DIVISIBILITY CRITERIA FOR CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS

Transcription

1 IVISIBILITY CRITERIA FOR CLASS NUMBERS OF IMAGINARY QUARATIC FIELS PAUL JENKINS AN KEN ONO Abstract. In a recent aer, Guerzhoy obtained formulas for certain class numbers as -adic limits of traces of singular moduli. Using earlier work by Bruinier and the second author, we derive a more recise form of these results using results of Zagier. Secifically, if < 4 is a fundamental discriminant and n is a ositive integer, then Tr( n d 4 ( ( 1 1 H( (mod n rovided that {, 3} and ( ( = 1, or {5, 7, 13} and Introduction and statement of results Throughout, let d 0, 3 (mod 4 be a ositive integer, and let Q d denote the set of ositive definite integral binary quadratic forms Q(x, y = ax + bxy + cy = [a, b, c] with discriminant = b 4ac (including imrimitive forms if there are any. The grou Γ := PSL (Z acts on Q d with finitely many orbits, and if ω Q is defined by if Q Γ [a, 0, a], ω Q = 3 if Q Γ [a, a, a], 1 otherwise, then the Hurwitz-Kronecker class number H( is given by (1.1 H(d = 1. ω Q Q Q d /Γ If < 4 is a fundamental discriminant, then H( is the class number of the ring of integers of the imaginary quadratic field Q(. ( Recently, Guerzhoy has obtained some interesting exressions for 1 ( H( as -adic limits of traces of singular moduli. To make this recise, we first recall some notation. For ositive definite binary quadratic forms Q, let α Q be the unique root of ate: August 31, Mathematics Subject Classification. 11R9, 11F37. Key words and hrases. Class numbers, singular moduli. The authors thank the National Science Foundation for their generous suort. The second author is grateful for the suort of a Packard and a Romnes Fellowshi. 1

2 PAUL JENKINS AN KEN ONO Q(x, 1 = 0 in the uer half of the comlex lane. If j(z is the usual SL (Z modular function j(z = E 4(z 3 (z = q q +, where q = e πiz, then define integers Tr(d by (1. Tr(d = Q Q d /Γ j(α Q 744 ω Q. The algebraic integers j(α Q are known as singular moduli. Guerzhoy roved (see Corollary.4 (a of [5] that if {3, 5, 7, 13} and < 4 is a fundamental discriminant, then one has the -adic limit formula ( ( (1.3 1 H( = 1 lim 4 n + Tr(n d. If ( = 1, then this result simly imlies that Tr( n d 0 -adically as n tends to infinity. Thanks to work of Boylan, Edixhoven and the first author (see [, 4, 6], it turns out that more is true. In articular, if is any rime and ( = 1, then (1.4 Tr( n d 0 (mod n. In earlier work [3], Bruinier and the second author obtained certain -adic exansions for H( in terms of the Borcherds exonents of certain modular functions with Heegner divisor. In his aer [5], Guerzhoy asks whether there is a connection between (1.3 and these results when ( 1. In this note we show that this is indeed the case by establishing the following congruences. Theorem 1.1. Suose that < 4 is a fundamental discriminant and that n is a ositive integer. If {, 3} and ( ( = 1, or {5, 7, 13} and 1, then ( ( H( Tr( n d (mod n. In articular, under these hyotheses n divides 4 1 n divides Tr( n d. ( 1 ( H( if and only if Three remarks. 1 Theorem 1.1 includes =. For simlicity, Guerzhoy chose to work with odd rimes, and this exlains the omission of = in (1.3. esite the uniformity of (1.4, it turns out that the restriction on in Theorem 1.1 is required. For examle, if = 11, n = 1 and = 15, then ( = 1, H( 15 =,

3 and we have IVISIBILITY CRITERIA FOR CLASS NUMBERS 3 Tr(11 15 = H( 15 (mod There are generalizations of Theorem 1.1 which hold for rimes {, 3, 5, 7, 13}. For examle, one may emloy Serre s theory [7] of -adic modular forms to derive more recise versions of Corollary.4 (b of [5].. The roof of Theorem 1.1 The roof of Theorem 1.1 follows by combining earlier work of Bruinier and the second author with results of Zagier and a combinatorial formula used earlier by the first author. We recall some necessary notation. Let M! be the sace of weight λ + 1 weakly holomorhic modular forms on Γ λ+ 1 0(4 with Fourier exansion f(z = a(nq n. ( 1 λ n 0,1 (mod 4 For 0 d 0, 3 (mod 4, we let f d (z be the unique form in M! 1 (.1 f d (z = q + A(, dq. 0< 0,1 (mod 4 with exansion The coefficients A(, d of the f d are integers. For comleteness, we set A(M, N = 0 if M or N is not an integer. These modular forms are described in detail in [8]. For fundamental discriminants < 4, Borcherds theory on the infinite roduct exansion of modular forms with Heegner divisor [1] imlies that q H( n=1 (1 q n A(n,d is a weight zero modular function on SL (Z whose divisor consists of a ole of order H( at infinity and a simle zero at each Heegner oint of discriminant. Using this factorization, Bruinier and the second author roved the following theorem. Theorem.1 ([3], Corollary 3. Let < 4 be a fundamental discriminant. If {, 3} and ( ( = 1, or {5, 7, 13} and 1, then as -adic numbers we have H( = 1 k A( k, d. 4

4 4 PAUL JENKINS AN KEN ONO Remark. The case when = 13 is not roven in [3]. However, thanks to the remark receding Theorem 8 of [7] on 13-adic modular forms with weight congruent to (mod 1, and Theorem of [3], the roof of Corollary 3 [3] still alies mutatis mutandis. Zagier identified traces of singular moduli with the coefficients A(, d as follows. Theorem. ([8], Corollary to Theorem 3. For all ositive integers d 0, 3 (mod 4, Tr(d = A(1, d. Combining Zagier s duality ([8], Theorem 4 between coefficients of modular forms in M! 1 and in M! 3 with the action of the Hecke oerators on these saces, the first author roved the following combinatorial formula. Lemma.3 ([6], Theorem 1.1. If is a rime and d,, n are ositive integers such that, 0, 1 (mod 4, then n 1 ( n k 1 ( ( A(, n d = n A( n, d + A, k d k+1 A ( k, d n 1 ( n k 1 ((( ( + k A( k, d. Remark. This result is stated for odd in [6], but the roof holds for = as well. Proof of Theorem 1.1. Under the given hyotheses, Theorem.1 imlies that (. n H( k A( k, d (mod n. By letting = 1 in Lemma.3, for these d and we find that ( ( n 1 (.3 1 k A( k, d = A(1, n d n A( n, d. Inserting this exression for the sum into (., we conclude that ( ( H( A(1, n d (mod n, which by Zagier s theorem is Tr( n d. References [1] R. E. Borcherds, Automorhic forms on O s+, (R and infinite roducts, Invent. Math. 10 (1995, no. 1, [] M. Boylan, -adic roerties of Hecke traces of singular moduli, Math. Res. Lett. 1 (005, no. 4,

5 IVISIBILITY CRITERIA FOR CLASS NUMBERS 5 [3] J. H. Bruinier and K. Ono, The arithmetic of Borcherds exonents, Math. Ann. 37 (003, no., [4] B. Edixhoven, On the -adic geometry of traces of singular moduli, International Journal of Number Theory 1 (005, [5] P. Guerzhoy, The Borcherds-Zagier isomorhism and a -adic version of the Kohnen-Shimura ma, Int. Math. Res. Not. (005, no. 13, [6] P. Jenkins, -adic roerties for traces of singular moduli, International Journal of Number Theory 1 (005, [7] J.-P. Serre, Formes modulaires et fonctions zêta -adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwer, 197, Sringer, Berlin, 1973, Lecture Notes in Math., Vol [8]. Zagier, Traces of singular moduli, Motives, olylogarithms and Hodge theory, Part I (Irvine, CA, 1998, Int. Press Lect. Ser., vol. 3, Int. Press, Somerville, MA, 00, UCLA Mathematics eartment, Los Angeles, California address: jenkins@math.ucla.edu eartment of Mathematics, University of Wisconsin, Madison, Wisconsin address: ono@math.wisc.edu