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Indivisibility of Class Numbers of Imaginary Quadratic Fields and Orders of Tate-Shafarevich Groups of Elliptic Curves with Complex Multiplication

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Indivisibility of Class Numbers of Imaginary Quadratic Fields and Orders of Tate-Shafarevich Groups of Elliptic Curves with Complex Multiplication INDIVISIBILITY OF CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS AND ORDERS OF TATE-SHAFAREVICH GROUPS OF ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION Winfried Kohnen and Ken Ono 1. Introduction and Statement of Results Since Gauss, ideal class groups of imaginary quadratic fields have been the focus of many investigations, and recently there have been many investigations regarding Tate- Shafarevich groups of elliptic curves. In both cases the literature is quite extensive, but little is known. Throughout D will denote a√ fundamental discriminant of a quadratic field. Let CL(D) denote the class group of Q( D), and let h(D) denote its order, i.e. the usual class number of primitive positive binary quadratic forms with discriminant D. One of the main problems deals with the structure of CL(D), and so one naturally studies the divisibility of h(D) by primes. Here we consider imaginary quadratic fields. Gauss’ genus theory precisely determines the parity of h(D), but the divisibility of h(D) by odd primes ` is much less well understood. In view of these difficulties, Cohen and Lenstra [C-L] gave heuristics describing the “expected” behavior of CL(D), and in par- ticular they predicted that the probability that ` - h(D) is ∞ Y 1 1 1 (1 − `−k) = 1 − − + + ··· . ` `2 `5 k=1 1991 Mathematics Subject Classification. Primary 11E41, 11F33; Secondary 11E25, 11G40. Key words and phrases. class numbers and Tate-Shafarevich groups of elliptic curves. The second author is supported by NSF grant DMS-9508976 and NSA grant MSPR-97Y012, and the first author thanks the Department of Mathematics at Penn State where his research was supported by the Shapiro fund. Typeset by AMS-TEX 1 2 WINFRIED KOHNEN AND KEN ONO Although extensive numerical evidence lends credence to these heuristics, little has been proved apart from the works of Davenport and Heilbronn [D-H] when ` = 3. They proved that if > 0, then for sufficiently large X > 0 # {−X < D < 0 | h(D) 6≡ 0 (mod 3)} 1 ≥ − . # {−X < D < 0} 2 Ankeny and Chowla, Humbert, and Nagell proved that if ` is an odd prime, then there are infinitely many D < 0 for which h(D) ≡ 0 (mod `), and recently M. R. Murty [M] extended these arguments and obtained the first nontrivial estimate for the number of such D. These results employ the useful fact that one can construct binary quadratic forms with prescribed order. For the complementary question, Hartung [Ha] proved that there are infinitely many D for which h(D) 6≡ 0 (mod 3), and he noticed that his method works for every prime `. More recent works by Bruinier [B], Horie and Onishi [Ho1, Ho2, Ho-On], Jochnowitz [J], and Ono and Skinner [O-S] address various refinements and generalizations. We go a step further by obtaining an estimate. This estimate is obtained from a more general result which is proved in §2 (see Theorem 3). Theorem 1. If ` > 3 is prime and > 0, then for all sufficiently large X > 0 √ 2(` − 2) X # {−X < D < 0 | h(D) 6≡ 0 (mod `)} ≥ √ − . 3(` − 1) log X Similar questions are of interest for Tate-Shafarevich groups X(E) of elliptic curves E/Q. Let E be an elliptic curve over Q given by the Weierstrass equation E : y2 = x3 + ax2 + bx + c, where a, b and c are integers. Moreover, let N(E) denote the conductor of E, and let rk(E) denote its rank. If d is a non-zero integer, then let E(d) denote the d-quadratic twist of E E(d): y2 = x3 + adx2 + bd2x + cd3. Let X(E(d)) denote the Tate-Shafarevich group of E(d), and let rk(E(d)) denote the rank of its Mordell-Weil group. If ` is an odd prime, then it is widely believed that a positive proportion of square-free d have the property that ` - #X(E(d)). By works of Bruinier [B], Jochnowitz [J], and Ono and Skinner [O-S], for modular E it is known that for almost every prime ` there are indeed infinitely many square-free d for which #X(E(d)) 6≡ 0 (mod `). These results have important implications for rank 1 elliptic curves (see [O-S], [K2]). In §3 we prove a general result (see Theorem 6 and Corollary 7) that implies the following result. CLASS NUMBERS AND TATE-SHAFAREVICH GROUPS 3 Theorem 2. If E/Q has complex multiplication, then for every prime ` E 0 √ X # {0 < |D| < X | rk(E(D)) = 0 and ` #X(E(D))} . - E,` log X All of these results are obtained by combining a theorem of Sturm on modular forms ‘mod p’ and classical facts regarding the behavior of the operators Up and Vp on spaces of half-integral weight modular forms. 2. The Class number case. We prove Theorem 1 by bounding the largest fundamental discriminant D that is a multiple of p for which ` - h(D). This is the content of the following theorem. −4 Theorem 3. Let ` > 3 be prime, and p any prime for which p 6≡ p (mod `). Then 3 there exists an integer 1 ≤ dp ≤ 4 (p+1) for which D := −pdp or −4pdp is a fundamental discriminant and h(D) 6≡ 0 (mod `). Proof of Theorem 3. Without loss of generality we may assume that p > 3. Let θ(z) := 2 P qn (throughout q := e2πiz) be the classical theta function, and define r(n) by n∈Z ∞ X r(n)qn := θ3(z) = 1 + 6q + 12q2 + 8q3 + 6q4 + 24q5 + .... n=0 Gauss proved that 12H(4n) if n ≡ 1, 2 (mod 4), 24H(n) if n ≡ 3 (mod 8), (1) r(n) = r(n/4) if n ≡ 0 (mod 4), 0 if n ≡ 7 (mod 8). Here if N ≡ 0, 3 (mod 4), then H(N) denotes the Hurwitz-Kronecker class number, i.e. the class number of quadratic forms of discriminant −N where each class C is counted with multiplicity 1/Aut(C). If −N = Df 2 where D is a negative fundamental discriminant, then H(N) is related to usual class numbers by the formula [p. 273, C]: h(D) X D (2) H(N) = µ(d) σ (f/d). w(D) d 1 d|f 4 WINFRIED KOHNEN AND KEN ONO √ th Here w(D) is half the number of units in Q( D), and σs(n) denotes the sum of the s powers of the positive divisors of n. 3 3 Define (Upθ )(z) and (Vpθ )(z) in the usual way, i.e. 3 X n X n (3a) (Upθ )(z) := r(pn)q = 1 + r(pn)q , n≥0 n≥1 3 3 X pn X pn (3b) (Vpθ )(z) := θ (pz) = r(n)q = 1 + r(n)q . n≥0 n≥1 3 3 4p Both Upθ and Vpθ are forms of weight 3/2 on Γ0(4p) with character • [Prop. 1.3, Sh]. 1 For k ∈ 2 Z and N ∈ N (with 4|N if k 6∈ Z) let Mk(N, χ) denote the space of modular forms of weight k on Γ0(N) with Nebentypus character χ (see [Sh] for definitions). If P∞ n g = n=0 a(n)q is a formal power series with integer coefficients, then define ord`(g) by ord`(g) := min{n | a(n) 6≡ 0 (mod `)}. By a theorem of Sturm [Th. 1, St], if g ∈ Mk(N, χ) has integer coefficients and k ord (g) > [Γ (1) : Γ (N)], ` 12 0 0 then g ≡ 0 (mod `), i.e. a(n) ≡ 0 (mod `) for all n. He proved this for integral k and trivial χ, but the general case obviously follows by taking an appropriate power of g. 3 It is well known that [Γ0(1) : Γ0(4p)] = 6(p + 1), hence κ(p) := 4 (p + 1) is the relevant 4p Sturm bound for forms in M3/2 4p, • . If p > 3, then κ(p) < p. Moreover, the 3 reduction of Vpθ is 3 p Vpθ ≡ 1 + 6q + ... (mod `). 3 3 −4 Now we claim that Upθ 6≡ Vpθ when p 6≡ p (mod `). To see this, it is sufficient p 3 3 to show that the coefficients of q in Upθ and Vpθ are not congruent modulo `, i.e. r(p2) 6≡ 6 (mod `). By (1) and (2) we know that −4 r(p2) = 12H(4p2) = 6 p + 1 − , p and the claim follows. Therefore there exists some 1 ≤ np ≤ κ(p) for which ` - r(pnp). By (1) and (2) again, the result follows. Q.E.D. CLASS NUMBERS AND TATE-SHAFAREVICH GROUPS 5 Deduction of Theorem 1 from Theorem 3. Let p1 < p2 < . be the primes in increasing −4 −4 order. If pi and pj are distinct primes for which pi 6≡ (mod `) and pj 6≡ , then pi pj in the notation from the proof of the theorem pinpi 6= pjnpj . If Di and Dj are the negative fundamental discriminants associated by (1) and (2), then 3 3 Di 6= Dj since npi ≤ 4 (pi + 1) < pi and npj ≤ 4 (pj + 1) < pj. Moreover, it is obvious that Di ≥ −3pi(pi + 1). Since there are only two arithmetic progressions of primes (mod 4`) that do not satisfy the given condition, the result now follows by Dirichlet’s theorem on primes in arithmetic progressions. Q.E.D. 3. The Tate-Shafarevich case In this section we shall prove Theorem 2. As in Theorem 3, we will depend heavily on Sturm’s theorem. First we give a few preliminaries and fix notation. Let E/Q be an elliptic curve with complex multiplication by K whose conductor is N(E), and let P∞ −s P∞ n L(E, s) = n=1 a(n)n be its Hasse-Weil L-function.
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