8 problems

Zariski-density conjecture. If $C$ contains an irreducible one-dimensional point, then the set of modular points contained in $C$ is Zariski dense.

Let $K/{\mathbb Q}_p$ be unramified, let $\overline\rho$ be absolutely irreducible, let $\lambda=(0,0)$ for each embedding, and let $t$ be a tame inertial type of level $f$. Let…

Let $R^{\operatorname{ps}}_p$ be the relevant pseudodeformation ring, let $E_{\mathfrak{B}}$ be the corresponding Hecke-module coefficient algebra, and let…

Integral local-global compatibility conjecture. The natural map from the unrestricted local deformation ring to the completed Hecke algebra factors through

Let $K$ be a non-archimedean local field of residue characteristic $l$, let $\mathbb{F}$ be a finite field of characteristic $p\ne l$, and let…

Let $K$ be a non-archimedean local field of residue characteristic $l$, let $\mathbb{F}$ be a finite field of characteristic $p\ne l$, and let…

Let $Q$ be an auxiliary Taylor–Wiles set, let $\overline{\rho}_{\mathfrak m}$ be the residual representation, let $\mu$ be the specified character, and let $r_{{\mathfrak m},Q}$ be…

Let $S$ be a finite set of primes not dividing $M\ell$. Fix $f\in S_2(\Gamma_0(N\Delta'\ell^2S),\psi)$ that is supercuspidal of type $\tau$ at $\ell$ and special at primes…