6 problems

Let $p$ be an odd prime, let $\mathcal{W}$ be the weight space for $\operatorname{GL}_2$, and let $\mathcal{X}$ be an eigencurve with weight map…

Let $p$ be an odd prime, let $N$ be coprime to $p$, and let $\psi_0$ be a character of $(\mathbb{Z}/p\mathbb{Z})^\times$ with $\psi_0(-1)=-1$. For $m\geq 2$, let $\psi$ be a charac…

Geometric level-raising conjecture. If $\mathcal{Z}\subset\mathcal{E}$ is a generically principal-series at $l$ component and there exists $x\in\mathcal{Z}$ such that…

Eigencurve intersection conjecture. If $x\in\mathcal{E}$ is such that $\pi(\rho_{x,l},N_x)$ is one dimensional, then there exist irreducible components…

Fix distinct primes $p$ and $l$, and an integer $N$ coprime to $pl$. Let $\mathcal{E}$ be the cuspidal eigencurve of tame level $\Gamma_0(Nl)$, parametrising overconvergent cuspida…

Let $\mathscr{E}^0$ be the semistable, distinct-eigenvalue locus of the eigencurve, let $\mathscr{V}^0$ be its rank-$2$ family of Galois representations, and set … For each…