5 problems

Let $E$ be the quadratic field under consideration, let $K_f$ be the chosen level, let $\mathcal{L}_{\bm{n}}(\mathbb{F})$ be the coefficient system, and let $\mathfrak{m}$ be a non…

A potentially crystalline representation is a representation of a local Galois group that becomes crystalline after a finite extension. A potentially diagonalizable representation…

Let $\overline{\rho}:G_K\rightarrow\mathrm{GSp}_4(\mathbf{F})$ be a continuous representation, and let $R_{\overline{\rho}}^{\lambda,\tau}$ be the potentially crystalline deformati…

Let $F$) be an imaginary quadratic field, let $\mathfrak{m}$ be a non-Eisenstein maximal ideal of residue characteristic $p>2$ associated to a residual Galois representation…

Existence Conjecture. For any $Q$ there is a surjection