30 problems

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…

Ulmer's conjecture. The action of $U_p$ on $S_k(\Gamma_0(Np),\mathbb Q_p)$ is semisimple. In particular, the polynomial

Let … where the locally convex inductive limit is over open neighborhoods $U\subset N\backslash P^{\operatorname{an}}_{\operatorname{dR}}$ containing the closure of the image of…

Bergdall–Pollack equidistribution conjecture. As $T$ tends to infinity, the set $Y_T\subseteq(-\infty,+\infty)$ is evenly distributed on $[0,1]$.

Big $P$-ordinary Hecke algebra conjecture. 1. For each tame character $\epsilon$, $\mathbf{T}^{P\text{-ord}}_{K^p,[\kappa,\tau],R,\epsilon}$ is finite free over…

Independence-of-weight conjecture. If $\kappa_1$ and $\kappa_2$ are dominant characters such that $[\kappa_1]=[\kappa_2]$, then there is a canonical isomorphism

Scalar $P$-ordinary classicality conjecture. The restriction

Density conjecture. The space

Let $r=s+p^t(p-1)d$, with $p mid d$, $s=b+c(p-1)$, and suppose that $2\leq bleq p$ and $1\leq cleq p-1$. Fix $ap$ such that $1<nu(ap)<p-1$, and let $tnu(ap)+c$. Suppose also that…

Let $r=s+p^t(p-1)d$, with $p mid d$, $s=b+c(p-1)$, and suppose that $2\leq bleq p$, $1\leq cleq p-1$, and $tgeq 2$. Let … where the coefficients $\alpha(i,l)$ are those defined in th…

Let $p$ be a prime, let $\tau\in\mathcal{H}_p$ be a real quadratic point of discriminant $D$ prime to $p$, and let $H_\tau$ be the narrow ring class field of the order defined by…

Overconvergence-rate conjecture. For all $i,j\geq 0$,

Let $p$ be an odd prime, let $\mathcal{W}$ be the weight space for $\operatorname{GL}_2$, and let $\mathcal{X}$ be an eigencurve with weight map…

Let $\bar\rho=\begin{pmatrix}\omega_1^{a+b+1}&*\\0&\omega_1^b\end{pmatrix}:\operatorname{I}_{\mathbb{Q}_p}\to\operatorname{GL}_2(\mathbb{F})$ be a character with…

Let $p$ be a prime number, let $N$ be a positive integer coprime to $p$, and normalize the $p$-adic valuation on $\overline{\mathbb{Q}}_p$ by $v_p(p)=1$. For $r\in\mathbb{Q}$ and a…

For $a=0,\dots,3$, define the ghost series … where … with the exponents $m_n^{(a)}(l)$ prescribed by the source. Set … and let…

Let $p$ be an odd prime, let $\mathscr{W}$ be the rigid analytic space parametrizing continuous characters of $\mathbb{Z}_p^\times$, and write $T(w)=w(\exp(p))-1$ for the $T$-coord…

Let $p$ be a prime and let $f$ be a modular function. Say that $f$ is $p$-adically annihilated if the sequence $f\mid U_p^n$ uniformly converges to $0$ in the $p$-adic limit as…

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 $b_n$ 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 $\mathbb{E}_{\mathcal{H}}$ be the eigenvariety constructed from a Hecke module, let $x$ be a point belonging to exactly one irreducible component of $\mathbb{E}_{\mathcal{H}}$,…

For each $a$, let $\mathcal{E}_{a,q}^{q}$, for $q=0,\ldots,a$, be the eigenvarieties constructed over the corresponding weight spaces, and let $\mathcal{W}_a$ be the weight space.…

Let $p$ be a prime, let $\mathcal{W}$ be the $p$-adic weight space, and let $\mathcal{W}_\eta$ be one of its components. For a weight $\kappa\in\mathcal{W}_\eta$, write…

Let $p$ and $\ell$ be primes, let … where … Let $\Phi\in\operatorname{Symb}^+_{\Gamma_0(p\ell)}({\mathcal D}_0)$ be an overconvergent eigensymbol with the same Hecke eigenvalues as…