19 problems

Let $p>3$ and $N$ be as above, and suppose that $p$ is $\Gamma_0(N)$-regular, meaning that the conditions on the eigenvalues of $T_p$ in weights up to $k_p$ hold as specified above…

Gouvêa's distribution conjecture. The set $X_T$ is equidistributed on $[0,1]$ as $T$ tends to infinity.

Let a $\mathbb{Z}_p$-tower be genus stable in the sense defined in the source, and let slope stable mean that the integer $n_0$ defined there is finite, so that the rescaled slopes…

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…

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

Folklore integrality conjecture. The slope $v_p(a_p(f))$ must be an integer. This is stated as a folklore conjecture in the paper's discussion of integrality of slopes. No resoluti…

Ghost conjecture. Whenever $\overline{\rho}$ is regular, the Newton polygons of the ghost series predict the slopes of weight-$k$ eigenforms. The conjecture is presented as a refin…

Let $\bar r_p$ be a reducible nonsplit generic residual representation of $\operatorname{Gal}_{\mathbb Q_p}$ with parameters $a\in\{1,\dots,p-4\}$ and $b\in\{0,\dots,p-2\}$. Let…

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…

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 $L(\chi_n,s)$ be the $L$-function of a primitive character of order $p^n$, let $\ell(n)$ be its degree, let $\ell_\alpha(n)$ count the slopes equal to a fixed rational number…

Let $(C_n)_{n\geq 0}$ be a $\mathbb{Z}_p$-tower of curves from algebraic geometry. Let $g_n$ be the genus of $C_n$, let $d_\alpha(n)$ be the slope-counting quantity defined in the…

Refined slope conjecture. At weight $k'$, the old slopes at weight $k$ occur with exactly the same multiplicity: for every old slope $\alpha$ in weight $k$,

Slope multiplicity conjecture. Under these hypotheses,

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

Boundary irreducibility conjecture. If $v_\chi(\alpha)\notin{\mathbb{Z}}$, then $\overline{V}_{k,\alpha,\chi}$ is irreducible.

Gouvêa's limiting-measure conjecture. As $k$ tends to infinity, these slopes converge to the measure that is uniform on

Let $p$ be an odd prime, let $U$ be as in the paper's setup, and let $t$ denote the associated multiplicity parameter. For trivial character, write … for the lesser slopes on the…