20 problems

Fix a positive integer $N$ and a prime $p mid N$. For each integer $k$, let $M{k,N}=Mk(Gamma1(N)capGamma0(p))$, let $Up$ be the Hecke operator on this space, and let $P{k,N}(X)$ be…

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

Let $k$ be the weight of a Drinfeld modular form of level $\Gamma_0(T)$, and let the slope of a $\mathrm{U}$-eigenvalue mean its valuation. A form is new if it belongs to the suita…

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…

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 $U$ be a sufficiently small level and let $\kappa=[k_1,k_2]\psi$ be a locally algebraic weight near the boundary. Let $V\in\{U_p,U_{\mathfrak p}\}$. For each…

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…

Fix a prime $p$ and an integer $N\geq 1$ not divisible by $p$. Let $G(w,t)\in\mathbf Z_p[[w,t]]$ be the ghost series, viewed as a series in $t$ with coefficients that are functions…

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…

Let $p$ be prime and let $k$ be the weight of a classical eigenform. Write $a_p$ for its $U_p$-eigenvalue and $v_p(a_p)$ for its slope. Buzzard–Gee slope conjecture. The condition…

Let $p$ be prime, and consider slopes of classical eigenforms of weight $k$. Strengthened forbidden-slope conjecture. Absolutely no slopes appear strictly between … and … This is e…

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

Gouvêa's equidistribution conjecture. As $k\to\infty$, the sets $\mathbf x_k$ become equidistributed on

The ghost conjecture. If $p$ is an odd $\Gamma_0(N)$-regular prime, or if $p=2$ and $N=1$, then

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