\(p^{\infty}\)-Selmer groups and rational points on CM elliptic curves
Let \(E/\mathbb{Q}\) be an elliptic curve, let \(L(E,s)\) be the \(L\)-function of \(E\), and let \(\mathrm{ord}_1( L(E,s))\) denote the order of vanishing of \(L(E,s)\) at \(s=1\). The celebrated theorem of Gross-Zagier, Kolyvagin, and Rubin states that if \(r:=\mathrm{ord}_1( L(E,s))\in \{0,1\}\), then the Mordell-Weil rank of \(E(\mathbb Q)\) is \(r\). Given a prime number \(p\), it is known as a \(p\)-converse of the celebrated theorem that if \(\mathrm{corank}_{\mathbb Z_p} \mathrm{Sel}_{p^\infty}(E/\mathbb Q)=1\), then \(\mathrm{ord}_1( L(E,s))=1\). Several \(p\)-converse theorems are available in the literature. The authors of the paper under review prove the following result, and a similar result for CM abelian varieties. Suppose that \(E\) has complex multiplication by an order of an imaginary quadratic field \(K\) of discriminant \(-D<0\), and that the Hecke character associated to \(E\) has conductor exactly divisible by the ideal generated by \(\sqrt{-D}\). Let \(p\) be a prime of good ordinary reduction for \(E\). Then, \(\mathrm{corank}_{\mathbb Z_p} \mathrm{Sel}_{p^\infty}(E/\mathbb Q)=1\) implies that \(\mathrm{ord}_1( L(E,s))=1\). In particular, if \(\mathrm{corank}_{\mathbb Z_p} \mathrm{Sel}_{p^\infty}(E/\mathbb Q)=1\), then the Mordell-Weil rank of \(E\) is \(1\), and the Tate-Shafarevich group of \(E\) is finite. This implies that if \(E\) has no rational \(p\)-torsion points, and the \(p\)-Selmer group of \(E\) has dimension \(1\), then \(\mathrm{ord}_1( L(E,s))=1\) and the \(p^\infty\)-torsion of the Tate-Shafarevich group of \(E\) is trivial. For prime \(p>3\), this result was first proved in [\textit{A. A. Burungale} and \textit{Y. Tian}, Invent. Math. 220, No. 1, 211--253 (2020; Zbl 1452.11068)]. However, the approach of the work under review is different, and has advantage of being generalized to CM elliptic curves over totally real fields, which will be explored in their follow-up paper. The \(p\)-converse theorem is also independently proved by Ressler and Yu, building on the approach of [loc. cit.], and it does not require the conductor hypothesis.
scientific article; zbMATH DE number 7598385
scientific article