Let [Formula: see text] be an elliptic curve defined over [Formula: see text] of conductor [Formula: see text], [Formula: see text] the Manin constant of [Formula: see text], and [Formula: see text] the product of Tamagawa numbers of [Formula: see text] at prime divisors of [Formula: see text]. Let [Formula: see text] be an imaginary quadratic field where all prime divisors of [Formula: see text] split in [Formula: see text], [Formula: see text] the Heegner point in [Formula: see text], and [Formula: see text] the Shafarevich–Tate group of [Formula: see text] over [Formula: see text]. Let [Formula: see text] be the number of roots of unity contained in [Formula: see text]. Gross and Zagier conjectured that if [Formula: see text] has infinite order in [Formula: see text], then the integer [Formula: see text] is divisible by [Formula: see text]. In this paper, we show that this conjecture is true if [Formula: see text], [Formula: see text] or [Formula: see text].

中文翻译：

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Gross 和 Zagier 的猜想：案例 E(ℚ)tor≅ℤ/2ℤ ⊕ ℤ/2ℤ, ℤ/2ℤ ⊕ ℤ/4ℤ or ℤ/2ℤ ⊕ ℤ/6ℤ

令 [Formula: see text] 为在导体 [Formula: see text] 的 [Formula: see text] 上定义的椭圆曲线，[Formula: see text] [Formula: see text] 的 Manin 常数 [Formula: see text] 和 [Formula: see text]文本] [公式：参见文本] 的多摩川数在 [公式：参见文本] 的素数除数处的乘积。令 [Formula: see text] 是一个虚构的二次域，其中 [Formula: see text] 的所有素因数在 [Formula: see text] 中分裂，[Formula: see text] [Formula: see text] 中的 Heegner 点，并且[公式：见文本] [公式：见文本] 上的 Shafarevich-Tate 群。令 [公式：见正文] 为 [公式：见正文] 中包含的单位根数。Gross 和 Zagier 推测，如果 [Formula: see text] 在 [Formula: see text] 中有无限阶，那么整数 [Formula: see text] 可以被 [Formula: see text] 整除。