A NOTE ON THE EXISTENCE OF CERTAIN INFINITE FAMILIES OF IMAGINARY QUADRATIC FIELDS

A NOTE ON THE EXISTENCE OF CERTAIN INFINITE FAMILIES OF IMAGINARY QUADRATIC FIELDS IWAO KIMURA ABSTRACT. Let l > 3 be an odd prime. Let S 0, S +, S be mutually disjoint nite sets of rational primes. For any sufficiently large real number X, we give a lower bound of the number of imaginary quadratic elds k which satisfy the following conditions: the discriminant of k is greater than X, the class number of k is not divisible by l, every q S 0 ramies, every q S + splits and every q S is inert in k, respectively. 1. INTRODUCTION In this paper, we give a lower bound of the number of certain families of imaginary quadratic elds whose absolute value of discriminants are less than given number (main result is corollary below). We briey review the investigations concerned with this kind of problems. Let us denote Z the ring of rational integers, Q the eld of rational numbers. For any rational prime l, we denote Z l the ring of l-adic integers. If k is an algebraic number eld of nite degree over Q, we denote h(k) the class number of k and D(k) the discriminant of k. We denote S the cardinality of any set S. Let ( / ) be Legendre-Kronecker symbol. Let us denote Q the set of all imaginary quadratic elds, Q + the set of all real quadratic elds. For any real number X > 0, let Q (X) = {k Q ; X < D(k)} and Q + (X) = {k Q + ; D(k) < X}. Note that Q (X) and Q + (X) are nite sets. 1991 Mathematics Subject Classication. primary: 11R29, secondary: 11R11. Partially supported by Grant-in-Aid for Young Scientists (B), 14740009, The Ministry of Education, Culture, Sports, Science and Technology, Japan. 1

2 IWAO KIMURA Hartung [13] proved that {k Q ; 3 h(k)} is an innite set, and remarked that his method (an application of Kronecker's class number relation) can be applied to the same statement in which 3 is replaced by any odd prime l. The case of l = 3 is also implied from Davenport-Heilbronn's investigation [10, 11] on mean 3-class numbers of quadratic elds. Kohnen and Ono [20] obtained lower bound of {k Q (X); l h(k)} where l 5 is any prime. This is quantitative renement of Hartung's result. Ono [26], Byeon [3] also obtained similar estimate for real quadratic elds for primes l 5. Cohen and Lenstra [6] conjectured the density {k Q (X); l h(k)} lim X Q (X) {k Q + (X); l h(k)} lim X Q + (X) = = (1 l n ), n 1 (1 l n ). n 2 The results cited above are far from these conjectured values. For any number eld k of nite degree over Q and any rational prime l, λ l (k), µ l (k) denote Iwasawa λ and µ invariants of the basic Z l -extension over k respectively. It is known by Iwasawa [18] that, if l h(k) and l does not split at all in k, then λ l (k) = µ l (k) = 0. Horie, Horie-Ônishi [14, 17, 15] proved that there exist innitely many k Q which satisfy l h(k) and certain ramication conditions (e.g. (D(k)/l) 1) if l is sufficientry large prime. The method they used involves l-adic Galois representation arising from Jacobians of certain modular curves and Trace formulae of Hecke operators acting on certain spaces of cusp forms. They deduced, by means of Iwasawa's theorem cited above, that {k Q ; λ l (k) = µ l (k) = 0} is an innite set (For any abelian number eld k and any rational prime l, it is known by Ferrero-Washington that µ l (k) = 0). Naito [23, 24] extended some parts of their results to the case of relative class numbers of CM-elds, and implied similar statements for relative λ and µ invariants.

A NOTE ON THE EXISTENCE OF CERTAIN INFINITE FAMILIES OF IMAGINARY QUADRATIC FIELDS3 Nakagawa-Horie [25] rened Davenport-Heilbronn's arguments and obtained estimates of lim inf X {k Q (X); 3 h(k), (D(k)/3) 1}, Q (X) and {k Q + (X); 3 h(k), (D(k)/3) 1} lim inf. X Q + (X) By Iwasawa's theorem, these values are lower bounds of and lim inf X (1) lim inf X {k Q (X); λ 3 (k) = µ 3 (k) = 0}, Q (X) {k Q + (X); λ 3 (k) = µ 3 (k) = 0}. Q + (X) It is longstanding conjecture (so-called Greenberg's conjecture [12]) that λ l (k) = 0 for any rational prime l if k is totally real. The real quadratic case of Nakagawa- Horie's investigation provides some evidence for the conjecture. Taya [29] improved their result and showed that the value of (1) 17. Nakagawa-Horie's result are also extended to more general situation, quadratic extensions over 24 xed number eld case by Horie and the author [16] and over xed function eld case by the author [19], using the theory of zeta functions associated to certain prehomogeneous vector space (the space of binary cubic forms) developed by Datskovsky and Wright [9]. Belabas and Fouvry [1] rened Davenport and Heilbronn's investigation and estimated {k Q + (X); 3 h(k) and D(k) is a rational prime}. Byeon [4] extended the investigation of Kohnen-Ono so that it covers the cases treated by Horie [14]. He obtained {k Q (X); λ l (k) = µ l (k) = 0} l,ε X log X for any odd prime l 5 and any real number ε > 0 (the suffix of means that an implied constant depends on them). Ono [26], Byeon [3, 5] also discussed the case of real quadratic elds and obtained similar lower bound. Their method relies on the fact that, the coefficients of θ 3 (z) where θ(z) = n Z q n2, are closely

4 IWAO KIMURA related to class numbers of quadratic forms, and Sturm's theorem [28] on congruence of modular forms. In this paper, we give quantitative version of the investigation of Horie-Ônishi [17] and Horie [15] (main result is corollary below). We use Eisenstein series of half integral weight constructed by Cohen [7] and Sturm's theorem. We note that Bruinier (Theorem 7 in [2]) gave similar result under some assumption. Acknowledgement. The author thanks anonymous referee for careful reading of manuscript. 2. RESULTS Theorem. Let l > 3 be an odd prime. Let S 0, S +, S be mutually disjoint nite sets of rational primes. Take an integer b > 0 which satises the following conditions: b is a fundamental discriminant, ( b/q) = 0, 1, 1 according as q S 0, S +, S respectively (where ( / ) is Legendre-Kronecker symbol). P = 4 q S 0 S + S q. For any prime p which satises p 2 Let 1 (mod P) and ( b/p) p (mod l), let κ = (1/2)pP 2 q pp(1 + q 1 ). Then there exists a natural number m p < κ which satises the following conditions: the class number of the imaginary quadratic eld Q( m p ) is not divisible by l, every prime q S 0 ramies, every prime q S + splits and every prime q S is inert, in Q( m p ), respectively. Corollary. Let l > 3 be an odd prime, ε > 0 be an arbitrary real number. Let S 0, S +, S and P be the same ones as in the theorem. Then, for any sufficiently large real number X > 0, we have {k Q (X); l h(k) and satises the following conditions ( ).} l,ε,p X log X. Conditions ( ): every prime q S 0 ramies, every prime q S + splits and every prime q S is inert in k, respectively (suffix of the inequality means that the implied constants depend on them). Remark. One of the application of this kind of results is, as mentioned above, to give an estimate of {k Q (X); λ l (k) = µ l (k) = 0}. We give another one.

A NOTE ON THE EXISTENCE OF CERTAIN INFINITE FAMILIES OF IMAGINARY QUADRATIC FIELDS5 B. Mazur [21] proved, among other things, that if there exists imaginary quadratic elds k such that 5 h(k) and (D(k)/11) 1, then the 5 primary part III(X 0 (11), k){5} of the Tate-Shafarevich group of a modular curve X 0 (11) of level 11 over k is 0. It thus follows from our corollary that {k Q (X); III(X 0 (11), k){5} = {0}} 3. PROOF OF RESULTS X log X. Let g(z) = n=0 a(n)q n be any formal power series of an indeterminate q of rational integer coefficients. For any rational prime p, we dene (U p g)(z), (V p g)(z) by the following formulae: (U p g)(z) = (V p g)(z) = a(pn)q n, n=0 a(n)q pn. n=0 We dene the order ord l (g) of g at rational prime l by ord l (g) = min{n a(n) 0 (mod l)}. Let h(z) = n=0 b(n)q n be another formal power series of rational integer coefficients, m be any rational integer. We denote g(z) h(z) (mod m) if and only if a(n) b(n) (mod m) for all n 0. For any half integer k 1 Z and natural number N (if k Z we assume that 2 4 N), let M k (N, χ) denote the space of modular forms of weight k, Nebentypus character χ, with respect to a congruence subgroup Γ 0 (N) (cf. Shimura [27]). If χ is trivial character, we write M k (N) instead of M k (N, 1). Let g(z) M k (N, χ), its Fourier expansion g(z) = n=0 a(n)q n. It is known that (U p g)(z), (V p g)(z) M k (N p, χ(p/ )). Sturm [28] proved that, if g(z) M k (N, χ) has rational integer coefficients and ord l (g) > κ(n, k) = k 12 [Γ 0(1) : Γ 0 (N)] = k 12 N (1 + q 1 ), q N

6 IWAO KIMURA then g(z) 0 (mod l). He proved this fact when k is an integer (detailed proof can be found in Murty [22]), but Kohnen-Ono loc. cit. pointed out that it is easy to verify this fact when k is a half integer. Let H(n) denote Hurwitz-Kronecker class number of integeral binary quadratic forms of discriminant n ( n 0, 1 (mod 4)). If n = D f 2 where D is a negative fundamental discriminant, then H(n) is related to the class number h(d) of an imaginary quadratic eld Q( D) by the well-known formula: H(n) = h(d) w(d) d f ( D ) ( ) f µ(d) σ 1. d d Here w(d) is half the number of roots of unity in Q( D), µ( ) is Möbius function, ( / ) is Kronecker-Legendere symbol, σ 1 ( ) is the sum of positive divisors of the arguments (cf. Cohen [8] Chapter 5.3). Let c, d Z with d 1. Suppose that c is quadratic non-residue modulo d (i.e., the equation x 2 c (mod d) has no solutions). Then we dene a function H c,d (z) by the following formula: H c,d (z) = H(n)q n (q = e 2π 1z ). n c (mod d) It is known that H c,d (z) M 3/2 (A) where A = 4d 2 (A can be taken A = d 2 if d is even) by Cohen [7]. We dene f (z) = 6H b,p (z) where b, P are the same ones as in the theorem. The coefficients of f (z) are rational integers since H(0) = 1/12 dose not appear in f (z). Lemma. Let p be a rational prime satisfying p 2 (mod l). Then we have 1 (mod P), p ( b/p) (U p f )(z) (V p f )(z) (mod l).

A NOTE ON THE EXISTENCE OF CERTAIN INFINITE FAMILIES OF IMAGINARY QUADRATIC FIELDS7 Proof. We see, by denition, (U p f )(z) = 6 (V p f )(z) = 6 pn b (mod P) n b (mod P) H(pn)q n M 3/2 (4P 2 p), H(n)q pn M 3/2 (4P 2 p). The bp-th coefficients of both (U p f )(z) and (V p f )(z) are H(bp 2 ) = (1 + p ( b/p))h(b) and H(b) respectively. Note that H(bp 2 ) appears in (U p f )(z) because bp 2 congruent modulo l. b (mod P). By our assumptions, these two coefficients are not Proof of Theorem. By the lemma and Sturm's theorem stated above, there exists a natural number n p < κ(4p 2 p, 3/2) = (1/2)P 2 p q pp(1+q 1 ) such that the n p -th coefficient of (U p f )(z) (V p f )(z) is not congruent to 0 (mod l), i.e., ( ) np H(pn p ) H (mod l). p If p n p, we read that the n p -th coefficient H(n p /p) of (V p f )(z) is 0. We thus have H(pn p ) 0 (mod l). Since pn p b (mod P), we also see that ( ) ( ) pnp b = = 0, 1, 1 according as q S 0, S +, S. q q On the other hand, if p n p, n p can be written as n p = pn p for some natural number n p. In this case, we see ( ( n )) H(pn p ) = H(p 2 n p p) = 1 + p H(n p p) H(n p) (mod l). H(n p) 0 (mod l) thus follows. Further we have n p = pn p /p 2 pn p (mod P) since p 2 1 (mod P). This shows that ( n ) ( ) ( ) p pnp b = = = 0, 1, 1 according as q S 0, S +, S. q q q Taking m p = pn p or n p according as p n p or not, we can prove the theorem.

8 IWAO KIMURA Proof of corollary. It is easy to see that the conditions on p stated in the theorem are equivalent to that p belongs some arithmetic progressions modulo P. Let p 1 < p 2 < be the primes contained in one of such arithmetic progressions modulo P in increasing order. Then, if i < j < k and D i, D j, D k are the discriminants of the imaginary quadratic elds Q( m pi ), Q( m p j ), Q( m pk ), then at least two of them are different by theorem 1. Moreover we plainly see that D i p i κ(4p 2 p i, 3/2). The result now follows by Dirichlet's theorem on arithmetic progressions. REFERENCES [1] K. Belabas and E. Fouvry. Sur le 3-rang des corps quadratiques de discriminant premier ou presque premier. Duke Math. J., 98(2):217 268, 1999. [2] Jan Hendrik Bruinier. Nonvanishing modulo l of Fourier coefficients of half-integral weight modular forms. Duke Math. J., 98(3):595 611, 1999. [3] Dongho Byeon. Class numbers and Iwasawa invariants of certain totally real number elds. J. Number Theory, 79(2):249 257, 1999. [4] Dongho Byeon. A note on basic Iwasawa λ-invariants of imaginary quadratic elds and congruence of modular forms. Acta Arith., 89(3):295 299, 1999. [5] Dongho Byeon. Indivisibility of class numbers and Iwasawa λ-invariants of real quadratic elds. Compositio Math., 126(3):249 256, 2001. [6] H. Cohen and H. W. Lenstra, Jr. Heuristics on class groups of number elds. In Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), pages 33 62. Springer, Berlin, 1984. [7] Henri Cohen. Sums involving the values at negative integers of L-functions of quadratic characters. Math. Ann., 217(3):271 285, 1975. [8] Henri Cohen. A course in computational algebraic number theory. Springer-Verlag, Berlin, 1993. [9] Boris Datskovsky and David J. Wright. Density of discriminants of cubic extensions. J. Reine Angew. Math., 386:116 138, 1988. [10] H. Davenport and H. Heilbronn. On the density of discriminants of cubic elds. Bull. London Math. Soc., 1:345 348, 1969. [11] H. Davenport and H. Heilbronn. On the density of discriminants of cubic elds. II. Proc. Roy. Soc. London Ser. A, 322(1551):405 420, 1971.

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10 IWAO KIMURA [28] Jacob Sturm. On the congruence of modular forms. In Number theory (New York, 1984 1985), LNM 1240, pages 275 280. Springer, Berlin, 1987. [29] Hisao Taya. Iwasawa invariants and class numbers of quadratic elds for the prime 3. Proc. Amer. Math. Soc., 128(5):1285 1292, 2000. DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, TOYAMA UNIVERSITY, GOFUKU 3190, TOYAMA CITY, TOYAMA 930 0885, JAPAN. E-mail address: iwao@sci.toyama-u.ac.jp