Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove that if $E$ has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes and if the rank of $E(\mathbb{Q})$ is one and the Tate-Shafarevich group of $E$ has finite order, then $\mathrm{ord}_{s=1}L(E,s)=1$. We also prove the corresponding result for the abelian variety associated with a weight two newform $f$ of trivial character. These, and other related results, are consequences of our main theorem, which establishes criteria for $f$ and $H^1_f(\mathbb{Q},V)$, where $V$ is the $p$-adic Galois representation associated with $f$, that ensure that $\mathrm{ord}_{s=1}L(f,s)=1$. The main theorem is proved using the Iwasawa theory of $V$ over an imaginary quadratic field to show that the $p$-adic logarithm of a suitable Heegner point is non-zero.

中文翻译：

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Gross、Zagier 和 Kolyvagin 定理的逆定理

令 $E$ 是 $\mathbb{Q}$ 上的半稳定椭圆曲线。我们证明，如果 $E$ 在至少一个奇素数上有非分裂乘法归约或在至少两个奇素数有分裂乘法归约，并且如果 $E(\mathbb{Q})$ 的秩为 1 并且 Tate- $E$的Shafarevich群有有限阶，则$\mathrm{ord}_{s=1}L(E,s)=1$。我们还证明了与权重两个新形式 $f$ 相关的阿贝尔变体的相应结果。这些以及其他相关结果是我们主要定理的结果，该定理建立了 $f$ 和 $H^1_f(\mathbb{Q},V)$ 的标准，其中 $V$ 是 $p$-adic Galois 表示与 $f$ 相关联，确保 $\mathrm{ord}_{s=1}L(f,s)=1$。