Indivisibility of Class Numbers of Real Quadratic Fields

INDIVISIBILITY OF CLASS NUMBERS OF REAL QUADRATIC FIELDS

Ken Ono

To T. Ono, my father, on his seventieth birthday.

Abstract. Let D denote the fundamental discriminant of a real quadratic ﬁeld, and let h(D) denote its associated class number. If p is prime, then the “Cohen and Lenstra Heuris- tics” give a probability that p - h(D). If p > 3 is prime, then subject to a mild condition, we show that √ X #{0 < D < X | p h(D)} p . - log X This condition holds for each 3 < p < 5000.

1. Introduction and Statement of Results Although the literature on class numbers of quadratic fields is quite extensive, very little is known. In this paper we consider class numbers of real quadratic fields, and as an immediate consequence we obtain an estimate for the number of vanishing Iwasawa λ invariants. √Throughout D will denote the fundamental discriminant of the quadratic number field D Q( D), h(D) its class number, and χD := · the usual Kronecker character. We shall let χ0 denote the trivial character, and | · |p the usual multiplicative p-adic valuation 1 normalized so that |p|p := p . Although the “Cohen-Lenstra Heuristics” [C-L] have gone a long way towards an ex- planation of experimental observations of such class numbers, very little has been proved. For example, it is not known that there√ are infinitely many D > 0 for which h(D) = 1. The presence of non-trivial units in Q( D) have posed the main difficulty. Basically, if

1991 Mathematics Subject Classiﬁcation. Primary 11R29; Secondary 11E41. Key words and phrases. class numbers. The author is supported by NSF grant DMS-9508976 and NSA grant MSPR-97Y012.

Typeset by AMS-TEX 1 2 KEN ONO √ (D) is the fundamental unit of Q( D), then the regulator R(D) := log((D)) complicates Dirichlet’s class number formula

2h(D)R(D) L(1, χD) = √ , D

one of the main vehicles for studying h(D). In view of these and other diﬃculties, it is natural to ask how often p - h(D), given a prime p. For the complementary question, M.R. Murty [M] has recently obtained lower bounds, although far from the Cohen and Lenstra expectation, for the number of D, both negative and positive, for which p | h(D). His results extend work of Ankeny and Chowla, Humbert, and Nagell where explicit elements of order p were constructed for inﬁnitely many discriminants. For D > 0, Cohen and Lenstra predict that the “probability” p - h(D) is

∞ p Y 1 1 1 (1 − p−i) = 1 − − − + ..., p − 1 p2 p3 p4 i=1

and for D < 0 they conjecture that the “probability” is

∞ Y 1 1 1 (1 − p−i) = 1 − − + + ··· . p p2 p5 i=1

Although extensive numerical evidence lends credence to these heuristics, apart from the works of Davenport and Heilbronn [D-H] when p = 3, little has been proved. Imaginary quadratic fields have been the focus of numerous investigations in this direction. For instance, one may see the works of Hartung [Ha], Horie and Onishi [Ho, Ho1, Ho-On], Jochnowitz [J], and Ono and Skinner [O-Sk]. These papers guarantee, under various conditions, that there are indeed infinitely many D < 0 for which p - h(D). In a recent paper, the author and Kohnen [Koh-O] have gone a step further by obtaining a lower bound for the number of −X < D < 0 for which p - h(D). In particular if p is prime and > 0, then for all sufficiently large X > 0 √ 2(p − 2) X # {−X < D < 0 | h(D) 6≡ 0 (mod p)} ≥ √ − . 3(p − 1) log X

Even less is known about the indivisibility of class numbers of real quadratic ﬁelds. There is work in this direction by Jochnowitz [J]. Subject to a mild condition regarding p−1 the existence of a suitable generalized Bernoulli number B 2 , χ , we go a step further by obtaining a lower bound for the number of 0 < D < X for which√p - h(D). As in [J], we also obtain information about Rp(D), the p-adic regulator of Q( D). CLASS NUMBERS OF REAL QUADRATIC FIELDS 3

Theorem 1. Let p > 3 be prime. If there is an odd fundamental discriminant D0 coprime to p for which

p−1 (i)(−1) 2 D0 > 0, p − 1 (ii) B , χD0 = 1, 2 p then ( ) √ R (D) X p # 0 < D < X h(D) 6≡ 0 (mod p), p | D, and √ = 1 p . D p log X

A lengthy MAPLE computation yields the following immediate corollary. Corollary 1. If 3 < p < 5000 is prime, then ( ) √ R (D) X p # 0 < D < X h(D) 6≡ 0 (mod p), p | D, and √ = 1 p . D p log X

This gives some evidence for a conjecture of Greenberg on Iwasawa invariants [G]. Let K = K0 be a number field, and let p be an odd prime. If Qn denotes the unique n n+1 subfield of degree p in the field Q(ζpn+1 ), the field of the p th roots of unity, then let Kn := KQn. These define the Zp cyclotomic extension of K

K = K0 ⊂ K1 ⊂ K2 ··· .

If Cln denotes the p-part of the class group of Kn, then

Cl0 ← Cl1 ← Cl2 ← · · · where each map is a norm. Iwasawa [Th. 13.13, W] proved that if n is suﬃciently large, then µ(K,p)pn+λ(K,p)n+ν(K,p) #Cln = p

where µ(K, p)√, λ(K, p), and ν(K, p) are ﬁxed integers, the “Iwasawa invariants.” If √K = Q( D) is√ a real quadratic ﬁeld, then Greenberg’s Conjectures [G] imply that λ(Q( D), p) = µ(√Q( D), p) = 0. By a theorem of Ferrero and Washington, it is indeed known that µ(Q( √D), p) = 0, but the complementary question regarding the vanishing of λ(D, p) := λ(Q( D), p) remains open. For p = 3, Horie and Nakagawa [Ho-N] used a theorem of Davenport and Heilbronn [D-H] and a criterion of Iwasawa to show that λ(D, 3) = 0 for a positive “proportion” of D. Moreover recent calculations for D < 10000 by Kraft and Schoof [K-S], Fukuda and Taya [F-T], and Ichimura and Sumida [Ic-Su] verify indeed that λ(D, 3) = 0 for all D < 104. Less is known when p > 3. Corollary 1 implies the following immediate result. 4 KEN ONO

Corollary 2. If 3 < p < 5000 is prime, then √ X # {0 < D < X | λ(D, p) = 0} . p log X

In §2 we shall present the essential preliminaries, and in §3 we shall prove these results.

2. Preliminaries H. Cohen [C] explicitly constructed half-integral weight Eisenstein series whose Fourier coefficients are given by generalized Bernoulli numbers for quadratic characters. These modular forms play a crucial role in this paper. Consult [Ko] for definitions and the standard facts about modular forms. Fix an integer r ≥ 2. If N 6≡ 0, 1 (mod 4), then let H(r, N) := 0. If N = 0, then let B H(r, 0) := ζ(1 − 2r) = − 2r . If N is a positive integer and Dn2 = (−1)rN where D is 2r the fundamental discriminant of a quadratic number field, then define H(r, N) by

X r−1 (1) H(r, N) := L(1 − r, χD) µ(d)χD(d)d σ2r−1(n/d). d|n

P ν r As usual σν (n) := d|n d . In particular, if D = (−1) N is the discriminant of a quadratic number ﬁeld, then

B(r, χ ) (2) H(r, N) = L(1 − r, χ ) = − D , D r where B(n, χ) is the nth generalized Bernoulli number with character χ. If (−1)rN = n2, then

X r−1 (3) H(r, N) = ζ(1 − r) µ(d)d σ2r−1(n/d). d|n

P∞ N 2πiz Proposition 1 [Th. 3.1, C]. If r ≥ 2 and Fr(z) := n=0 H(r, N)q (q := e ), then Fr(z) ∈ M 1 (Γ0(4), χ0). r+ 2 If D > 0, then let Lp(s, χD) denote the Kubota-Leopoldt p-adic√ Dirichlet L-function with character χD, and let Rp(D) denote the p-adic regulator of Q( D) [Ch. 5,W]. Proposition 2. If p > 3 is prime and D is an odd fundamental discriminant coprime to p−1 p−1 2 p for which (−1) D > 0, then H 2 ,D is p-integral and p − 1 p−1 2h(Dp)Rp(Dp) H , (−1) 2 D ≡ p (mod p) 2 Dp CLASS NUMBERS OF REAL QUADRATIC FIELDS 5

p−1 where Dp := (−1) 2 Dp.

p−1 p−1 p−1 p−1 2B( ,χD ) 2 2 Proof. By (2) we ﬁnd that H 2 , (−1) D = L 1 − 2 , χD = − p−1 , and it is well known to be p-integral by a theorem of Carlitz [Ca]. Now Dp is a positive fundamental discriminant, and by the construction of the Kubota- Leopoldt p-adic L-function Lp(s, χD) [Th. 5.11, W]

p−1 − p−1 p − 1 2B( , χ · ω 2 ) L 1 − , χ = − 2 Dp , p 2 Dp p − 1

p−1 − 2 where ω is the usual Teichm¨uller character. It is easy to see that χDp · ω = χD, and so p − 1 2B( p−1 , χ ) p − 1 L 1 − , χ = − 2 D = H ,D . p 2 Dp p − 1 2 p−1 By the Kummer congruences [Cor. 5.13, W], we ﬁnd that Lp(1, χDp ) ≡ Lp 1 − 2 , χDp (mod p), and so the claim now follows from the p-adic class number formula [Th. 5.24, W] 2h(D )R (D ) L (1, χ ) = p p p . p Dp p Dp

Q.E.D.

Remark 1. Although one might suspect that the Eisenstein series Fp−1(z) would be better to work with, the obvious generalization of the proof of Theorem 1 does not work.

p−1 p−1 2 In view of Proposition 2, our objective is to study its coefficients H 2 , (−1) D (mod p). Unfortunately there is a minor technical difficulty which arises. The coefficients p−1 p−1 2 H 2 , 0 and H 2 , pn are not p-integral. However this does not pose too much trouble as the next two propositions indicate. Proposition 3. If p > 3 is prime, then there exists an integer α(p) coprime to p for which

(i) α(p)pF p−1 (z) ∈ [[q]], 2 Z

(ii) α(p)pF p−1 (z) ≡ Θ(pz) (mod p) 2

P∞ n2 4 9 where Θ(z) := n=−∞ q = 1 + 2q + 2q + 2q + · · · ∈ M1/2(Γ0(4), χ0). Proof. By (1), (2), and a theorem of Carlitz [Ca], it turns out that the only coeﬃcients of p−1 p−1 2 F p−1 (z) which are not necessarily p-integral are H , 0 and H , pn . Therefore 2 2 2 p−1 if n 6= p, then pH 2 , n ≡ 0 (mod p). 6 KEN ONO

Since

p − 1 B H , 0 = ζ(1 − (p − 1)) = − p−1 , 2 p − 1 p − 1 p − 1 2B( p−1 , Ψ ) H , p = L 1 − , Ψ = − 2 p 2 2 p p − 1

q p−1 where Ψp is the Kronecker Dirichlet character for Q (−1) 2 p , the proof now follows from (1) and the Claussen and Von Staudt theorem [Th. 5.10, W] once one checks that

p−1 H 2 , p p−1 ≡ 2 (mod p), H 2 , 0

and X p−3 µ(d)Ψp(d)d 2 σp−2(n/d) ≡ 1 (mod p). d|n The second congruence follows easily, and the ﬁrst congruence follows easily from the deﬁnitions of Bernoulli numbers and generalized Bernoulli numbers. Q.E.D.

3. Proof of results Theorem 1 follows easily from the following result. Theorem 2. Let p > 3 be prime, and suppose there is an odd fundamental discriminant D0 coprime to p for which

p−1 (i)(−1) 2 D0 > 0, p − 1 (ii) B , χD0 = 1. 2 p

Then there is an arithmetic progression rp (mod tp) with (rp, tp) = 1, and a constant κ(p) such that for each prime ` ≡ rp (mod tp) there is an integer 1 ≤ d` ≤ κ(p)` for which

(i) D` := d``p is a fundamental discriminant,

(ii) h(D`) 6≡ 0 (mod p),

Rp(D`) (iii) √ = 1. D` p CLASS NUMBERS OF REAL QUADRATIC FIELDS 7

Proof of Theorem 2. Fix at the outset an integer α(p) satisfying the conditions of Propo- 2 sition 3. Deﬁne Fp(z) ∈ Mp/2(Γ0(4p ), χ0) by

X p − 1 n Fp(z) := α(p)F p−1 (z) − α(p) Vp|Up|F p−1 (z) = α(p) H , n q . 2 2 2 (n,p)=1

D0 2 2 Let Q 6= p be any prime for which Q = −1, and deﬁne Gp(z) ∈ Mp/2(Γ0(4p Q ), χ0) by · X n p − 1 G (z) := F (z) ⊗ = α(p) H , n qn. p p Q Q 2 (n,p)=1

2 4 Finally deﬁne Gp(z) ∈ Mp/2(Γ0(4p Q ), χ0) by

· Gp(z) ⊗ Q − Gp(z) X p − 1 (4) G (z) := = α(p) H , n qn. p 2 2 n (n,p)=1, (Q)=−1

D0 It is easy to see that 0 6≡ Gp(z) (mod p) since the coefficient of q in Gp(z) is non- p−1 p−1 2 p−1 zero by hypothesis. Moreover the coefficients H 2 , 0 ,H 2 , n , and H 2 , pn , among others, have been annihilated. In particular, by (1), Proposition 2, and Proposition 3 every remaining non-zero coefficient is p-integral and contains information about the class number and p-adic regulator of some real quadratic field in which p ramifies. 2 4 4` If ` is prime, then define (U`|Gp)(z) and (V`|Gp)(z) ∈ Mp/2 Γ0(4p Q `), · in the usual way (see [Sh]), i.e.

∞ X X p − 1 (5a) (U |G )(z) := u (n)qn = α(p) H , `n qn, ` p p,` 2 n=1 `n ( Q )=−1, (`n,p)=1 ∞ X X p − 1 (5b) (V |G )(z) := v (n)qn = α(p) H , n q`n. ` p p,` 2 n=1 n (Q)=−1, (n,p)=1

P∞ n If g = n=0 a(n)q has integer coeﬃcients, then deﬁne ordp(g) by

ordp(g) := min{n | a(n) 6≡ 0 (mod p)}.

By a theorem of Sturm [St], if g ∈ Mk(Γ0(N), χ) has integer coeﬃcients and

k ord (g) > [Γ (1) : Γ (N)], p 12 0 0 8 KEN ONO

then g ≡ 0 (mod p). He proved this for integral k and trivial χ, but the general case obviously follows by taking an appropriate power of g. Deﬁne κ(p) by

κ(p) := p2Q3(p + 1)(Q + 1)/4.

2 4 3 Since [Γ0(1) : Γ0(4p Q `)] = 6pQ (p + 1)(Q + 1)(` + 1), if ` is suﬃciently large and 2 4 4` g ∈ Mp/2 Γ0(4p Q `), · has integer coeﬃcients and has the property that

(6) ordp(g) > κ(p)`,

then by Sturm’s theorem g ≡ 0 (mod p). ` n Suppose that ` 6= p is a prime for which Q = 1. If Q 6= −1 or (n, p) 6= 1, then by (5a) and (5b)

(7) up,`(n`) = vp,`(n`) = 0.

n For those n with Q = −1 and (n, p) = 1

p − 1 (8a) u (n`) = α(p)H , n`2 . p,` 2 p − 1 (8b) v (n`) = α(p)H , n . p,` 2

` If ` is suﬃciently large for which Q = 1, then by (1), (4), (7) and (8a) we ﬁnd for every n ≤ κ(p) that p−3 p−2 p − 1 (9) u (n`) = α(p)(1 − χ (`)` 2 + ` )H , n . p,` Dn 2

q p−1 Here Dn denotes the fundamental discriminant of the ﬁeld Q (−1) 2 n . n Let Sp denote the set of those Dn with n ≤ κ(p) for which Q = −1 and (n, p) = 1. 2 Dnm For all other Dn with n ≤ κ(p) it is clear the the coeﬃcients of q in Gp(z) are zero, and so they do not play a role in the ensuing analysis. There is a progression rp (mod tp) with (rp, tp) = 1 and p | tp for which

(i) χDn (`) = 1 for every prime ` ≡ rp (mod tp) and Dn ∈ Sp, ` (ii) = 1 for every prime ` ≡ r (mod t ), Q p p r (10) (iii) p = −1. p CLASS NUMBERS OF REAL QUADRATIC FIELDS 9

By (8a), (8b) and (9), for every prime ` ≡ rp (mod tp) we ﬁnd that

p−3 2 p−2 up,`(n`) ≡ α(p)(1 − rp + rp )vp,`(n`) (mod p) for all n ≤ κ(p). Since vp,`(n) = 0 if ` - n, by (6) if there are no n ≤ κ(p)` coprime to ` for which p − 1 u (n) = α(p)H , n` 6≡ 0 (mod p), p,` 2

p−3 2 then U`|Gp ≡ α(p)(1 − rp + rp−2)V`|Gp (mod p). By (1), the multiplicative property for H(r, N), and the deﬁnition of up,`(N) and vp,`(N), if ` ≡ rp (mod tp), then

p−3 p−5 p − 1 u (D `3) ≡ α(p)(1 − r 2 − r 2 + rp−2 + rp−3)H ,D (mod p), p,` 0 p p p p 2 0 p−3 p − 1 v (D `3) ≡ α(p)(1 − r 2 + rp−2)H ,D (mod p). p,` 0 p p 2 0

3 3 Since α(p)H(p − 1,D0) 6≡ 0 (mod p), we ﬁnd that up,`(D0` ) 6≡ vp,`(D0` ) (mod p) if and only if p−5 p−3 2 rp 6≡ rp (mod p). This is obviously satisﬁed in view of (10). Therefore

U`|Gp 6≡ V`|Gp (mod p), and so there must be an integer 1 ≤ n ≤ κ(p)` coprime to ` for which

p − 1 u (n) = α(p)H , n` 6≡ 0 (mod p). p,` 2

By (1) and Proposition 2 there is a positive fundamental discriminant D` := d``p with d` ≤ κ(p)` for which 2h(D )R (D ) √` p ` 6≡ 0 (mod p). D` The proof is now complete in view of a theorem of Coates [p. 78,W] that asserts that

Rp(D`) √ ≤ 1. D` p

Q.E.D. 10 KEN ONO

Proof of Theorem 1. In the notation from Theorem 2, if ` ≡ rp (mod tp) is prime, then there is an integer 1 ≤ d` ≤ κ(p)` for which D` := d``p is a fundamental discriminant with the desired properties. Let ì denote these primes in increasing order. If j < k < l and D`j = D`k = D`l , then `j`k`l | D`j . However this can only occur for finitely many j, k, and l since D ≤ κ(p)`2p. Hence by Dirichlet’s theorem on primes in arithmetic `j j √ progressions we find that the number of D < X obtained in this way is p π( X). Q.E.D.

Proof√ of Corollary 2. Iwasawa [I] proved that if there is only one prime lying above p in Q( D) and p - h(D), then p cannot divide the class number of any field in the Iwasawa tower. Since p | D for the relevant discriminants in Theorem 1, it follows that p is ramified and the condition above is satisfied. Q.E.D.

Acknowledgements The author thanks L. Washington for bringing several new references to his attention, and he thanks the referee for making suggestions which improved the presentation in the paper.

References [Ca] L. Carlitz, Arithmetic properties of generalized Bernoulli numbers, J. reine. angew. Math. 202 (1959), 174-182. [C] H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), 271-285. [C–L] H. Cohen and H. W. Lenstra, Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983, Springer Lect. Notes 1068 (1984), 33-62. [D–H] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields II, Proc. Roy. Soc. Lond. A 322 (1971), 405-420. [F-T] T. Fukuda and H. Taya, The Iwasawa λ-invariants of Zp extensions of real quadratic fields, Acta. Arith. 69 (1995), 277-292. [G] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), 263-284. [Ha] P. Hartung, Proof of the existence of infinitely many imaginary quadratic fields whose class number is not divisible by 3, J. Number Th. 6 (1974), 276-278. [Ho1] K. Horie, A note on basic Iwasawa λ-invariants of imaginary quadratic fields, Invent. Math. 88 (1987), 31-38. [Ho2] K. Horie, Trace formulae and imaginary quadratic fields, Math. Ann. 288 (1990), 605-612. [Ho-N] K. Horie and J. Nakagawa, Elliptic curves with no rational points, Proc. Amer. Math. Soc. 104 (1988), 20-24. [Ho–On] K. Horie and Y. Onishi, The existence of certain infinite families of imaginary quadratic fields, J. Reine und ange. Math. 390 (1988), 97-133. [Ic-Su] H. Ichimura and H. Sumida, On the Iwasawa invariants of certain real abelian fields, II, Internat. J. Math. 7 (1996), 721-744. CLASS NUMBERS OF REAL QUADRATIC FIELDS 11

[I] K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257-258. [J] N. Jochnowitz, Congruences between modular forms of half integral weights and implica- tions for class numbers and elliptic curves, (preprint). [Ko] N. Koblitz, Introduction to the theory of elliptic curves and modular forms, Springer Verlag, New York, 1984. [Koh-O] W. Kohnen and K. Ono, Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication, Invent. Math. (to appear). [K-S] J. Kraft and R. Schoof, Computing Iwasawa modules of real quadratic number fields, Com- positio Math. 97 (1995), 135-155. [M] M. R. Murty, Private communication. [O-Sk] K. Ono and C. Skinner, Fourier coefficients of half-integral weight modular forms modulo `, Ann. Math. 147, No. 2 (1998), 451-468. [Sh] G. Shimura, On modular forms of half-integral weight, Ann. Math. 97 (1973), 440-481. [St] J. Sturm, On the congruence of modular forms, Springer Lect. Notes 1240 (1984), 275-280. [W] L. Washington, Introduction to cyclotomic fields, Springer Verlag, New York, 1982.

Department of Mathematics, Penn State University, University Park, Pennsylvania 16802 E-mail address: [email protected]