6 problems

Ghost conjecture. All $U_p$-slopes are given by the ghost series.

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…

Fix a prime $p$ and an integer $N\geq 1$ not divisible by $p$. Let $G(w,t)\in\mathbf Z_p[[w,t]]$ be the ghost series, viewed as a series in $t$ with coefficients that are functions…

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

The ghost conjecture. If $p$ is an odd $\Gamma_0(N)$-regular prime, or if $p=2$ and $N=1$, then