Plus/minus Selmer groups and anticyclotomic \(\mathbb{Z}_p\)-extensions
Fix an elliptic curve \(E/\mathbb{Q}\), a prime number \(p\) of good supersingular reduction for \(E\), and a quadratic imaginary field \(K/\mathbb{Q}\) such that all primes dividing the conductor \(N\) of the elliptic curve and the prime \(p\) are split in \(K\). Let \(K_\infty\) be the anticyclotomic \(\mathbb{Z}_p\)-extension of \(K\), and suppose that both primes of \(K\) above \(p\) are totally ramified in \(K_\infty\). Let \(\Lambda=\mathbb{Z}_p[\![G_\infty]\!]\) be the Iwasawa algebra of \(G_\infty=\mathrm{Gal}(K_\infty/K)\), \(\mathrm{Sel}_{p^\infty}(E/K_\infty)\) the \(p\)-primary Selmer group of \(E\) over \(K_\infty\) and let \(\mathrm{Sel}_{p^\infty}^\pm(E/K_\infty)\) denote Kobayashi's restricted \(\pm\)-Selmer groups of \(E\) over \(K_\infty\). It is known thanks to results by Kim and Longo-Vigni that \(\mathrm{corank}_\Lambda\left(\mathrm{Sel}_{p^\infty}^\pm(E/K_\infty)\right)=1\) for each choice of sign. The main new results are the following: \begin{itemize} \item[1.] \(\mathrm{corank}_\Lambda\left(\mathrm{Sel}_{p^\infty}(E/K_\infty)\right)=2\Leftrightarrow \mathrm{corank}_\Lambda\left(\mathrm{Sel}_{p^\infty}^\pm(E/K_\infty)\right)=1\) for each choice of sign \(\pm\); \item[2.] the \(\mu\)-invariant of the Pontryagin dual of \(\mathrm{Sel}_{p^\infty}^\pm(E/K_\infty)\) are zero, for each choice of sign \(\pm\), under some mild hypothesis on \(p\) (excluding only a finite explicit set of such primes \(p\)). \end{itemize}
scientific article; zbMATH DE number 7370425
scientific article