11 problems

Let $\overline U$ be a compact open subgroup of $\overline G(\mathbb A)$ such that, for every place $w\in \operatorname{Spl}^S$, its local component is the maximal compact subgroup…

Assume the indefinite Shimura-variety setup of the source. Let $\mathcal{S}_0^\circ$ and $\mathcal{S}_0^\bullet$ be the relevant Shimura varieties, let $\Omega_\lambda$ be the loca…

Clozel–Harris–Taylor's generalized Ihara lemma. Then $V$ is Whittaker-generic.

Assume that $\epsilon(\Pi_0\times\Pi_1)=-1$. Let $a=a(\emptyset,\{\mathfrak l\})$, and let $\mho^a_0/\mathsf m^{\{\mathfrak l\}}_0$ denote the map introduced in the surrounding con…

Let $K$ and $v$ be as in the paper's level-lowering setup, and let $\mathfrak{m}\subseteq\mathbb{T}^S(K)$ be a non-Eisenstein maximal ideal. Let … be the level-lowering map. Ihara'…

Weak Ihara conjecture. The $k[G(F_{v_0}^+)]$-module $\langle G(F_{v_0}^+)\cdot f\rangle$ is generic. This is explicitly presented as a weaker conjecture than the preceding weak Iha…

Ihara's lemma. Then $\pi$ is generic. This is the precise weak Ihara lemma attributed in the source to Clozel, Harris, and Taylor. The introduction explains that the corresponding…

Let $F$ be a totally real field, let $D$ be a quaternion algebra over $F$ ramified at exactly one infinite place, and let $\mathfrak{p}$ be a finite place at which $D$ is unramifie…

Let $F=F^+E$ be a CM field with $E/{\mathbb Q}$ quadratic imaginary, and let $\overline G$ be the associated unitary similitude group. Let $S$ be a finite set of places and let…

Let $G$, $F^+$, $F$, $T$, $S_{\ell}$, $S(B)$, $U^v$, $S_{a,\{\rho_x\},\varnothing}$, and $\bar{k}$ have the meanings used in the paper. Let a maximal ideal be non-Eisenstein when i…

Let $q$ be a prime number such that $q\not|N\Delta\ell$. Fix a maximal non-Eisenstein ideal $\mathfrak m$ of the Hecke algebra ${\bf T}^{\widehat\psi}(N)$ acting on…