9 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 $p\geq 7$ 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…

Slope multiplicity conjecture. Under these hypotheses,

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