3 problems

Let $E$ be an elliptic curve over ${\mathbb{Q}}$, let $K$ be an imaginary quadratic field, let $p$ be a prime, and let $K_{\infty}/K$ be the anticyclotomic ${\mathbb{Z}_p}$-extensi…

Let $k$ be a totally real number field, let $p$ be a prime, let $E$ be a finite extension of $\mathbb Q_p$, and let $\chi:G_k\to E$ be an even Artin character realizable over $E$.…

Let $k$ be a totally real number field, let $p$ be a prime, let $E$ be a finite extension of $mathbb Q_p$, and let $chi$ be an even Artin character of $G_k$ realizable over $E$. Le…