4 problems

Let $E_i$ be $p$-ordinary elliptic curves over $\mathbf Q$ of square-free conductors $M_i$, and let $\boldsymbol{E}=E_1\times E_2\times E_3$. Write $L(\boldsymbol{E},s)$ for the de…

Let $V$ be a geometric $p$-adic representation of $\mathrm{G}_{\mathbb{Q}}$ that is semistable at $p$ and critical in the sense of Deligne. Write…

Let $M$ be an irreducible motive, pure of weight $0$ over $\mathbb{Q}$, such that $s=0$ is a critical integer in the sense of Deligne and $L(M,0)\neq 0$. Let $p$ be a prime, let…

Let $M$ be a critical motive with nonzero complex value $L(0,M)$, let $V$ be a semistable representation of $G_{\mathbb{Q}_p}$, and let $D$ be a regular filtered $(\phi,N)$-submodu…