer congruent to 0 or 3 moduo 4 so that d is the discriminant of an order in an imaginary quadratic fied. Denote by Q d the set of positive definite integra binary quadratic forms Q(x, y = ax 2 + bxy + cy 2 with discriminant d = b 2 4ac. To each Q Q d, et α Q be the unique compex number in the upper haf pane which is a root of Q(x, 1; the singuar moduus j(α Q depends ony on the equivaence cass of Q under the action of Γ := P SL 2 (Z. Define ω Q {1, 2, 3} by Let J(z be the Hauptmodu ω Q := 2 if Q Γ [a, 0, a], 3 if Q Γ [a, a, a], 1 otherwise. J(z := j(z 744 = q q q 2 +. Zagier [15] defined the trace of the singuar modui of discriminant d as t(d := Q Q d /Γ J(α Q ω Q = Q Q d /Γ j(α Q 744 ω Q Z. Zagier has shown that t(d has some interesting properties. Namey, the foowing resut (see Theorem 1 in [15] shows that the t(d s are Fourier coefficients of a hafintegra weight moduar form Mathematics Subject Cassification. Primary 11F33, 11F37; Secondary 11F50. 1

2 2 ROBERT OSBURN Theorem 1.1. Let θ 1 (z and E 4 (z be defined by n 3 q n E 4 (z := q n, θ 1 (z := η2 (z η(2z = and et g(z be defined by Then n=1 n= g(z : = q g(z = θ 1(zE 4 (4z η 6 (4z ( 1 n q n2 = 1 2q + 2q 4 2q <d 0,3 (mod 4 t(dq d = q q q q 7 i.e., g(z is a moduar form of weight 3 2 on Γ 0(4, hoomorphic on the upper haf pane and meromorphic at the cusps. Now what about divisibiity properties of t(d as d varies? In this direction, Ahgren and Ono [1] recenty proved the foowing resut which shows that these traces t(d satisfy congruences based on the factorization of primes in certain imaginary quadratic fieds. Theorem 1.2. If d is a positive integer for which an odd prime spits in Q( d, then t( 2 d 0 (mod. Recenty, Kim [11] and Zagier [15] defined an anaogous trace of singuar modui by repacing the j-function by a moduar function of higher eve, in particuar by the Hauptmodu associated to other groups of genus 0. Let Γ 0 (N be the group generated by Γ 0 (N and a Atkin-Lehner invoutions W e for e N, i.e., e is a positive divisor of N for which gcd(e, N/e = 1. There are ony finitey many vaues of N for which Γ 0 (N is of genus 0 (see [8], [9], or [14]. In particuar, there are ony finitey many prime vaues of N. For such a prime p, et jp be the corresponding Hauptmodu. For these primes p, Kim and Zagier define a trace t (p (d (see Section 3 beow in terms of singuar vaues of jp. The goa of this paper is to prove that the same type of congruence hods for t (p (d, namey Theorem 1.3. Let p be a prime for which Γ 0 (p is of genus 0. If d is a positive integer such that d is congruent to a square moduo 4p and for which an odd prime p spits in Q( d, then t (p ( 2 d 0 mod. 2. Preiminaries on Moduar and Jacobi forms We first reca some facts about haf-integra weight moduar ( forms (see [12], [13]. If a b f(z is a function of the upper haf-pane, λ 1 2 Z, and GL + c d 2 (R, then we define the sash operator by ( a b ( f(z λ := (ad bc λ az + b 2 (cz + d c d λ f cz + d

3 CONGRUENCES 3 ( Here we take the branch of the square root having non-negative rea part. If γ = a b Γ c d 0 (4, then define ( c j(γ, z := cz + d, d where { 1 if d 1 (mod 4, ɛ := i if d 1 (mod 4. If k is an integer and N is an odd positive integer, then et M k+ 1 (Γ 0(4N denote the 2 infinite dimensiona vector space of neary hoomorphic moduar forms of weight k on Γ 0 (4N. These are functions f(z which are hoomorphic on the upper haf-pane, meromorphic at the cusps, and which satisfy ɛ 1 d (1 f(γz = j(γ, z 2k+1 f(z for a γ Γ 0 (4N. Denote by M + k+ 1 2 (Γ 0 (4N the Kohnen pus-spaces (see [13] of neary hoomorphic forms which transform according to (1 and which have a Fourier expansion of the form a(nq n. ( 1 k n 0,1 (mod 4 We reca some properties of Hecke operators on M + k+ 1 2 (Γ 0 (4N. If is a prime such that N, then the Hecke operator T k+ 1 2,4N ( 2 on a moduar form f(z := a(nq n M + (Γ k+ 1 0 (4N 2 is given by f(z T k+ 1 2,4N(2 := ( 1 k n 0,1 (mod 4 ( 1 k n 0,1 (mod 4 ( ( a( 2 ( 1 k n n + k 1 a(n + 2k 1 a(n/ 2 q n ( where is a Legendre symbo. Let us now reca some facts about Jacobi forms (see [6]. A Jacobi form on SL 2 (Z is a hoomorphic function φ : H C C satisfying ( aτ + b φ cτ + d, z = (cτ + d k cz2 2πiN e cτ+d φ(τ, z cτ + d φ(τ, z + λτ + µ = e 2πiN(λ2τ+2λz φ(τ, z ( a b for a SL c d 2 (Z and (λ, µ Z 2, and having a Fourier expansion of the form (q = e 2πiτ, ζ = e 2πiz φ(τ, z = c(n, rq n ζ r. n=0 r Z r 2 4Nn Here k and N are the weight and index of φ, respectivey. Let J k,n denote the space of Jacobi forms of weight k and index N on SL 2 (Z. By Theorem 2.2 in [6], the coefficient c(n, r depends ony on 4Nn r 2 and r mod 2N. By definition c(n, r = 0 uness 4Nn

4 4 ROBERT OSBURN r 2 0. If we drop the condition 4Nn r 2 0, we obtain a neary hoomorphic Jacobi form. Let J k,n! be the space of neary hoomorphic Jacobi forms of weight k and index N. 3. Traces Let Γ 0 (N be the group generated by Γ 0 (N and a Atkin-Lehner invoutions W e for e N, that is, e is a positive divisor ( of N for which gcd(e, N/e = 1. W e can be represented by a matrix of the form ex y 1 e with x, y, z, w Z and xwe yzn/e = 1. There Nz ew are ony finitey many vaues of N for which Γ 0 (N is of genus 0 (see [8], [9], or [14]. In particuar, if we et S denote the set of prime vaues for such N, then S = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}. For p S, et j p be the corresponding Hauptmodu with Fourier expansion q a 1 q + a 2 q Let us now define a trace in terms of the j p s. Let d be a positive integer such that d is congruent to a square moduo 4p. Choose an integer β mod 2p such that β 2 d mod 4p and consider the set Q d,p,β = {[a, b, c] Q d : a 0 mod p, b β mod 2p}. Note that Γ 0 (p acts on Q d,p,β. Assume that d is not divisibe as a discriminant by the square of any prime dividing p, i.e. not divisibe by p 2. Then we have a bijection via the natura map between and Q d,p,β Γ 0 (p Q d Γ as the image of the root α Q, Q Q d,p,β, in Γ 0 (p H corresponds to a Heegner point. We coud then define a trace t (p,β (d as the sum of the vaues of jp with Q running over a set of representatives for Q d,p,β Γ 0 (p. As t (p,β (d is independent of β, we define the trace t (p (d (see Section 8 of [15] or Section 1 of [11] t (p (d = Q jp(α Q Z ω Q where the sum is over Γ 0 (p representatives of forms Q = [a, b, c] satisfying a 0 mod p. Remark 3.1. For p = 2, we have t (2 (4 = 1 2 j 2 ( 1+i 2 = 52, t(2 (7 = j2 ( = 23, t (2 (8 = j2 ( 2 2 = 152. For p = 3, we have t(3 (3 = 1 3 j 3 ( = 14, t (3 (11 = j3( = 22. Moreover by the tabe in Section 8 of [15], we have:

5 CONGRUENCES 5 d t (2 (d t (3 (d t (5 (d The empty entries correspond to d which are not congruent to squares moduo 4p. By the discussion in Section 8 of [15] or Section 2.2 in [11], there exist forms φ p J! 2,p uniquey characterized by the condition that their Fourier coefficients c(n, r = B(4pn r 2 depend ony on r 2 4pn and where B( 1 = 1, B(d = 0 if d = 4pn r 2 < 0, 1 and B(0 = 2. Define g p (z as g p (z := q 1 + d 0 B(dq d. By the correspondence between Jacobi forms and haf-integra weight forms (Theorem 5.6 in [6], g p (z M + 3 (Γ 0 (4p. As the dimension of J 2,p is zero, we have that for every 2 integer d 0 such that d is congruent to a square moduo 4p, there exists a unique f d,p M + 1 (Γ 0 (4p with Fourier expansion 2 f d,p (z = q d + A(D, dq D. 0<D 0,1 (mod 4 An expicit construction of f d,p can be found in the appendix of [10] and the uniqueness of f d,p foows from the discussion at the end of Section 2 in [10]. The foowing resut reates the Fourier coefficients A(1, d and B(d and shows that the traces t (p (d are Fourier coefficients of a neary hoomorphic Jacobi form of weight 2 and index p (see Theorem 8 in [15] or Lemma 3.5 and Coroary 3.6 in [11]. Theorem 3.2. Let p be a prime for which Γ 0 (p is of genus 0. (i Let d = 4pn r 2 for some integers n and r. Let A(1, d be the coefficient of q in f d,p and B(d be the coefficient of q n ζ r in φ p. Then A(1, d = B(d. (ii For each natura number d which is congruent to a square moduo 4p, et t (p (d be defined as above. We aso put t (p ( 1 = 1, t (p (d = 0 for d < 1. Then t (p (d = B(d. 4. Proof of Theorem 1.3 Proof. The proof requires the study of Hecke operators T k+ 1 2,4p ( 2 on the forms g p (z and f d,p (z. Define integers A (d and B (d by A (d := the coefficient of q in f d,p T 1 2,4p(2,

6 6 ROBERT OSBURN B (d := the coefficient of q d in g p (z T 3 2,4p(2. From equation (19 of [15], we have Aso note that we have g p (z T 3 2,4p(2 = q 1 + q 2 + A (d = A(1, d + A( 2, d. 0<d 0,3 (mod 4 ( ( B( 2 d d + B(d + B(d/ 2 q d ( and so B (d = B( 2 d d + B(d + B(d/ 2. Now suppose p is in S and d is a positive integer such that d is a square moduo 4p and for which an odd prime p spits in Q( ( d. Then d = 1. By Theorem 3.2 and the above cacuations, we have t (p ( 2 d = B( 2 d ( d = B (d + B(d + B(d/ 2 B (d + B(d mod A (d + B(d mod A(1, d + A( 2, d + B(d mod B(d + B(d mod 0 mod. Exampe 4.1. We now iustrate Theorem 1.3. non-negative integer s, we have t (2 (3 2 (24s mod 3. If p = 2 and = 3, then for every In particuar, if we want to compute t (2 (207, then we are interested in φ 2 J! 2,2. By Theorem 9.3 in [6] and the discussion preceding Tabe 8 in [15], J! 2,2 is the free poynomia agebra over C[E 4 (τ, E 6 (τ, 1 ] (E 4 (τ 3 E 6 (τ 2 on two generators a and b where = E 4(τ 3 E 6 (τ 2. The Fourier expansions of a and 1728 b begin a = (ζ 2 + ζ 1 + ( 2ζ 2 + 8ζ ζ 1 2ζ 2 q + (ζ 3 12ζ ζ 56 + q 2 + (8ζ 3 56ζ ζ q 3 +. b = (ζ ζ 1 + (10ζ 2 64ζ ζ + 10ζ 2 q + (ζ ζ 2 513ζ q 2 + ( 64ζ ζ ζ q 3 +. The coefficients for a and b can be obtained using Tabe 1 or the recursion formuas on page 39 of [6]. The representation of φ 2 in terms of a and b is: φ 2 = 1 12 a(e 4(τb E 6 (τa.

7 CONGRUENCES 7 By Theorem 3.2, we have t (2 (207 = B(207. As 8n r 2 = 207 has a soution n = 29 and r = 5, then B(207 is the coefficient of q 29 ζ 5 which is Thus t (2 (207 = mod 3. Remark 4.2. (1 Zagier actuay defined t (N (d and proved part (ii of Theorem 3.2 for a N such that Γ 0 (N is of genus 0 (see Section 8 in [15]. One might be abe to prove part (i of Theorem 3.2 in the case N is squarefree. If so, then a congruence, simiar to Theorem 1.3, shoud hod for t (N (d, N squarefree. If N is not squarefree, then C. Kim has kindy pointed out part (i of Theorem 3.2 does not hod. For exampe, if N = 4 and d = 7, one can construct f 7,4 (see the appendix in [11] and compute that f 7,4 = q 7 55q + 0q q 9 +. Thus A(1, 7 = 55. But B(7 = 23 (see Remark 3.1. (2 We shoud note that Theorem 1.3 is an extension of the simpest case of Theorem 1 in [1]. Ono and Ahgren have aso proven congruences for t(d which invove ramified or inert primes in quadratic fieds. In fact, they prove that a positive proportion of primes yied congruences for t(d (see parts (2 and (3 of Theorem 1 in [1]. It woud be interesting to see if such congruences hod for t (p (d or t (N (d. (3 The Monster M is the argest of the sporadic simpe groups of order Ogg [14] noticed that the primes dividing the order of M are exacty those in the set S. The monster M acts on a graded vector agebra V = V 1 n 1 V n (see Frenke, Lepowsky, and Meurman [7] for the construction. For any eement g M, et T r(g V n denote the trace of g acting on V n for each n. Then T r(g V 1 = 1 and T r(g V n Z for every n 1. The Thompson series is defined by: T g (z = q 1 + n 1 T r(g V n q n. The authors in [4] study Thompson series evauated at imaginary quadratic arguments, i.e. singuar modui of Thompson series. It is possibe to define a trace of singuar modui of Thompson series. A natura question is do we have congruences for these traces? Acknowedgments The author woud ike to thank Imin Chen, Chang Heon Kim, Ken Ono, and Noriko Yui for their vauabe comments. References [1] S. Ahgren, K. Ono, Arithmetic of Singuar Modui and Cass Equations, to appear in Compositio Math. [2] R. E. Borcherds, Automorphic forms on O s+2,2 (R and infinite products, Invent. Math. Vo 120, (1995, [3] R. E. Borcherds, Automorphic forms on O s+2,2 (R + and generaized Kac-Moody agebras, Proc. Int. Congress of Mathematicians (Zürich, 1994 (1995, [4] I. Chen, N. Yui, Singuar vaues of Thompson series, in Groups, Difference Sets and Monster (K. T. Arau et a., Eds., pp , de Gruyter, Berin, [5] D. Cox, Primes of the Form x 2 + ny 2, John Wiey & Sons, Inc, New York, [6] M. Eicher, D. Zagier, The Theory of Jacobi forms, Progress in Math. 55, Birkhäuser-Verag, [7] I. Frenke, J. Leopowsky, A. Meurman, Vertex Operator Agebras and the Monster, Academic Press, New York [8] R. Fricke, Die Eiptische Funktionen und Ihre Anwendungen, 2-ter Tei, Teubner, Leipzig 1922.

8 8 ROBERT OSBURN [9] R. Fricke, Lehrbuch der Agebra III (Agebraische Zahen, Vieweg, Braunschweig [10] C.H. Kim, Borcherds products associated with certain Thompson series, Compositio Math. 140 (2004, [11] C.H. Kim, Traces of singuar vaues and Borcherds products, preprint, [12] N. Kobitz, Introduction to eiptic curves and moduar forms, Springer-Verag, [13] W. Kohnen, Newforms of haf-integra weight, J. reine angew. Math 333, (1982, [14] A. Ogg, Automorphismes de courbes moduaires, Séminaire Deange-Pisot-Poitou (Théorie des nombres 16e année (1974/75, No. 7, [15] D. Zagier, Traces of singuar modui, Motives, Poyogarithms and Hodge Theory, Part I. Internationa Press Lecture Series, editors F. Bogomoov and L. Katzarkov, Internationa Press, Somervie (2002, Department of Mathematics & Statistics, Queen s University, Kingston, Ontario, Canada K7L 3N6 E-mai address: osburnr@mast.queensu.ca