3 problems

Let $p\geq 7$ be prime, let $N$ be coprime to $p$, and let $\mathrm{K}_p=\operatorname{GL}_2(\mathbb{Z}_p)$. Let $\mathcal{O}_{\mathbb{C}_p}$ be the ring of integers of…

Let $\bar\rho=\begin{pmatrix}\omega_1^{a+b+1}&*\\0&\omega_1^b\end{pmatrix}:\operatorname{I}_{\mathbb{Q}_p}\to\operatorname{GL}_2(\mathbb{F})$ be a character with…

The $p=2$ ghost conjecture. If $2$ is $\Gamma_0(N)$-regular, then