TATE-SHAFAREVICH GROUPS OF THE CONGRUENT NUMBER ELLIPTIC CURVES Ken Ono Abstract. Using elliptic modular functions, Kronecker proved a number of recurrence relations for suitable class numbers of positive binary quadratic forms. For instance if F (N) denotes the number of uneven classes of positive binary quadratic forms with determinant N, then X F (m) + ( ) k F (m k ) σ(m), k where σ(m) : P d+ m χ (d + ) (see [p. 08, (III),D]). In this note we derive similar relations, assuming the Birch and Swinnerton-Dyer Conjecture, for the orders of Tate-Shafarevich groups of the congruent number elliptic curves E N : y x 3 N x. Assuming the Birch and Swinnerton-Dyer Conjecture, if E N has rank 0, then q explicit finite linear combination of X(E N ) where N < N. q X(E N ) is a simple The relations. If N is an odd square-free integer, then let E (N) and E (N) denote the elliptic curves over Q E i (N) : y x 3 4 i N x, and let r i (N) denote the rank of E i (N). Similarly let X i (N) denote the Tate-Shafarevich group X(E i (N)). If q : e πiz, η(z) : q /4 n ( qn ), and Θ(z) : n Z, then let f qn (z) S 3 (8, χ 0) and f (z) S 3 (8, χ ) be eigenforms given by f (z) : η(8z)η(6z)θ(z) f (z) : η(8z)η(6z)θ(4z) : a (n)q n. n a (n)q n. Throughout χ t : ( t ) shall denote Kronecker s character for Q( t). Both forms lift, via the Shimura correspondence, to the cusp form associated to the curve y x 3 x n n a(n)q n : η (4z)η (8z) q ( q 4n ) ( q 8n ). n 99 Mathematics Subject Classification. Primary G40. The author is supported by NSF grants DMS-9508976 and DMS-9304580. Typeset by AMS-TEX
KEN ONO Consequently we obtain the following multiplicative formulae for square-free t : a (tm ) a (t) ( ) t χ (d)µ(d) a(m/d), d d m a (tm ) a (t) ( ) t () χ (d)µ(d) a(m/d). d d m Given a i (t), the integers a i (tm ) follow immediately from () since () a(n) x Z, y 0 4x + (y + ) N ( ) x+y (y + ). This can be deduced by explicitly computing the Hecke Grössencharacter of y x 3 x, or by computing the relevant Jacobstahl sums [Ch. 6, B-E-W], or by classical q series identities [Th. 3,M-O]. Tunnell [T] proved that if N is an odd square-free integer, then (3) L(E i (N), ) i Ω a i (N) 4, i N where Ω : dx.6. Therefore assuming the Birch and Swinnerton-Dyer x x3 x Conjecture, E i (N) has rank 0 if and only if a i (N) 0. In addition if a i (N) 0, then (4) Xi (N) a i(n) τ(n) where τ(n) denotes the number of divisors of N. If the functions T i (t, m) are defined by (5) T (t, m) : { sign(a (t))τ(t) d m χ (d)µ(d) ( t d) a(m/d) if a (t) 0 0 if a (t) 0, (6) T (t, m) : { sign(a (t))τ(t) d m χ (d)µ(d) ( t d) a(m/d) if a (t) 0, 0 if a (t) 0, then by (), (4), (5), and (6), if t is an odd square-free integer, then (7) a i (tm ) T i (t, m) X i (t). For convenience we define the sets S (N) and F(N), the indices for the first explicit Kronecker relation: { S (N) : (m, k) Z N k ( ) } N k + k 3 odd, m Z + square-free, r m 0 F(N) : { (x, y) x Z, y 0, and 4x + (y + ) N }.
Theorem. If N is a positive integer, then a (N ) + a (N (k + ) ) k x Z,y 0 8x +(y+) N CONGRUENT NUMBERS 3 ( ) y (y + ) + x Z,y 0, 6x +4(y+) N Proof Theorem. If F (z) : n A (n)q n : η(4z)η(8z)θ(z) that F (z) C (z) + η (8z)η (6z) ( ) x+y (y + ). k0 q (k+) where C (z) n b(n)qn is the newform associated to the elliptic curve y x 3 + x., then it turns out In particular all three forms are in S (64) and the identity follows from the standard dimension counting argument. In this case checking the identity for the first 9 terms suffice. Therefore we find that A (N) b(n) + a(n/). Using [Ch. 6, B-E-W], or [Th. 3, M-O], it turns out that b(n) ( ) y (y + ). (x,y) F(N) Assuming the Birch and Swinnerton-Dyer Conjecture, E (t) for t odd and square-free, has rank 0 if and only if a (t) 0. The proof now follows immediately from () and (7). Using the previous discussion we obtain the following immediate corollary. Corollary. Assuming the Birch and Swinnerton-Dyer Conjecture, if N is a positive square-free integer for which E (N ) has rank 0, then T (N, ) ( ) ( ) N k N k X (N ) + T m, m X m (x,y) F(N) (m,k) S (N) ( ) y (y + ) + (x,y) F(N/) ( ) x+y (y + ). Corollary. Assuming the Birch and Swinnerton-Dyer Conjecture, if N is a positive square-free integer for which E (N ) has rank 0 and ord p (N) is odd for some prime p 3 (mod 4), then X (N ) τ(n ) (m,k) S (N) ( ) ( N k N k T m, m X m We now define the index sets S (N), H(N), and I(N) for the second Kronecker relation: S (N) : {(m, k) Z + N ( ) } 4k N k m Z + square-free, r m 0, ). H(N) : { (x, y) x Z, y 0, and 6x + (y + ) N }, I(N) : { (x, y) x, y 0, and 4(x + ) + (y + ) N }.
4 KEN ONO Theorem. If N is a positive integer, then a (N) + a (N 4k ) k x Z,y 0 6x +(y+) N ( ) x+y χ (y + )(y + ) 4 x,y 0 4(x+) +(y+) N ( ) x+ χ (y + )(x + ). Proof Theorem. If F (z) n A (n)q n : f (z)θ(4z), then it is easy to deduce that F (z) : A (n)q n A (n)q n n,3,7,,3,5 (mod 6) is the newform associated to the elliptic curve n 5,9 (mod 6) y x 3 x. p ( x 3 ) x The proof now follows from the explicit Jacobstahl sums which can be found in p x0 [6.., 6.., B-E-W]. As immediate corollaries we obtain: Corollary 3. Assuming the Birch and Swinnerton-Dyer Conjecture, if N is an odd squarefree integer for which E (N) has rank 0, then T (N, ) ( ) ( ) N 4k N 4k X (N) + T m, m X m (x,y) H(N) (m,k) S (N) ( ) x+y χ (y + )(y + ) 4 (x,y) I(N) ( ) x+ χ (y + )(x + ). Corollary 4. Assuming the Birch and Swinnerton-Dyer Conjecture, if N is a positive odd squarefree integer for which E (N) has rank 0 and ord p (N) for some prime p 3 (mod 4), then X (N)) 4 ( ) ( ) N 4k N 4k τ(n) T m, m X m. (m,k) S (N) We conclude with an application to the following question due to Kolyvagin. Kolyvagin s question. If E/Q is an elliptic curve and p is prime, are there infinitely many quadratic twists E D for which X(E D ) 0 (mod p)? Corollary 5. If p is prime, then there are infinitely many square-free integers N and M for which r (N) 0 and X (N) 0 (mod p), r (M) 0 and X (M) 0 (mod p). Proof. If p, then this is a standard application of descents. By Rubin s theorem, if p is odd and p divides X i (N) when a i (N) 0, then p a i (N). The result now follows easily from the unconditional recurrences for a i (N) in Theorems and.
CONGRUENT NUMBERS 5 Remarks Using the fact that X () X () (i.e. via Rubins theorem [R] and (4) ), Corollaries and 3 conditionally capture the orders of all the Tate-Shafarevich groups of rank 0 congruent number curves. The only feature that may appear to be a mystery are the signs of a i (t) which are part of T i (t, m). However one can easily deduce these signs from the recurrence relations since Xi (N) is always a positive integer. Therefore these relations are closed in the sense that no additional information is required apart from the fact that a i (). The existence of these Kronecker-type formulae is not necessary for obtaining Corollary 5. In a forthcoming paper, the author and C. Skinner [O-S] show how to obtain such results, in a more general setting, in the absence of Kronecker-type formulae. N. Jochnowitz [J] also obtains such results via a completely different argument. The Kronecker relations presented here have the pleasant property that they are explicit and only depend on the traces of Frobenius of the elliptic curves y x 3 x, y x 3 + x, y x 3 x. In particular the E (N) and E (N) are simply twists of these special curves. It is of some interest to classify those rare elliptic curves E for which one can obtain Kronecker formulae for orders of Tate-Shafarevich groups of families of twists, especially those formulae which only depend on the Frobenius of special twists of E. References [B-E-W] B.C. Berndt, R. J. Evans, and K. S. Williams, Gauss and Jacobi sums, Wiley Publ., (to appear). [C] J.E. Cremona, Algorithms for elliptic curves, Cambridge Univ. Press, 99. [D] L. E. Dickson, History of the theory of numbers, Vol. 3, G. E. Strechert & Co., 934. [M-O] Y. Martin and K. Ono, Eta-quotients and elliptic curves, Proc. Amer. Math. Soc., (to appear). [J] N. Jochnowitz, Congruences between modular forms of half integral weights and implications for class numbers and elliptic curves, (preprint). [O-S] K. Ono and C. Skinner, Fourier coefficients of half-integral weight modular forms mod l, (preprint). [R] K. Rubin, Tate-Shafarevich groups and L functions of elliptic curves with complex multiplication, Invent. Math. 89 (987), 57-560. [S] J. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 986. [T] J.B. Tunnell, A classical Diophantine problem and modular forms of weight 3/, Invent. Math. 7 (983), 33-334. School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540 E-mail address: ono@math.ias.edu Department of Mathematics, Penn State University, University Park, Pennsylvania 680 E-mail address: ono@math.psu.edu