17 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 $\bar r$ be an absolutely irreducible residual Galois representation and write $\bar r_p=\bar r|_{\rho_{\text{Gal}_{\boldsymbol{\bf Q}_p}}}$. Assume that $\bar r_p$ is reducibl…

For a prime $p$, let $U_p$ act on a space of modular forms, and let the slope of an eigenform be the $p$-adic valuation of its $U_p$ eigenvalue. Let the ghost series be the formal…

Coleman–Mazur halo conjecture. The part of the eigencurve over the boundary of weight space should be a disjoint union of countably infinitely many connected components, the weight…

Let $p$ be a rational prime, let $N$ be a positive integer prime to $p$, let $\mathcal{W}$ be the $p$-adic weight space for $\mathit{GL}_{2,\mathbb{Q}}$, and let $\mathcal{C}_N$ be…

Coleman's conjecture. There should exist an $\mathbb{F}_p((X))$-Banach space $\overline{M}_{\overline{\kappa}_\eta}^{\dagger}$ of overconvergent modular forms of weight…

Let $\mathscr{C}$ be the eigencurve, let $\mathscr{V}$ be the sheaf of modular-form data, and let $\mathscr{H}_{\mathrm{Iw}}^{1}(\mathbf{Q},\mathscr{V})$ be the corresponding Iwasa…

Let $N$ be the tame level and let $x\in\mathscr{C}(N)(\overline{\mathbf{Q}_{p}})$ be a crystalline point of conductor $N$, corresponding to a normalized cuspidal elliptic newform…

Let $\mathcal{W}$ be weight space with coordinate $T$, let $r<1$, and let $\mathcal{W}^{\geq r}$ be the sub-annulus where $r\leq |T|<1$. Put … Here $\operatorname{wt}$ is the weigh…

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…

Paulin's level lowering conjecture. Let $Z\subset\mathcal{C}^{0,\mathrm{full}}(\ell)$ be an irreducible component that is generically special at $\ell$. If there exists $x\in Z$ su…

Kisin's conjecture. The canonical inclusion is an equality:

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…

Assume the representability conjecture for triangulordinary filtrations in rank $2$ and rank $1$, so that the morphism $p_\sigma\colon\mathscr{E}^0(\sigma)\to\mathscr{E}^0$ exists,…

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…