Optimal curves differing by a 3-isogeny by Dongho Byeon and Donggeon Yhee (Seoul) 1. Introduction. For a positive integer N , let X 1 (N ) = H * /Γ 1 (N ) and X 0 (N ) = H * /Γ 0 (N ) denote the usual modular curves. Let C denote an isogeny class of elliptic curves defined over Q of conductor N . For i = 0, 1, there is a unique curve E i ∈ C and a parametrization φ i : X i (N ) → E i such that for any E ∈ C and parametrization φ i : X i (N ) → E, there is an isogenyIt seems that for most isogeny classes C, E 0 and E 1 are the same. However, there are examples where they differ. For example, E 0 = X 0 (11) and E 1 = X 1 (11) differ by a 5-isogeny. Stein and Watkins [SW] have made a precise conjecture about when E 0 and E 1 differ by a 2-isogeny or a 3-isogeny, based on numerical observations. For the 3-isogeny case, the conjecture is the following.Conjecture (Stein and Watkins). For i = 0, 1, let E i be the X i (N )optimal curve of an isogeny class C of elliptic curves defined over Q of conductor N . Then the following statements are equivalent:(A) There is an elliptic curve E ∈ C given by E : y 2 + axy + y = x 3 with discriminant a 3 − 27 = (a − 3)(a 2 + 3a + 9), where a is an integer such that no prime factors of a − 3 are congruent to 1 modulo 6 and a 2 + 3a + 9 is a power of a prime number. (B) E 0 and E 1 differ by a 3-isogeny.Remark. This conjecture has to be modified because (B) does not imply (A) in general. For example, let C be the isogeny class consisting of the two elliptic curves 396C1 and 396C2 in Cremona's table. Then E 0 = 396C1 and E 1 = 396C2 differ by a 3-isogeny, but (A) is not true in this case.In this paper, we prove the following theorem.

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Let E be an elliptic curve defined over Q of conductor N, let M be the Manin constant of E, and C be the product of local Tamagawa numbers of E at prime divisors of N. Let K be an imaginary quadratic field in which each prime divisor of N splits, P K be the Heegner point in E(K), and X(E/K) be the Tate-Shafarevich group of E over K. Also, let 2u K be the number of roots of unity contained in K. In [11], Gross and Zagier conjectured that if P K has infinite order in E(K), then the integer u K ·C · M · (#X(E/K)) 1/2 is divisible by #E(Q) tors . In this paper, we show that this conjecture is true.

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