11 problems

Liu–Wan–Xiao's arithmetic-progression conjecture. These slopes should be finite unions of arithmetic progressions whose initial terms are the slopes in explicit classical weight-tw…

Let $p$ be prime, let $r\in(\frac{1}{p+1},\frac{p}{p+1})$, and let $h$ be an $r$-overconvergent modular function on the region $X_0(1)_{\ge p^{-r}}$. Its spectral expansion is the…

Let $p\in\{2,3,5\}$, let $r$ range over some open interval containing $\frac{1}{2}$, and let $\mathcal{S}_k(r)$ be the space on which the overconvergent $U$ operator acts. A factor…

Let $f$ be the modular form in the paper, let $\varrho_f$ be its associated Galois representation, and let $\operatorname{ad}^0\varrho_f$ denote its trace-zero adjoint representati…

Let $f$ be an overconvergent eigenform of weight $k$, and let $\rho_f$ be its associated Galois representation. Say that $\rho_f$ has Hodge–Tate weights $0,k-1$ when its Hodge–Tate…

Nearly-overconvergent universal eigenform conjecture. There is an affinoid neighbourhood

Let $[F:\mathbb{Q}]=g$, let $U$ be a sufficiently small level, and let $(\kappa,r)$ be a locally algebraic weight near the boundary. For a compact operator…

Let $\kappa$ be a weight in a component $\mathcal{W}_\chi$ of weight space, let $w(\kappa)=\kappa(\gamma)-1$, and let $s_1(\kappa),s_2(\kappa),\ldots$ be the slopes of $U_p$ acting…

Let $b_n$ be the coefficient of the $n$-th basis vector in the Kolberg-basis expansion of $h=\iota^*(\Theta_1^{-1}\mathcal{F}^{[\mathfrak{p}_1]})$. Coefficient-bound conjecture. On…

Let $b_n$ be the coefficient of the $n$-th basis vector in the Kolberg-basis expansion of the form $h=\iota^*(\Theta_1^{-1}\mathcal{F}^{[\mathfrak{p}_1]})$. Overconvergence conject…

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…