11 problems

Let $x:\mathT^S(K^p)[1/p]\to\overline{\mathbf Q}_p$ be a continuous ring homomorphism, let $\mathfrak m_x=\ker(x)$, and suppose that the global Langlands program supplies a corresp…

Let $G$ be a reductive group over $\mathbb{Q}$ admitting a Shimura datum $(G,X)$, and let $(\operatorname{Sh}_{K,\mathbb{C}})_K$ be the associated inverse system of Shimura varieti…

Generalized Emerton conjecture. For every finite set $S$, there is a factorization

Emerton's conjecture. If $\rho$ is unramified outside finitely many places, then

Galois-infinitesimal character conjecture. The center $Z(\mathfrak{g})$ acts on this representation via $\zeta^C_\rho$.

Let $\rho$ be the Galois representation attached to an automorphic representation of a unitary group of semisimple rank $2$, and let $\tau_v\subset\pi_v$ be the $K$-type correspond…

Let $F$ be a number field, $p$ a prime, $n$ a positive integer, $U\subset\operatorname{GL}_n({\mathbb A}_F^\infty)$ a good subgroup, and…

Let $F$ be the number field, $p$ the fixed prime, $U$ a product subgroup as above, $S$ a finite set of finite places containing those dividing $p$, and $\Lambda$ the ordinary weight…

Let $X$ be the parameter space appearing in the global setting, let $L$ be the coefficient field, and let $\Sigma_p$ be the set of places above $p$. For a generic ordinary point…

Let $\pi$ be an irreducible infinite-dimensional smooth admissible representation of $\mathrm{GL}_n(\mathbf{Q}_p)$, and let $\pi^{\vee}=\operatorname{Hom}(\pi,\mathbf{F}_p)$. Let…

Let $K=\mathrm{SL}_n(\mathbf{Z}_p)$ and let $K(p)$ be its principal congruence subgroup. Write $\Lambda=\mathbf{Z}_p[[K(p)]]$. For a finitely generated $\Lambda$-module $M$, let it…