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A converse to a theorem of Gross, Zagier, and Kolyvagin. (English) Zbl 1447.11071

The celebrated result of Gross-Zagier and Kolyvagin is that the Birch and Swinnerton-Dyer conjecture holds if the analytic rank of an elliptic curve over \(\mathbb{Q}\) is \(0\) or \(1\). The paper under review states that the converse of this result holds under some conditions. More precisely, for a semistable elliptic curve \(E\) over \(\mathbb{Q}\), the author proves that, if \(E\) has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes, one has \[ \operatorname{rank}_{\mathbb{Z}}E(\mathbb{Q})=1\quad \text{ and }\quad \sharp Ш(E)<\infty \Longrightarrow \operatorname{ord}_{s=1}L(E,s)=1. \] In the case of the algebraic rank \(0\), this implication is established essentially by the solution of the Iwasawa Main conjecture for \(\mathrm{GL}_2\) and, in the case of the algebraic rank \(1\), it is established for elliptic curves having complex multiplication by the work of Rubin, Bertrand, and Perrin-Riou.
As in the case of complex multiplications, the method of the proof is essentially a \(p\)-adic one. However, unlike the proof of K. Rubin [Invent. Math. 107, No. 2, 323–350 (1992; Zbl 0770.11033)] which uses the \(p\)-adic Gross-Zagier formula, the author of this paper uses the general Gross-Zagier formula for the derivation of a \(L\)-function by Yuan, Zhang, and Zhang [X. Yuan et al., The Gross-Zagier formula on Shimura curves. Princeton, NJ: Princeton University Press (2013; Zbl 1272.11082)]. Then, he shows the non-vanishing of the Néron-Tate height-pairing of suitable Heegner points by using the Iwasawa theory and concludes the analytic rank of the related \(L\)-functions is \(1\).
It is needless to say that the result of this paper gives the essential development for the Birch and Swinnerton-Dyer conjecture and further, the main idea plays a crucial role in the proof of M. Bhargava and the author [J. Ramanujan Math. Soc. 29, No. 2, 221–242 (2014; Zbl 1315.11045)] concerning the positivity of the proportion of elliptic curves which have both algebraic and analytic rank \(1\).

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R23 Iwasawa theory
11G05 Elliptic curves over global fields
11G07 Elliptic curves over local fields
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