54 problems

Let ${\mathcal W}$ be the weight space, viewed as a union of open unit discs, and let ${\mathcal E}$ be the Coleman–Mazur eigencurve over ${\mathcal W}$. Coleman's spectral halo co…

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…

Gouvêa's conjecture. All small slopes of overconvergent modular forms come from forms in

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 $p$ 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 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 $[0,1]$.

Bounded $q$-expansion conjecture.

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

Vector-valued $P$-ordinary classicality conjecture. The restriction

Scalar $P$-ordinary classicality conjecture. The restriction

Density conjecture. The space

Let $\mathcal{U}$ be a neighbourhood in weight space over which a Hida family $\underline{\Pi}$ is defined, and suppose the associated Eichler--Shimura isomorphism exists after bas…

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

Let $K$ be a real quadratic field, let $p$ be a finite prime that is ramified or inert in $K$, and let $\tau$ be a real multiplication point in the Drinfeld $p$-adic upper half pla…

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

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 $f_\beta=E_{r+2}^{\operatorname{crit}}(\psi,\tau)$ be the critical-slope $p$-stabilisation of a $p$-decent Eisenstein series, and suppose the equivalent conditions in the prece…

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 $U_5$ act in the case studied in the paper, and let an appropriate variant of the ghost series be the power series associated with the observed ghost pattern of the upper minor…