13 problems

Let $X$ be a complex variety, let $\mathcal{M}$ be a holonomic $\mathcal{D}$-module on $X$, and let $S(\mathcal{M})$ be the union, over germs of hypersurfaces $Z$ of $X$, of the se…

Let $A$ be an integer matrix, let $\mathcal{M}_A(\beta)$ be the projectivized $A$-hypergeometric system with parameter $\beta$, and let $Y$ be a coordinate variety. For a filtratio…

Let $k\geq12$ be even, let $S_k=S_k(\Gamma_0(1),\mathbf{Q})$ be the space of level-$1$ cusp forms of weight $k$, and let $m=\dim S_k$. For a polynomial with rational coefficients,…

Slope-bound conjecture. For all $i>d$, the $q$-slopes of these eigenvalues are contained in

Given a positive even integer $k$, let $\nu_\mathcal{L}^{\varepsilon}(k,p,N)$ be the finite sequence of slopes of the $p$-adic $\mathcal{L}$-invariants attached to forms in…

Slope stability conjecture. The following hold: (1) the $q$-slopes are equidistributed in $[0,1]$ as $n\to\infty$; (2) for each fixed rational $\alpha\in[0,1]$, there is a polynomi…

Let $V\in\{U_p,U_{\mathfrak p}\}$, let $\kappa_{\min,\chi\tau^t}$ be the smallest classical weight associated with the character $\chi\tau^t$, and let…

Let $p$ be a prime and $N=1$. For a $p$-new eigenform $f\in S_k(\Gamma)$ of slope $h$, let $\mathscr L_f$ be its $\mathscr L$-invariant and define … Here $\widetilde m_p(h)$ is a n…

Modified ghost conjecture. For each $\kappa\in\mathcal W$,

Let $X$ be a complex manifold and let $f$ be a holonomic $f$-module on $X$. For each hypersurface $Z$ of $X$, consider the analytic slopes of $f$ along $Z$, namely the real numbers…

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…

Slope-distribution conjecture. The probability that an element of $R_k$ chosen with uniform distribution lies in the interval

Gouvêa–Mazur conjecture. Let $k$, $l$ be integers, $n$ be a positive integer, and $\alpha<n$. Then