Comositio Mathematica 16: 49^56, 001. 49 # 001 Kluwer Academic Publishers. Printed in the Netherlands. Indivisibility of Class Numbers and Iwasawa l-invariants of Real Quadratic Fields ONGHO BYEON School of Mathematics, Korea Institute for Advanced Study, 07-43 Cheongryangri-dong, ongdaemun-gu, Seoul 130-01, Korea; e-mail: dhbyeon@kias.re.kr (Received: 7 October 1999; acceted: 1 May 000) Abstract. Let > 0 be the fundamental discriminant of a real quadratic eld, and h its class number. In this aer, by re ning Ono's idea, we show that for any rime > 3, ]f0 < < j h 60(mod)g >> log : Mathematics Subject Classi cation (000). 11R9. Key words. class numbers, Iwasawa l-invariants. 1. Introduction Let > 0 be the fundamental discriminant of the real quadratic eld Q,and h its class number. Let be rime, Z the ring of -adic integers, and l Q the Iwasawa l-invariant of the cyclotomic Z -extension of Q. Let R denote the -adic regulator of Q, and jj denote the usual multilicative -adic valuation normalized so that jj ˆ 1=. Although the `Cohen^Lenstra heuristics' [3] redict that for any rime,thereare in nitely many real quadratic elds Q with 6jh, it is roved only for the case < 5000 ([4, 14]). On the other hand, Greenberg [6] conjectured that l Q ˆ 0 for any real quadratic eld Q and any rime number. However, very little is known (cf. [14]). In articular, Greenberg recently asked the question whether there exist in nitely many real quadratic elds Q with slitting and l Q vanishing for a given odd rime (cf. [17]). This roblem is solved only for the case ˆ 3 ([17]). In this direction, in this aer we shall rove the following theorem: THEOREM 1.1. Let > 3 be rime and d ˆ 1or 1. If d ˆ 1, then for any 3 (mod 4), and if d ˆ 1, then for any, ] 0 < < j h 60(mod ; ˆ d; and jr j ˆ 1 >> log :
50 ONGHO BYEON For the case = ˆ 1, i.e., remains rime in the real quadratic eld Q, and 6jh, we have l Q ˆ 0 by a criterion of Iwasawa [1]. Further for the case = ˆ1, i.e., slits in the real quadratic eld Q, 6jh, and jr j ˆ 1=, wealsohavel Q ˆ 0 by a criterion of Fukuda and Komatsu [5]. Thus, by Theorem 1.1 we immediately have the following theorem: THEOREM 1.. Let > 3 be rime and d ˆ 1 or 1. If d ˆ 1, then for any 3 (mod 4), and if d ˆ 1, then for any, ] 0 < < j l Q ˆ 0; ˆ d >> log : To rove Theorem 1.1, rst we shall re ne Ono's idea [14] and rove the following theorem. THEOREM 1.3. Let > 3 be rime and d ˆ 1or 1. If there is a fundamental discriminant 0 corime to of a real quadratic eld Q 0 such that 0 (i) ˆ d; (ii) h 0 60mod; (iii) jr 0 j ˆ 1 ; then for each d, ] 0 < < j h 60(mod ; ˆ d; and jr j ˆ 1 >> log : Finally, we shall show that the condition in Theorem 1.3 holds for any 3(mod 4) if d ˆ 1andforany if d ˆ 1. Remark. Similar works for imaginary quadratic elds can be found in [1, 7^9, 13, 15].. Proof of Theorem 1.3 To rove Theorem 1.3, we shall basically follow the roof of Theorem 1 in [14]. Consult [14] for more details. Let be the fundamental discriminant of a quadratic number eld, w : ˆ = the usual Kronecker character, and w 0 the trivial character. Let M k G 0 N ; w denote the sace of modular forms of weight k on G 0 N with character w. Letr and N be nonnegative integers with r. If N 6 0; 1 (mod 4), then let H r; N ˆ0: If
CLASS NUMBERS AN IWASAWA l-invariants 51 N ˆ 0, then let H r; 0 : ˆ z 1 r. Ifn ˆ 1 r N, then de ne H r; N by H r; N : ˆ L 1 r; w m d w d d r 1 s r 1 n=d ; djn where s n n : ˆ Pdjn dn. Cohen [] roved that for every r, F r z : ˆ H r; N q N q: ˆ e iz M r 1 G 0 4 ; w 0 : N 0 By the similar arguments as in the roof of Proosition in [14], which use the construction of the Kubota^Leooldt -adic L-function L s; w, the Kummer congruences, and the -adic class number formula (cf. [18]), we have the following roosition. PROPOSITION.1. Let be an odd rime number and 6ˆ 1 be the fundamental discriminant of a real quadratic eld. Then H 1 ; is -integral and H 1 ; h R (mod : Let e > 1 be the fundamental unit of the real quadratic eld Q.Then R ˆlog e.let > 3berimeand a rime ideal of Q over. Let n ; be a non negative integer satisfying that n ; j e N 1 1 but n ; 1 6je N 1 1; where N is the absolute norm of Q. Note that n ; 1. Since je N 1 1j ˆjlog e N 1 j, we have that jr j ˆ n ; ; if is unramified; n ; = ; if is ramified: Thus, by Proosition.1 we immediately have the following roosition: PROPOSITION.. Let > 3 be rime and 6ˆ 1 be the fundamental discriminant of the real quadratic eld Q in which is unrami ed. Then H 1 ; = is -integral and H 1 ; h R (mod : Let d ˆ 1or1.Let > 3 be rime and de ne G z M 1 1 G 0 4 ; w 0 by G z : ˆ F 1 z ˆ 1 n H 1 ; n q n ; nˆ0
5 ONGHO BYEON and A d z M 1 1 G 0 4 4 ; w 0 by A d z : ˆ G z dg z ˆ H 1 ; n q n : n ˆd Similary, for a rime Q 6ˆ, de ne C d z M 1 G 0 4 4 Q 4 ; w 1 0 by C d z : ˆ H 1 ; n q n : n ˆd; Q n ˆ 1 If l 6ˆ is rime, then de ne U l jc d z and V ljc d z M 1 G 1 0 4 4 Q 4 l ; 4l by U l jc d 1 z : ˆ u d ;l n qn ˆ H 1 ; ln q n ; nˆ1 n ˆd; Q n ˆ 1 V l jc d 1 z : ˆ v d ;l n qn ˆ nˆ1 n ˆd; n Q ˆ 1 H 1 ; n q ln : By the similar arguments as in the roof of Proosition 3 in [14] and Proosition., we know that there exist a Z corime to such that a = U l jc d z and a = V l jc d z have integer Fourier coef cients. Now we assume that there is a fundamental discriminant of real quadratic eld of Q 0 for which 0 H 1 ; 0 ˆ d and 6 0 (mod : Let n be the fundamental discriminant of the real quadratic eld Q n and S denote the set of those n with n W k : ˆ 1 1 3 Q 3 1 Q 1 =4 for which n ˆ 1 Q and n ˆ d: Let l be a suf ciently large rime satisfying w 0 l ˆ1and (1) w n l ˆ1 for every n S ; () l l ˆ 1 and ˆ 1; Q (3) l 6 1 (mod : Then by the roerties of l and the similar arguments in the roof of Theorem in [14], which use a theorem of Sturm [16] on the congruence of modular forms, we have
CLASS NUMBERS AN IWASAWA l-invariants 53 that there must be an integer 1 W n W k l corime to l for which a ud ;l n ˆa H 1 ; nl 60 (mod : Thus, by Proosition., we have the following roosition: PROPOSITION.3. Let > 3 be rime and d ˆ 1 or 1. Assume that there is a fundamental discriminant 0 corime to of a real quadratic eld Q 0 such that (i) 0 ˆ d; (ii) h 0 60mod; (iii) jr 0 j ˆ 1 ; If l is a suf ciently large rime satisfying w 0 l ˆ1 and (1), (), (3), then for each d, there is a ositive fundamental discriminant l : ˆ d l l with d l W k l such that h l 60 (mod ; l ˆ d; and jr l j ˆ 1 : Proof of Theorem 1.3. Let r (mod t ) be an arithmetic rogression with r ; t ˆ1 and jt such that for every rime l r (mod t ), l satis es w 0 l ˆ1 and (1), (), (3). Then, by the similar arguments as in the roof of Theorem 1 in [14], which use irichlet's theorem on rimes in arithmetic rogression, Theorem 1.3 easily follows from Proosition.3. & 3. Proof of Theorem 1.1 Theorem 1.1 follows immediately from Theorem 1.3 and the following roosition. PROPOSITION 3.1. Let > 3 be rime and d ˆ 1 or 1. If d ˆ 1, then for any 3 (mod 4), let be the fundamental discriminant of the real quadratic eld Q 1 and if d ˆ 1, then for any, let be the fundamental discriminant of the real quadratic eld Q 4. Then for each d, satis es the condition in Theorem 1.3, i.e., 0 (i) ˆ d; (ii) h 0 60mod; (iii) jr 0 j ˆ 1 ; To rove Proosition 3.1, we need the following lemmas:
54 ONGHO BYEON LEMMA 3. (L. K. Hua [10]). Let be the fundamental discriminant of the real quadratic eld Q and L s; w be the irichlet L-function with character w. Then L 1; w < log 1: LEMMA 3.3 ([11]). Let be an odd rime and a rime ideal of the real quadratic eld Q over. If a is an element of Q such that a n 1modbut a n 6 1mod for some integer n, then we have a N 1 6 1 mod. Proof of Proosition 3.1. (i). If d ˆ 1, then since f ˆ 4 ˆ 4 ˆ 1 f Z ; we have = ˆ1forany. Ifd ˆ 1, then since f ˆ 1 ˆ 1 f Z ; we have = ˆ 1forany3(mod4). (ii) irichlet's class number formula says that L 1; w h ˆ : log e By Lemma 3., we have that h < log < log ; 4 log e log =4 because e > =. Let be the fundamental discriminant of Q 1 or Q 4. Thenby easy comutation, we have h < if 11: Since we can also easily check that h < ; if < 11, we rove that 6jh for any. (iii) Let d ˆ 1and be the fundamental discriminant of the real quadratic eld Q 4. Let e > 1 be the fundamental unit of Q 4 and a: ˆ 4 =. Since N Q =Q a ˆ 1anda > 1, a ˆ e j for some odd j > 0. Since! 0 1! 1 4 1 1 ˆ @ 4 A; we have that a 1 ˆ e j 1 1mod, but a 1 ˆ e j 1 6 1mod :
CLASS NUMBERS AN IWASAWA l-invariants 55 Thus, by Lemma 3.3 and the discussion above in Proosition., we have that jr j ˆ 1. Now, we consider the case d ˆ 1and is the fundamental discriminant of Q 1. In this case, if we let a: ˆ 1, then by the same method, we can also rove that jr j ˆ 1. & Acknowledgements The author would like to thank K. Ono for kindly answering his questions and J. H. Oh for helful discussion. References 1. Byeon,.: A note on the basic Iwasawa l-invariants of imaginary quadratic elds and congruence of modular forms, Acta Arith. 89 (1999), 95^99.. Cohen, H.: Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 17 (1975), 71^85. 3. Cohen, H. and Lenstra, H. W.: Heuristics on class grous of number elds, In: Number Theory (Noordwijkerhout 1983), Lecture Notes in Math. 1068, Sringer, New York, 1984,. 33^6 4. avenort, H. and Heilbronn, H.: On the density of discriminants of cubic elds II, Proc. Roy. Soc. London Ser. A 3 (1971), 405^40. 5. Fukuda, T. and Komatsu, K.: On the l-invariants of Z -extensions of real quadratic elds, J. Number Theory 3 (1986), 38^4. 6. Greenberg, R.: On the Iwasawa invariants of totally real number elds, Amer.J.Math. 98 (1976), 63^84. 7. Hartung, P.: Proof of the existence of in nitely many imaginary quadratic elds whose class number is not divisible by 3, J. Number Theory 6 (1974), 76^78. 8. Horie, K.: A note on basic Iwasawa l-invariants of imaginary quadratic elds, Invent. Math. 88 (1987), 31^38. 9. Horie, K. and Onishi, Y.: The existence of certain in nite families of imaginary quadratic elds, J. Reine Angew. Math. 390 (1988), 97^113. 10. Hua, L.-K.: On the least solution of Pell's equation, Bull. Amer. Math. Soc. 48 (194), 731^735. 11. Ichimura, H.: A note on Greenberg's conjecture and the abc conjecture, Proc. Amer. Math. Soc. 16 (1998), 1315^130. 1. Iwasawa, H.: A note on class numbers of algebraic number elds, Abh. Math. Sem. Univ. Hamburg 0 (1956), 57^58. 13. Kohnen, W. and Ono, K.: Indivisibility of class numbers of imaginary quadratic elds and orders of Tate^Shafarevich grous of ellitic curves with comlex multilication, Invent. Math. 135 (1999), 387^398. 14. Ono, K.: Indivisibility of class numbers of real quadratic elds, Comositio Math. 119 (1999), 1^11. 15. Ono, K. and Skinner, C.: Fourier coef cients of half-integral weight modular forms modulo l, Ann. Math. 147 (1998), 453^470. 16. Sturm, J.: On the congruence of modular forms, In: Lecture Notes in Math. 140, Sringer, New York, 1984,. 75^80.
56 ONGHO BYEON 17. Taya, H.: Iwasawa invariants and class numbers of quadratic elds for the rime 3, Proc. Amer. Math. Soc. 18 (000), 185^19. 18. Washington, L.: Introduction to Cyclotomic Fields, Grad. Texts in Math. 83, Sringer, Berlin, 198.