8 problems

In the indefinite case, let $\mathcal{S}_0^\circ$ be the relevant Shimura variety, let $\widetilde{\mathcal{S}}_0^\circ$ be its special fiber, let $\mathrm{S}'_0$ parameterize the…

Let $\mathfrak{m}$ be a maximal ideal of the Hecke algebra, and assume Hypotheses and. The relevant special fibre is denoted by ${{\Sh}_{1,n-1}^{{\rm{ss}}}}$, its ambient Shimura v…

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…

Level-raising sufficiency conjecture. If this condition holds at $p$, then $\rho$ arises from a weight-$k$ newform in $\mathcal{S}_k(\Gamma_0(Np))$. This is the precise conjecture…

Let $k$ be an even integer with $k\geq4$, let $l>k+1$, and let $N$ be squarefree. Consider the representation $\mathbf{1}\oplus\chi_l^{k-1}$, where $\chi_l$ is the mod $l$ cyclotom…

Let $k\geq4$ be even, let $l$ be a prime with $l>k+1$, and consider the representation $\mathbf{1}\oplus\chi_l^{k-1}$, where $\chi_l$ is the mod $l$ cyclotomic character. A prime…

Geometric level-raising conjecture. If $\mathcal{Z}\subset\mathcal{E}$ is a generically principal-series at $l$ component and there exists $x\in\mathcal{Z}$ such that…

Fix distinct primes $p$ and $l$, and an integer $N$ coprime to $pl$. Let $\mathcal{E}$ be the cuspidal eigencurve of tame level $\Gamma_0(Nl)$, parametrising overconvergent cuspida…